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ratio and fractions problem solving

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Unit: Ratios and rates

Intro to ratios.

Visualize ratios

Equivalent ratios

Ratio application

Intro to rates

About this unit

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How To Simplify Ratios With Fractions

Simplifying ratios that involve fractions.

Ratios are used as a way of comparing values. Typically expressed in the form A:B, they tell us how much of one thing we have compared to another. For example, if we have a bowl of fruit containing two apples and six oranges, we can express this as 2:6.

Ratios are useful because they allow us to easily scale values and express quantities in a way that is simple to understand. As such, you’ll find them used in everything from official statistics to grocery store labels.

When ratios appear in an everyday context, they are usually given in their lowest form. In a numerical reasoning test , you may be required to simplify a ratio. This process is straightforward enough when working with whole numbers, but is complicated when working with ratios that involve fractions.

Take, for example, the ratio 1 5/12:7/3. As each side, or term, of the ratio is a fraction, we first need to manipulate these and identify a common denominator that will allow us to find the ratio’s lowest form. We’ll walk through this process below.

A step-by-step guide to simplifying ratios with fractions

Taking the example above of the ratio 1 5/12:7/3, there are four steps we need to work through to express this in its simplest form.

Step 1: Convert mixed fractions to improper fractions

In our example, the antecedent term of the ratio is the mixed fraction 1 5/12, so we first need to convert this to an improper fraction (if there are no mixed fractions in the ratio, you can skip this step).

To do this, we first multiply the whole number by the denominator of the fractional part; in this case, 1 x 12 = 12.

We then add this result to the numerator of the fractional part, so 12 + 5 = 17. This number now becomes our new numerator, which we place above our original denominator, giving us the improper fraction 17/12.

Step 2: Convert both fractions using the lowest common denominator

Our ratio is now 17/12:7/30, so our next task is to find the lowest common denominator of our two fractions.

By listing multiples of both 12 and 30, we find the lowest common denominator is 60.

To keep the values of our fractions the same, we need to multiply both the numerator and the denominator by the same amount. So:

12 x 5 = 60 and 17 x 5 = 85, giving us 85/60

30 x 2 = 60 and 7 x 2 = 14, giving us 14/60

Step 3: Write the numerators as a ratio

Now we have a common denominator, we can omit this and write out our ratio using the numerators.

This gives us the ratio 85:14

Step 4: Simplify the ratio

The final step is to simplify the ratio by dividing both sides by the highest common factor.

In our example, the highest common factor is 1, so the ratio is already in its lowest form.

We can therefore say that 1 5/12:7/30 = 85:14.

Example question 1

Simplify the ratio 12/17:15/68

Here we have two proper fractions, so we can move straight onto step two and find the lowest common denominator. Since 68 is a multiple of 17, this is our answer:

17 x 4 = 68

Now multiply the numerator by the same amount to keep the value of the fraction:

12 x 4 = 48

This leaves us with 48/68:15/68.

We can omit the common denominator and write our ratio using both numerators:

Now look for the highest common factor, in this case 3, and divide both terms of the ratio to simplify:

Answer: 12/17:15/68 = 16:5

Example question 2

Simplify the ratio 3 2/9:5/18

Here we have a mixed fraction, so we first need to convert this to an improper fraction. Multiply the whole number by the denominator of the fractional part:

Add this to the numerator:

27 + 2 = 29

Place this over the original denominator, giving you the ratio 29/9:5/18.

Now find the lowest common denominator of the two fractions, in this case, 18. Keep the value of the fractions by multiplying both numerator and denominator by the same amount:

9 x 2 - 18 and 29 x 2 = 59

This gives us 58/18:5/18

Omit the denominators and write out the ratio using the numerators:

The highest common factor here is 1, so we cannot simply any further.

Answer: 3 2/9:5/18 = 58:5

Tips for simplifying ratios that include fractions

Practise simplifying ratios with whole numbers.

Before you start working with ratios that involve fractions, first master the art of simplifying ratios with whole numbers .

You’ll need to understand this process to express a ratio in its lowest form, so make this the starting point of your revision. You need to work at speed in a numerical reasoning test, and the quicker you are at this process, the faster you’ll be overall.

Learn to convert mixed fractions

A ratio containing both mixed and proper fractions can be intimidating, causing you to draw a blank and eating into your time allowance.

This is easily avoidable if you’re confident in converting mixed fractions to improper fractions. All you need to remember are two simple rules: multiply the whole by the denominator, and add to the numerator.

Know your times tables

Quick multiplication is key when looking for a lowest common denominator. Knowing your times tables is also useful for identifying the highest common factor across the terms of a ratio.

Pay attention to detail

Always ensure you understand the question and watch out for costly mistakes. If your question is presented as a word problem, write it out as a numerical equation, checking each part of the ratio is placed correctly.

For example, ‘what is 3/6 to 5/8 in its simplest form?’ should be written 3/6:5/8, with the first figure mentioned as the antecedent term.

Equally, when converting fractions, double-check your numerators and denominators and, when writing your final answer, ensure the terms of your ratio are consistent with the question.

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Welcome to our Ratio Word Problems page. Here you will find our range of Fifth Grade Ratio Problem worksheets which will help your child apply and practice their Math skills to solve a range of ratio problems.

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Here you will find a range of problem solving worksheets about ratio.

The sheets involve using and applying knowledge to ratios to solve problems.

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Ratio and Probability Problems

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Here you will find our selection of free 5th grade math word problems.

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There are a range of different logic problems from 1st through 5th grade!

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Here you will find a range of fraction word problems to help your child apply their fraction learning.

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Sat / act prep online guides and tips, complete guide to fractions and ratios in act math.

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In essence, fractions and ratios represent pieces of a whole by comparing those pieces either to each other or to the whole itself. Don’t worry if that sentence makes no sense right now. We’ll break down all the rules and workings of these concepts throughout this guide--both how these mathematical concepts work in general and how they will be presented to you on the ACT.

Whether you are an old hat at dealing with fractions, ratios, and rationals, or a novice, this guide is for you. This guide will break down what these terms mean, how to manipulate these kinds of problems, and how to answer the most difficult fraction, ratio, and rational number questions on the ACT.

What are Fractions?

$${\a\piece}/{\the\whole}$$

Fractions are pieces of a whole. They are expressed as the amount you have (the numerator) over the whole (the denominator).

Amy’s cat gave birth to 8 kittens. 5 of the kittens had stripes and 3 had spots. What fraction of the litter had stripes?

$5/8$ of the litter had stripes. 5 is the numerator (top number) because that was the amount of striped kittens, and 8 is the denominator (bottom number) because there are 8 kittens total in the litter (the whole).

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Special Fractions

There are several different kinds of "special fractions" that you must know in order to solve the more complex fraction problems. Let us go through each of these:

A number over itself equals 1

$47/47 = 1$

${xy}/{xy} = 1$

A whole number can be expressed as itself over 1

$17 = 17/1$

$108 = 108/1$

$xy = {xy}/1$

0 divided by any number is 0

$0/{xy} = 0$

Any number divided by 0 is undefined

Zero cannot act as a denominator. For more information on this check out our guide to advanced integers . But, for now, all that matters is that you know that 0 cannot act as a denominator.

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Reducing Fractions

If you have a fraction in which both the numerator and the denominator can be divided by the same number (called a “common factor”), then the fraction can be reduced. Most of the time, your final answer will be presented in its most reduced form.

In order to reduce a fraction, you must find the common factor between each piece of the fraction and divide both the numerator and the denominator by that same amount. By dividing both the numerator and the denominator by the same number, you are able to maintain the proper relationship between each piece of your fraction.

So if your fraction is $5/25$, then it can be written as $1/5$. Why? Because both 5 and 25 are divisible by 5.

$25/5 = 5$.

So your final fraction is $1/5$.

Adding or Subtracting Fractions

You can add or subtract fractions as long as their denominators are the same. To do so, you keep the denominator consistent and simply add the numerators.

$2/11 + 6/11 = 8/11$

But you CANNOT add or subtract fractions if your denominators are unequal.

$2/11 + 4/5 = ?$

So what can you do when your denominators are unequal? You must make them equal by finding a common multiple (number they can both multiply evenly into) of their denominators.

$2/11 + 4/5$

Here, a common multiple (a number they can both be multiplied evenly into) of the two denominators 11 & 5 is 55.

To convert the fraction, you must multiply both the numerator and the denominator by the amount the denominator took to achieve the new denominator (the common multiple).

Why multiply both? Just like when we reduced fractions and had to divide the numerator and denominator by the same amount, now we must multiply the numerator and denominator by the same amount. This process keeps the fraction (the relationship between numerator and denominator) consistent.

To get to the common denominator of 55, $2/11$ must be multiplied by $5/5$. Why? Because $11 * 5 = 55$.

$(2/11)(5/5) = 10/55$.

To get to the common denominator of 55, $4/5$ must be multiplied by $11/11$. Why? Because $5 *11 = 55$.

$(4/5)(11/11) = 44/55$.

Now we can add them, as they have the same denominator.

$10/55 + 44/55 = 54/55$

We cannot reduce $54/55$ any further as the two numbers do not share a common factor.

So our final answer is $54/55$.

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Here, we are not being asked to actually add the fractions, just to find the least common denominator so that we could add the fractions.

Because we are being asked to find the least amount of something, we should start at the smallest number and work our way down (for more on using answer choices to help solve your problem in the quickest and easiest way, check out our article on plugging in answers ).

Answer choice A is eliminated, as 40 is not evenly divisible by 12.

120 is evenly divisible by 8, 12, and 15, so it is our least common denominator.

So our final answer is B , 120.

Multiplying Fractions

Luckily it is much simpler to multiply fractions than it is to add or divide them. There is no need to find a common denominator when multiplying--you can just multiply the fractions straight across.

To multiply a fraction, first multiply the numerators. This product becomes your new numerator.

Next, multiply your two denominators. This product becomes your new denominator.

$2/3 * 3/4 = (2 * 3)/(3 * 4) = 6/12$

And again, we reduce our fraction. Both the numerator and the denominator are divisible by 6, so our final answer becomes:

Special note: you can speed up the multiplication and reduction process by finding a common factor of your cross multiples before you multiply.

$2/3 * 3/4$ => $1/1 * 1/2$ => $1/2$.

Both 3’s are multiples of 3, so we can replace them with 1 ($3/3 = 1$).

Our other cross multiples are 2 and 4, which are both multiples of 2, so we were able to replace them with 1 and 2, respectively ($2/2 = 1$ and $4/2 = 2$).

Because our cross multiples had factors in common, we were able to reduce the cross multiples before we even began. This saved us time in reducing the final fraction at the end.

Take note that we can only reduce cross multiples when multiplying fractions, never while adding or subtracting them! It is also a completely optional step , so do not feel obligated to reduce your cross multiples--you can always simply reduce your fraction at the end.

Dividing Fractions

In order to divide fractions, we must first take the reciprocal (the reversal) of one of the fractions. Afterwards, we simply multiply the two fractions together as normal.

Why do we do this? Because division is the opposite of multiplication, so we must reverse one of the fractions to turn it back into a multiplication question.

${1/3} ÷ {3/8} => {1/3} * {8/3}$ (we took the reciprocal of $3/8$, which means we flipped the fraction upside down to become $8/3$)

${1/3} * {8/3} = 8/9$

Now that we've seen how to solve a fraction problem the long way, let's talk short cuts.

Decimal Points

Because fractions are pieces of a whole, you can also express fractions as either a decimal point or a percentage.

To convert a fraction into a decimal, simply divide the numerator by the denominator. (The $/$ symbol also acts as a division sign)

$3/10 => 3 + 10 = 0.3$

Sometimes it is easier to convert a fraction to a decimal in order to work through a problem. This can save you time and effort trying to figure out how to divide or multiply fractions.

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This is a perfect example of a time when it might be easier to work with decimals than with fractions. We’ll go through this problem both ways. 

Fastest way--with decimals:

Simply find the decimal form for each fraction and then compare their sizes. To find the decimals, divide the numerator by the denominator.

$5/3 = 1.667$

$7/4 = 1.75$

$6/5 = 1.2$

$9/8 = 1.125$

We can clearly see which fractions are smaller and larger now that they are in decimal form. In ascending order, they would be:

$1.125, 1.2, 1.667, 1.75$

Which, when converted back to their fraction form, is:

$9/8, 6/5, 5/3, 7/4$

So our final answer is A .

Slower way--with fractions:

Alternatively, we could compare the fractions by finding a common denominator of each fraction and then comparing the sizes of their numerators.

Our denominators are: 3, 4, 5, & 8.

We know that there are no multiples of 4 or 8 that end in an odd number (because an even number * an even number = an even number), so a common denominator for all must end in 0. (Why? Because all multiples of 5 end in 0 or 5.)

Multiples of 8 that end in 0 are also multiples of 40 (because $8 * 5 = 40$). 40 is not divisible by 3 and neither is 80, but 120 is.

120 is divisible by all four digits, so it is a common denominator.

Now we must find out how many times each denominator must be multiplied to equal 120. That number will then be the amount to which we multiply the numerator in order to keep the fraction consistent.

$120/3 = 40$

$5/3$ => ${5(40)}/{3(40)}$ => $200/120$

$120/4 = 30$

$7/4$ => ${7(30)}/{4(30)}$ => $210/120$

$120/5 = 24$

$6/5$ => ${6(24)}/{5(24)}$ => $144/120$

$120/8 = 15$

$9/8$ => ${9(15)}/{8(15)}$ => $135/120$

Now that they all share a common denominator, we can simply look to the size of their numerators and compare the smallest and the largest. So the order of the fractions from least to greatest would be:

$135/120, 144/120, 200/120, 210/120$

Which, when converted back into their original fractions, is:

So once again, our final answer is A .

As you can see, we were able to solve the problem using either fractions or decimals. How you chose to approach these types of problems is completely up to you and depends on how you work best, as well as your time management strategies.

Percentages

After you convert your fraction to a decimal, you can also turn it into a percentage (if the need arises).

To get a percentage, multiply your decimal point by 100.

So 0.3 can also be written as 30%, because $0.3 * 100 = 30$.

0.01 can be written as 1% because $0.01 * 100 = 1$, etc.

Be mindful of your decimals and percentages and don't mix them up! 0.1 is NOT the same thing as 0.1%.  

Mixed Fractions

Sometimes you may be given a mixed fraction on the ACT. A mixed fraction is a combination of a whole number and a fraction.

For example, $5{1/3}$ is a mixed fraction. We have a whole number, 5, and a fraction, $1/3$.

You can turn a mixed fraction into an ordinary fraction by multiplying the whole number by the denominator and then adding that product to the numerator. The final answer will be ${\the \new \numerator}/{\the \original \denominator}$.

$5{1/3}$  

$(5)(3) = 15$

$15 + 1 = 16$

So your final answer = $16/3$

You must convert mixed fractions into non-mixed fractions in order to multiply, divide, add, or subtract them with other fractions.

A cobbler charges a flat fee of 45 dollars plus 75 dollars per hour to make a pair of shoes. How many hours of labor was spent making the shoes if the total bill was $320?

If the total bill was 320 dollars and the flat fee was 45 dollars, we must subtract the flat fee from the total bill in order to find the number of hours the cobbler worked.

$320 - 45 = 275$

So the cobbler worked 275 dollars’ worth of hours. In order to find out how many hours that is, we must divide the earnings by the hourly fee.

$275/75 = 3{50/75}$

75 was able to go evenly into 225, leaving 50 out of 75 left over.

Because 50 and 75 share a common denominator of 25, we can reduce $3{50/75}$ to:

So our final answer is B , $3{2/3}$

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What are Ratios?

Ratios are used as a way to compare one thing to another (or multiple things to one another).

If Piotr has exactly 2 grey scarves and 7 red scarves in a drawer, the ratio of grey scarves to red scarves is 2 to 7.

Expressing Ratios

Ratios can be written in three different ways:

No matter which way you write them, these are all ratios comparing A to B.

Different Types of Ratios

Just as a fraction represents a part of something out of a whole (written as: ${\a \part}/{\the \whole}$), a ratio can be expressed as either:

${\a \part}:{\a \different \part}$

$\a \part:\the \whole$

Ratios compare values, so they can either compare individual pieces to one another or an individual piece to the whole.

If Piotr has exactly 2 grey scarves and 7 red scarves in a drawer, the ratio of grey scarves to all the scarves in the drawer is 2 to 9. (Why 9? Because there are 2 grey and 7 red scarves, so together they make $2 + 7 = 9$ scarves total.)

Reducing Ratios

Just as fractions can be reduced, so too can ratios.

Danielle collects toy racecars. 12 of them are blue and 4 of them are yellow. What is the ratio of of blue cars to yellow cars in her collection?

Right now, the ratio is $12:4$. But they have a common denominator of 4, so this ratio can be reduced.

So the cars have a ratio of $3:2$

Increasing Ratios

Because you can reduce ratios, you can also do the opposite and increase them. In order to do so, you must multiply each piece of the ratio by the same amount (just as you had to divide by the same amount on each side to reduce the ratio).

So the ratio of $3:2$ can also be

$3(2):2(2) = 6:4$

$3(3):2(3) = 9:6$

$3(4):2(4) = 12:8$

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Though this presents itself as a geometry problem, we don’t need to know any geometry in order to solve it--we only need to know about ratios.

We have two triangles in a ratio of 2:5 and the smaller triangle has a hypotenuse of 5 inches. This means that we need to increase each side of the ratio by the amount it takes 2 to go into 5.

$5/2 = 2.5$

So we must increase each side of the ratio by a matter of 2.5

$2(2.5):5(2.5)$

Our new, increased ratio is 5:12.5, which means that the larger hypotenuse is 12.5.

Our final answer is K.

Expand ratios, reduce them--go wild!

Finding the Whole

If you are given a ratio comparing two parts ($\piece:\another \piece$), and you are told to find the whole amount, simply add all the pieces together.

It may help you to think of this like an algebra problem wherein each side of the ratio is a certain multiple of x. Because each side of the ratio must always be divided or multiplied by the same amount to keep the ratio consistent, we can think of each side as having the same variable attached to it.

For example, a ratio of $6:7$ can be:

$6(1):7(1) = 6:7$

$6(2):7(2) = 12:14$

And so on, just as we did above.

But this means we could also represent $6:7$ as:

Why? Because each side must change at the same rate. And in this case, our rate is $x$.

So if you were asked to find the total amount, you would add the pieces together.

$6x + 7x = 13x$. The total amount is $13x$. In this case, we don’t have any more information, but we know that the total MUST be either 13 or any number divisible by 13.

So let’s take a look at another problem.

Clarissa has a jewelry box with necklaces and bracelets. The necklaces and bracelets are in a ratio of 4:3. What is NOT a possible number of total pieces of jewelry Clarissa can have in the box?

In order to find out how many pieces of jewelry she may have total, we must add the two pieces of our ratio together.

So $4x + 3x = 7x$

This means that the total number of jewelry items in the box has to either be 7 or any multiple of 7. Why? Because $4:3$ is the most reduced form of the ratio of jewelry items in the box. This means she could have:

$4(1):3(1) = 7$ jewels in the box (7 jewelry pieces total)

$4(2):3(2) = 8:6$ jewels in the box (14 jewelry pieces total)

$4(3):3(3) = 12:9$ jewels in the box (21 jewelry pieces total)

And so forth. We don’t know exactly how many jewelry items she has, but we know that it must be a multiple of 7.

This means our answer is A , 12. There is no possible way that she can have 12 jewels in the box, because 12 is not a multiple of 7 and one cannot have half a bracelet (unless something has gone terribly wrong).

You may also be asked to find the number of individual pieces in your ratio after you are given the whole. This is exactly the opposite of what we did above.

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The is the exact same process as finding the whole, but in reverse.

We know we must add the pieces of our ratio to find our multiple of 30. And we also know our ratio is $2:3$. So let us add these together. 

$2x + 3x = 5x$

Together, our ratio components add up to $5x$. And there are 30 feet total. So:

This means that we must multiply each side of our ratio by 6 in order to get the exact amount of wood used. This means that each piece is:

$2(6):3(6)$

Which means our shorter piece is 12 feet long.

Our final answer is H , 12.

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Rational and Irrational Numbers

A rational number is any number that can be written as a fraction of two integers (where the denominator does NOT equal to 0).

All other numbers are considered irrational.

Rational Numbers:

$7/2, 5, 1/212, 0.66666667$

Why is 5 a rational number? Because it can be expressed as the fraction $5/1$.

Why is 0.6666667 a rational number? Because it can be expressed as the fraction $2/3$

Irrational Numbers:

$π, √2, √3$

Why is $π$ irrational? Because there is no fraction of two integers that can properly express it (through 22/7 comes awfully close).

(Hint: if the decimals continue on forever without repeating, the number is irrational)

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Here, we are being asked to find the single rational number. Even if you didn’t know what a rational number meant, you might be able to figure this problem out just by finding the answer choice that stands out the most. But since you DO know what rational and irrational numbers are, it makes the problem even easier.

Many square roots are irrational (unless they are roots of perfect squares like $√16 = 4$). We can immediately eliminate answer choices A, B, and C, as they are not perfect squares and so are irrational.

We can also eliminate answer choice D. When we reduce the fraction, we get $√{1/5}$, and this would also get us an irrational number.

This leaves us with answer choice E.

We can see that both the numerator and the denominator of the fraction $64/49$ inside the square root sign are perfect squares. Since the fraction is under the root sign, let us take the square root of each of these.

So our final fraction would look like:

$√{64/49}$ => $8/7$

Because our final fraction is represented as a fraction with two integers, this is a rational number.

So our final answer is E .

How to Solve Fraction, Ratio, and Rational Number Questions

When you are presented with a fraction or ratio problem, take note of these steps to find your solution:

1) Identify whether the problem involves fractions or ratios

A fraction will involve the comparison of a $\piece/\whole$.

A ratio will almost always involve the comparison of a $\piece:\piece$ (or, very rarely , a $\piece:\whole$).

You can tell when the problem is ratio specific as the question text will do one of three things:

If the questions wants you to give an answer as a ratio comparing two pieces, make sure you don’t confuse it with a fraction comparing a piece to the whole!

2) If a ratio question asks you to change or identify values, first find the sum of your pieces

In order to determine your total amount (or the non-reduced amount of your individual pieces), you must add all the parts of your ratio together. This sum will either be your complete whole or will be a factor of your whole, if your ratio has been reduced.  

3) When in doubt try to use decimals

Decimals can make it much easier to work out problems rather than using fractions. So do not be afraid to convert your fractions into decimals to get through a problem more quickly and easily.

4) Remember your special fractions

Always remember that a number over 1 is the same thing as the original number, and that when you have a number over itself, it equals 1.

Test Your Knowledge

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4) How many irrational numbers are there between 1 and 8?

Answers: B, J, D, E

Answer Explanations:

1) For this problem, we must combine our like terms in order to eventually isolate $k$ (for more on this, check out our guide to ACT single variable equations).

We know that, when adding fractions, we must give them the same denominator, so we can manipulate our fractions to have matching denominators and solve from there.

Alternatively, we could again use decimal points instead of fractions. We will go through both ways here. 

Method 1--Fractions

We have ${1/3}k$ and ${1/4}k$ that we must add. They share a common multiple of 12, so let us convert them to fractions out of 12. 

$1/3$ => ${1(4)}/{3(4)}$ => $4/12$

$1/4$ => ${1(3)}/{4(3)}$ => $3/12$

Now that they have the same numerator, we can combine them to be:

$4/12 + 3/12 = 7/12$

So our equation is:

${7/12}k = 1$

Now we must divide both sides by $7/12$, which means that we must inverse and multiply. 

$k = 1(12/7)$

So our final answer is B. 

Method 2--Decimals

Instead of using and converting fractions, we also could have used decimals instead. 

$1/3$ => $0.333$

$1/4$ => $0.25$

Because they are decimals, we can simply add them together to be:

$0.333k + 0.25k = 0.583k$

$0.58k = 1$

$k = 1/0.583$

$k = 1.715$

Now, simply convert the answer choices to decimals to find one that matches. In this case answer choice A would be far too small, and answers D and E are whole numbers, so they can all be eliminated. Answer choice C would be $7/2 = 3.5$. 

This leaves us with answer choice B:

$12/7 = 1.714$

So our final answer is, again, B. 

2) This question specifically asks for a rational number answer, but it is a bit deceptive, as a quick glance shows us that all the answer choices are rational numbers.  This means you can ignore this stipulation for the time being.

Again, we can solve this problem in one of two ways--via fractions or via decimals. We will go through both methods. 

We are trying to find a rational fraction halfway between $1/5$ and $1/3$, so let us convert them into fractions with the same denominator. 

A common multiple of 3 and 5 is 15, so let us make that their new denominator. 

$1/5$ => ${1(3)}/{5(3)}$ => $3/15$

$1/3$ =>  ${1(5)}/{3(5)}$ => $5/15$

Well the rational number exactly halfway between $3/15$ and $5/15$ is $4/15$. 

So our answer is J, $4/15$.

Again, if fractions aren't your favorite, you can always feel free to use decimals. 

First, convert $1/5$ and $1/3$ into decimals. 

$1/5 = 0.2$

$1/3 = 0.333$

Now, find the decimal halfway between them:

${0.2 + 0.333}/2 = 0.2665$ (For more on this process, check out our guide to ACT mean, median, and mode )

Now, let us find the answer choice that, when converted into a decimal, matches our answer. 

If you know your decimals, then you know that $1/2 = 0.5$ and $1/4 = 0.25$, so these can be eliminated. 

We are now left with $2/15$, $4/15$, and $8/15$. The smart thing to do here is to pick the middle value and then go up or down if the mid value is too small or too large. So if we test $4/15$, we get:

$4/15 = 0.2666$

Success! We nailed it at the mid value, no need to try the others. 

Our final answer is, again, J.  

3) Even though this problem may, at first glance, look like a fraction problem, it is a ratio problem. We can tell because the question specifically asks for the ratios of the boys' sandwich consumption.

If you're not paying attention, you can easily make a mistake and treat the question as a fraction problem when ratios are written using the "/" symbol. 

So we have Jerome, who eats half the sandwich and Kevin, who eats one third, and Seth, who eats the rest. Now we can do this problem several ways, but let us pick two of the most straightforward--ratio and fraction manipulation or plugging in your own numbers (for more on this strategy, check out guide to plugging in numbers). 

Method 1--Ratio and Fraction Manipulation

Because we are not told the portion of the sandwich that Seth ate, we must find it. Fractions represent pieces of the whole and the whole is 1 (because anything over itself = 1). So let us add our two fractions and subtract that sum from 1 to find Seth's share of the sandwich. 

$1/2 + 1/3$. First, we must convert these fractions to ones with a shared denominator. Both 2 and 3 are multiples of 6, so we will use 6 as our new denominator. 

$1/2$ => ${1(3)}/{2(3)}$ => $3/6$

$1/3$ =>  ${1(2)}/{3(2)}$ => $2/6$

Now, let us add them together and subtract their sum from 1. 

$3/6 + 2/6 = 5/6$

$1 - 5/6 = 1/6$

So Seth ate $1/6$ of the sandwich. 

And because these fractions now all share a common denominator, we can simply compare their numerators to find their ratio of sandwich shares (remember, ratios compare parts to other parts). 

So the sandwich eating fractions are:

$3/6, 2/6,$ and $1/6$

When we just look at the numerators, the ratio is:

Our final answer is D , $3:2:1$. 

Method 2--Plugging in Numbers

Instead of working exclusively with fractions and ratios, let's try the problem again using whole numbers. We know that Jerome ate $1/2$ and sandwich and Kevin ate $1/3$, so let's give the sandwich an actual length value that is a shared multiple of those two numbers (note: our sandwich length does not have to be a multiple of 2 and 3--it can be anything we want. It simply makes our lives easier to use a common multiple, as that way we can work with integers.)

So let us say that the sandwich is 12 feet long. 

If Jerome ate half of it, then he ate:

$12/2 = 6$ feet of sandwich. 

If Kevin ate one third of it, then he ate:

$12/3 = 4$ feet of sandwich.

If we add them together, they ate:

$6 + 4 = 10$ feet of sandwich.

Which means that Seth ate:

$12 - 10 = 2$ feet of sandwich.

Now let us compare their shares of 6, 4, and 2.

We know that ratios can be reduced if each of the values shares a common factor. In this case, they can all be divided by 2, so let us reduce the ratio.

$6:4:2$ => $3:2:1$

Again, our final answer is D , $3:2:1$

4) This question asks you to find the amount of irrational numbers between two real numbers, and the simple answer is that there are infinitely many. (Note: there is also an infinite amount of rational numbers between any two real numbers as well!). 

Why is this true? Think of it this way:

The square root of 1 is rational, because it equals 1, which can be written as $1/1$.

But the square root of 1.01 is irrational. And so is the square root of 1.02, and the square root of 1.03....None of these numbers can be written as ${\an \integer}/{\an \integer}$ (which you can tell because their decimals continue without repeating), and yet they all sit between 1 and 8 on a number line. 

So our final answer is E , more than 7 (and, in fact, infinite). 

Hurray and huzzah, you did it!

The Take-Aways

Don’t let fractions, ratios, and/or rational numbers intimidate you. Once you’ve mastered the basics behind how they behave, you’ll be able to work your way through many of the toughest fraction and ratio problems the ACT can put in your way

The biggest point to look out for, when dealing with fractions and ratios, is not to mix them up ! Always pay strict attention to times when you are comparing pieces to pieces or pieces to the whole. Though it can be easy to make a mistake during the test, don’t let yourself lose a point due to careless error.

What’s Next?

For you, fractions are a breeze, ratios were a snap, and rationals?--Forget about it! Luckily for you, there is plenty more to tackle before test day. We have guides aplenty for the many math topics covered on the ACT , including trigonometry , integers , and  solid geometry . 

Running out of time during ACT Math practice? Check out our article on how to finish your math section before it's pencil's down.  

Don't know what score to aim for? Make sure you have a good grasp of what kind of score would best suit your goals and current skill level , and how to improve it from there. 

Trying to push your score to the top? Look to our guide on how to get a perfect score , written by a 36 ACT-scorer. 

Want to improve your ACT score by 4 points? 

Check out our best-in-class online ACT prep program . We guarantee your money back if you don't improve your ACT score by 4 points or more.

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Ratios as Fractions

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So many times we are given a ratio with which items are to be shared, like a ratio between boys and girls in your class.

Do you know that this could be converted to fractions and can be used in many ways?

In this article, you would be learning how ratios can be represented as fractions .

Ratios as fractions meaning

Ratio as a fraction occurs when rations are written in the form of fractions.

The ratio X : Y is expressed as a fraction in the form X Y .

Express the following ratios as fractions .

The antecedent of the ratio is the numerator of the fraction while the consequent of the ratio is the denominator of the fraction.

a. The ratio 1 : 2 becomes

1 : 2 = 1 2

b. The ratio 5 : 6 becomes

5 : 6 = 5 6

c. The ratio 3 : 2 becomes

3 : 2 = 3 2

d. The ratio 13 : 12 becomes

13 : 12 = 13 12

Ratios as fractions properties

Below are several properties of ratios as fractions and some direct examples of each property.

a. The antecedent of the ratio is the numerator of the fraction while the consequent of the ratio is the denominator of the ratio. This applies only when the fraction is not in the simplified form.

In the ratio 2:3, 2 is the antecedent while 3 is the consequent.

When 2:3 is converted to a fraction, it becomes 2 3 .

Note that the antecedent (2) is now the numerator of the fraction while the consequent (3) is now the denominator of the fraction.

b. A ratio can only be converted to a fraction when the consequent is not a factor of the antecedent, otherwise a whole number is formed.

10:5 cannot be converted to a fraction because 5 (consequent) is a factor of 10 (antecedent). Thus, when converted to a fraction 10:5 is simplified to a whole number which is 2.

c. When a ratio is converted to a fraction, the fraction must be reduced to its simplified form.

For example, 6:10 being converted to a fraction becomes 6 10 and has to be further simplified to be 3 5 .

d. Ratios as fractions have no unit because both the antecedent and consequent are of same units .

given the ratio of two distances is 5cm is to 7cm. By conversion it becomes,

5 c m : 7 c m = 5 c m 7 c m = 5 7

e. Since ratios have no units , it means that if the antecedent has a unit, the consequent must have the same unit so that the ratio is unit free.

We consider the ratio of two distances is 50 cm: 1 m, hence before even putting this ratio into a fraction, we need to ensure that both antecedent and consequent have the same unit.

1 m = 100 c m 50 c m : 1 m = 50 c m : 100 c m = 50 c m 100 c m = 1 2

f. The antecedent and the consequent of a ratio should be expressed as whole numbers.

The ratio 4 . 5 : 3 . 5 would be converted by multiplying all through by 2 to become 9 : 7 which is 9 7

Another way to convert the antecedent and the consequent into whole numbers is to multiply them by 10, to get

4 . 5 : 3 . 5 = 45 : 35 = 45 35 = 9 7

g. Ratios containing more than one quantity cannot be expressed as a fraction. The ratio a:b:c should not express as a fraction unless each quantity is expressed as a fraction of the total quantities.

If biscuits are shared in the ratio 2:1:3 among three people, we cannot represent it as a fraction like

However, we may be asked to express the value of the first as a ratio of the total and hence we have

t o t a l = 2 + 1 + 3 = 6 f i r s t : t o t a l = 2 : 6 = 2 6 = 1 3

Ratios as fractions in simplest form

There are several methods used to write ratios as fractions in the simplest form.

Using the highest common factor (HCF)

When simplifying ratios in the form of fractions , we divide through by the highest common factor of the numerator and the denominator. Note that this is a once-and-for-all process because the highest common factor divides the numerator and denominator once to arrive at the final answer.

Simplify the following ratio

Express the ratio as a fraction,

18 : 24 = 18 24 .

Find the HCF between the numerator and the denominator. The HCF between 18 and 24 is 6.

Divide the numerator and the denominator by the HCF,

18 ÷ 6 24 ÷ 6 = 3 4

So the answer is,

Using the lowest common factor (LCF)

When you are to simplify ratios that are in the form of fractions , in this case, you need to find the lowest common factor between the numerator and denominator. Thereafter, you should divide the fraction by using the lowest common factor continuously until there is no common factor between them. This is a more rigorous method when dealing with ratios of large numbers.

Simplify the following

Express the ratio as a fraction. Thus,

18 : 24 = 18 24

Divide by using the lowest common factor. The lowest common factor between 18 and 24 is 2, Thus;

18 24 = 9 12

Between 9 and 12 what is the lowest common factor? The lowest common factor is 3. So divide through by 3,

Between 3 and 4 what is the lowest common factor? There is no common factor between 3 and for. Therefore our answer is;

Knowing these properties would enable us to understand problems regarding ratios as fractions much better.

Ratios as fractions calculations

We may encounter problems where ratios are not expressed as fractions . All we have to do is to carefully convert details to the right expression as a ratio.

Afterward, we convert such a ratio to its corresponding fraction, then we make sure to simplify the fraction to find the final answer depending on the requirement of the question.

Till now, we have only dealt with ratios of parts, now we expand more on ratios out of a whole.

Ratios out of a whole

Sometimes we are required to find the ratio between a part of the ratio and the total ratio. In order to do so, we find the sum of all the values which make up the ratio before expressing the value as a ratio of the total. We will see this in the following example.

Two teenagers Bill and Jill share a loaf of bread in the ratio 2:3. What is the fraction of the whole bread did Jill take?

Jill's share is 3. The total ratio is,

The fraction of the whole bread Jill take is,

J i l l ' s s h a r e : t o t a l s h a r e = 3 : 5 = 3 5

Examples of ratios as fractions

The best way to understand how ratio as fractions is calculated is through examples. You shall also be able to see word problems involving ratios as fractions hereafter.

In a cinema, the ratio of horror to sci-fi to comedy movies is 2:3:7. Express horror movies as a fraction of all kinds of movies being viewed in the cinema.

We are told that the ratio of these kinds of movies is,

Find the total of the ratio,

2 : 3 : 7 = 2 + 3 + 7 = 12 .

We are asked to express horror movies as a fraction of all the movies, and horror movies have 2 among the ratio. Thus divide the horror movies' quantity by the total ratio,

h o r r o r m o v i e s t o t a l m o v i e s = 2 12

Simplify, by dividing by the HCF which is 2,

2 ÷ 2 12 ÷ 2 = 1 6 .

One-fifth of Kohe's books are torn. What is the ratio of the books untorn to those torn?

Kohe's books comprise both those that are torn and untorn.

Whenever you are given a fraction to express proportion or ratio , note that the sum of all items is 1. In this case, we have

t o r n + u n t o r n = 1 1 5 + u n t o r n = 1 u n t o r n = 1 - 1 5 u n t o r n = 4 5

This implies that four-fifth of Kohe's books are untorn. But you are asked to find the ratio of untorn to torn. Thus,

u n t o r n : t o r n = 4 5 : 1 5

Recall that neither the antecedent nor the consequent of a ratio should be a fraction. Thus multiply through by 5;

( 4 5 × 5 ) : ( 1 5 × 5 ) = 4 : 1

Thus, the ratio of untorn to torn is 4:1.

A bag contains billiard balls of three colors white, blue and amber in the ratio 4:5:6 respectively. That fraction of the ball is not amber?

We first find the total ratio,

4 : 5 : 6 = 4 + 5 + 6 = 15

We next find the portion of the ratio that is not amber. Since amber is 6 and the total is 15, the amount that is not amber is,

N o t a m b e r = 15 - 6 = 9

Now you know the proportion that is not amber, you can now find the fraction that is not amber out of the total ratio to be;

f r a c t i o n n o t a m b e r = v a l u e n o t a m b e r t o t a l r a t i o v a l u e f r a c t i o n n o t a m b e r = 9 15

Simplify by dividing through by 3,

f r a c t i o n n o t a m b e r = 9 ÷ 3 15 ÷ 3 f r a c t i o n n o t a m b e r = 3 5

Ratios as Fractions - Key takeaways

Frequently Asked Questions about Ratios as Fractions

--> how do you convert a ratio to fraction .

A ratio is converted to a fraction by placing the antecedent of the ratio as the numerator of the fraction and placing the consequent of the ratio as the denominator of the fraction.

--> Can ratios be written as fractions? 

Yes! Ratios can be written as fractions.

--> What is an example representing ratios as fractions? 

An example of representing a ratio to fraction would be converting 2:3 to 2/3.

--> What is the method for solving ratio as fraction? 

The method of solving ratio as fraction is to first convert the ratio to fraction. afterwards either the HCF or LCF method is applied to simplify the fraction obtained.

--> How do you write a ratio as a fraction? 

You write ratio as a fraction in the fraction for a/b where both a and b are whole numbers and values for the ratio a:b.

Final Ratios as Fractions Quiz

How would you describe ratios as fractions?

Show answer

Ratios as fractions deal with the expression of ratios in the form of fractions. 

Show question

A man's monthly income is spent on transport and accommodation in the ratio 3:2. If he earns 4000 pounds monthly, how much is spent on accommodation?

1600 pounds

What is the antecedent in the ratio 7:9?

If clearing a plot of land among 3 labourers were to be share in the ratio 3:2:1, how many square meters would be cleared by each labourer if a plot is 600 meters square?

300m 2 , 200m 2 and 100m 2

If 400 out of 1000 fowls have hatched eggs, what is the ratio of the hatched to unhatched flock?

If one-third of Brown's pocket money is spent on video games, two-fifth in buying biscuits during lunch breaks and the rest is used for transportation. What is the ratio of his pocket money is used in transportation to that used in buying biscuits?

If one-third of Brown's pocket money is spent on video games, two-fifth  in buying biscuits during lunch breaks and the rest is used for transportation. What is the ratio with which he shares his spending on video games, biscuits and transport?

of the users don't pass the Ratios as Fractions quiz! Will you pass the quiz?

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How to Solve Fraction Questions in Math

Last Updated: February 24, 2023 References Approved

This article was co-authored by Mario Banuelos, PhD and by wikiHow staff writer, Sophia Latorre . Mario Banuelos is an Assistant Professor of Mathematics at California State University, Fresno. With over eight years of teaching experience, Mario specializes in mathematical biology, optimization, statistical models for genome evolution, and data science. Mario holds a BA in Mathematics from California State University, Fresno, and a Ph.D. in Applied Mathematics from the University of California, Merced. Mario has taught at both the high school and collegiate levels. There are 7 references cited in this article, which can be found at the bottom of the page. wikiHow marks an article as reader-approved once it receives enough positive feedback. This article has 17 testimonials from our readers, earning it our reader-approved status. This article has been viewed 1,116,600 times.

Fraction questions can look tricky at first, but they become easier with practice and know-how. Start by learning the terminology and fundamentals, then pratice adding, subtracting, multiplying, and dividing fractions. [1] X Research source Once you understand what fractions are and how to manipulate them, you'll be breezing through fraction problems in no time.

Doing Calculations with Fractions

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Practicing the Basics

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Tip: Typically, you'll need to convert mixed numbers to improper fractions if you're multiplying or dividing them.

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To solve a fraction multiplication question in math, line up the 2 fractions next to each other. Multiply the top of the left fraction by the top of the right fraction and write that answer on top, then multiply the bottom of each fraction and write that answer on the bottom. Simplify the new fraction as much as possible. To divide fractions, flip one of the fractions upside-down and multiply them the same way. If you need to add or subtract fractions, keep reading! Did this summary help you? Yes No

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Ratio To Fraction

Here we will learn about ratios to fractions, including using ratios to find fractions and using fractions to find ratios.

There are also ratio to fractions worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is a ratio to a fraction?

A ratio to fraction is a way of writing a ratio as a fraction. A ratio compares how much of one thing there is compared to another. It can be written using a ‘:’, the word ‘to’ or as a fraction .

In order to convert ratios to fractions when we have the ratio a:b, where both values are parts of the total, we can say that for the ratio  \frac{a}{a+b} and \frac{b}{a+b}.

In the diagram below is a bar model that represents the ratio of blue:red as 3:2 (3 to 2). There are 3 blue blocks, 2 red blocks which means there are 5 blocks in total.

ratio and fractions problem solving

The fraction for blue is \frac{3}{2+3}=\frac{3}{5}.

The fraction for red is \frac{2}{2+3}=\frac{2}{5}.

Here the total number of shares in the ratio is equal to a+b (the denominator of each fraction) and the numerator is the part of the ratio we are interested in. 

Note: if it is possible to simplify the fraction (or the ratio) then simplify it but remember to keep using whole numbers (integers).

Step-by-step guide: How to work out ratios (coming soon)

What is a ratio to a fraction?

How to find a fraction given a ratio

In order to find a fraction given a ratio:

Add the parts of the ratio for the denominator.

State the required part of the ratio as the numerator.

Explain how to find a fraction given a ratio

Explain how to find a fraction given a ratio

Ratio to fraction worksheet

Get your free ratio to fraction worksheet of 20+ questions and answers. Includes reasoning and applied questions.

Related lessons on ratio

Ratio to fraction is part of our series of lessons to support revision on ratio . You may find it helpful to start with the main ratio lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Ratio to fraction examples

Example 1: standard question.

Ann and Bob share a box of cookies in the ratio of 3:4. What fraction of the cookies does Bob receive?

2 State the required part of the ratio as the numerator.

Bob receives 4 parts.

Solution: \frac{4}{7}

Example 2: picture

The diagram below shows part of the repeating pattern of red and yellow beads on a bracelet.

ratio and fractions problem solving

What fraction of the beads in the bracelet are red?

As there are 5 yellow beads and 3 red beads in each repeat, the ratio of yellow to red beads is 5:3.

The total of the parts is 5+3=8.

There are 3 red beads in each repeat.

Solution: \frac{3}{8}

Example 3: three part ratio

A tin of paint requires white, yellow and blue paint in the ratio 5:3:2.

What fraction of the tin is blue paint?

Blue is the final value in the ratio (2).

Solution: \frac{2}{10}=\frac{1}{5}

Example 4: bar modelling

A group of friends want to watch a film at the cinema. Below is a bar model to represent the friends’ choices. The friends who want to watch a romantic comedy are in red, those who want to watch a science fiction film are represented by yellow, and the others want to watch the latest action blockbuster.

What fraction of the group of friends do not want to watch a science fiction film? Write your answer in its simplest form.

ratio and fractions problem solving

The total number of shares in this ratio is 3+4+5=12.

As the question wants the number of people who do not want to watch a science fiction film, this would be equal to 3+5=8 people.

Solution: \frac{8}{12}=\frac{2}{3}

How to find a ratio given a fraction

In order to find a ratio given a fraction:

Subtract the numerator from the denominator of the fraction.

State the parts of the ratio in the correct order.

Explain how to find a ratio given a fraction

Explain how to find a ratio given a fraction

Fraction to ratio examples

Example 5: standard question.

\frac{3}{4} of a school of fish are male. The rest are female. Write the ratio of females to males in the school.

1 part is female, 3 parts are male.

Solution: 1:3

Example 6: worded problem

Pepper the cat spends roughly \frac{3}{5} of each day sleeping. State the ratio of hours awake to hours asleep to hours asleep for Pepper.

Pepper is awake for 2 hours for every 3 hours he is asleep.

Solution: 2:3

Example 7: three part ratio

A dental practice sells 3 different packages of dental care plans: Basic, Premium, and Family. \frac{1}{4} of their members have the Basic plan, whilst \frac{1}{5} have a Premium plan. State the ratio of Basic : Premium : Family.

Before we calculate this value, we need to know the sum of the two fractions we currently have. The common denominator will also help us state the sum of parts in the ratio

\frac{1}{4}+\frac{1}{5}=\frac{5}{20}+\frac{4}{20}=\frac{9}{20}

We can now calculate the difference between the numerator and the denominator:

The ratio asks us for Basic : Premium : Family. The three fractions we will use are: \frac{5}{20}, \frac{4}{20}, and \frac{9}{20}. Since the fractions have the same denominator we can use the numerators to write the ratio.

B:P:F=5:4:9

The numbers in the ratio have no common factors, so it is in its simplest form.

Solution: 5:4:9

Common misconceptions

For example, the ratio 2:3 is expressed as the fraction \frac{2}{3} and not \frac{2}{5}. This is a misunderstanding of the sum of the parts of the ratio. Be careful with what the question is asking as the denominator may be the part, or the whole amount.

Make sure that all the units in the ratio are the same. For example, in example 6, all the units in the ratio were in millilitres. We did not mix ml and l in the ratio.

The numerator is incorrectly stated from the ratio. For example, the number of mugs to glasses in a kitchen is written as the ratio 4:3 respectively. Write the fraction of mugs in the kitchen. The correct answer is \frac{4}{7}. The “mugs” value is the first number in the ratio, m:g=4:3 as we use the same order as the written sentence.

The parts of the ratio are written in the wrong order. For example the number of dogs to cats is given as the ratio 12:13 but the solution is written as 13:12.

Practice ratio to fraction questions

1. The ratio of tyrannosaurus rex to velociraptor fossils is 3:8. What fraction of the fossils are tyrannosaurus rex? Give your answer as a fraction in its simplest form.

GCSE Quiz False

Total number of fossils = 3+8=11

3 out of 11 are T-Rex fossils so \frac{3}{11}

2. A tailor sells silk and polyester ties in a ratio of 8:7. Calculate the fraction of ties that are silk.

8 out of 15 ties are silk so \frac{8}{15}.

3. The HEX colour \#428715 used for websites is made from the ratio of red to green to blue as 66:135:21 respectively. What fraction of the colour is green? Simplify your answer.

ratio and fractions problem solving

135 out of 222 are green so \frac{135}{222} = \frac{45}{74}.

4. There are 52 cards in a deck. \frac{12}{52} cards are picture cards. State the ratio of picture cards to non-picture cards in its simplest form.

picture:not picture

12:40=3:10 simplified

5. A football team won \frac{3}{5} of their matches in the league. They did not draw any matches. Write the ratio of wins : losses.

Win : Lose = 3:2

6. The number of beetles (B), slugs (S) and worms (W) eaten by a hedgehog was recorded over one night. \frac{1}{6} of the insects eaten were beetles, and \frac{2}{9} were slugs. Give the ratio of insects eaten by the hedgehog B:S:W in the simplest form.

Ratio to fraction GCSE questions

1. A market stall sells apples, pears and bananas only. The ratio of apples to pears is given as mixed numbers 5:2:1. Write down the fraction of the apples sold.

2. (a) The school orchestra is playing in a concert. 30\% of ticket sales were sold to members of the public (P), \ \frac{3}{5} sales were to family and friends of the musicians (F), and the rest were sold to staff at the school (S). Write the number of ticket sales as a ratio in its simplest form P:F:S

(b) If 440 tickets were sold to the public and staff, how many tickets were sold to members of the public only?

3. (a) Complete the frequency tree using the information provided.

Ratio to Fraction GCSE Question 3a Image 1

(b) Write the ratio of people that wear glasses to the total number of people in the simplest form.

Ratio to Fraction GCSE Question 3a Image 2

38+24=62 wear glasses

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The ratio calculator performs three types of operations and shows the steps to solve:

This ratio calculator will accept integers, decimals and scientific e notation with a limit of 15 characters.

Simplify Ratios:

Enter A and B to find C and D. (or enter C and D to find A and B) The calculator will simplify the ratio A : B if possible. Otherwise the calculator finds an equivalent ratio by multiplying each of A and B by 2 to create values for C and D.

Compare Ratios and Solve for the Missing Value:

Enter A, B and C to find D. The calculator shows the steps and solves for D = C * (B/A)

Enter A, B and D to find C. The calculator shows the steps and solves for C = D * (A/B)

Evaluate Equivalent Ratios:

Enter A, B, C and D. Is the ratio A : B equivalent to the ratio C : D? The calculator finds the values of A/B and C/D and compares the results to evaluate whether the statement is true or false.

Convert Ratio to Fraction

A part-to-part ratio states the proportion of the parts in relation to each other. The sum of the parts makes up the whole. The ratio 1 : 2 is read as "1 to 2." This means of the whole of 3, there is a part worth 1 and another part worth 2.

To convert a part-to-part ratio to fractions:

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Home / United States / Math Classes / 5th Grade Math / Problem Solving using Fractions

Problem Solving using Fractions

Fractions are numbers that exist between whole numbers. We get fractions when we divide whole numbers into equal parts. Here we will learn to solve some real-life problems using fractions. ...Read More Read Less

Table of Contents

ratio and fractions problem solving

What are Fractions?

Types of fractions.

Solved Examples

Equal parts of a whole or a collection of things are represented by fractions . In other words a fraction is a part or a portion of the whole. When we divide something into equal pieces, each part becomes a fraction of the whole.

For example in the given figure, one pizza represents a whole. When cut into 2 equal parts, each part is half of the whole, that can be represented by the fraction  \(\frac{1}{2}\) . 

Similarly, if it is divided into 4 equal parts, then each part is one fourth of the whole, that can be represented by the fraction \(\frac{1}{4}\) .

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Proper fractions

A fraction in which the numerator is less than the denominator value is called a  proper fraction.

For example ,  \(\frac{3}{4}\) ,  \(\frac{5}{7}\) ,  \(\frac{3}{8}\)   are proper fractions.

Improper fractions 

A fraction with the numerator higher than or equal to the denominator is called an improper fraction .

Eg \(\frac{9}{4}\) ,  \(\frac{8}{8}\) ,  \(\frac{9}{4}\)   are examples of improper fractions.

Mixed fractions

A mixed number or a mixed fraction is a type of fraction which is a combination of both a whole number and a proper fraction.

We express improper fractions as mixed numbers.

For example ,  5\(\frac{1}{3}\) ,  1\(\frac{4}{9}\) ,  13\(\frac{7}{8}\)   are mixed fractions.

Unit fraction

A unit fraction is a fraction with a numerator equal to one. If a whole or a collection is divided into equal parts, then exactly 1 part of the total parts represents a unit fraction .

new2

Fractions with Like and Unlike Denominators

Like fractions are those in which two or more fractions have the same denominator, whereas unlike fractions are those in which the denominators of two or more fractions are different.

For example,  

\(\frac{1}{4}\)  and  \(\frac{3}{4}\)  are like fractions as they both have the same denominator, that is, 4.

\(\frac{1}{3}\)  and  \(\frac{1}{4}\)   are unlike fractions as they both have a different denominator.

Operations on Fractions

We can perform addition, subtraction, multiplication and division operations on fractions.

Fractions with unlike denominators can be added or subtracted using equivalent fractions. Equivalent fractions can be obtained by finding a common denominator. And a common denominator is obtained either by determining a common multiple of the denominators or by calculating the product of the denominators.

There is another method to add or subtract mixed numbers, that is, solve the fractional and whole number parts separately, and then, find their sum to get the final answer.

Fractions can be Multiplied by Using:

Division operations on fractions can be performed using a tape diagram and area model. Also, when a fraction is divided by another fraction then we can solve it by multiplying the dividend with the reciprocal of the divisor. 

Let’s Take a Look at a Few Examples

Addition and subtraction using common denominator

( \(\frac{1}{6} ~+ ~\frac{2}{5}\) )

We apply the method of equivalent fractions. For this we need a common denominator, or a common multiple of the two denominators 6 and 5, that is, 30.

\(\frac{1}{6} ~+ ~\frac{2}{5}\)

= \(\frac{5~+~12}{30}\)  

=  \(\frac{17}{30}\) 

( \(\frac{5}{2}~-~\frac{1}{6}\) )

= \(\frac{12~-~5}{30}\)

= \(\frac{7}{30}\)

Examples of Multiplication and Division

Multiplication:

(\(\frac{1}{6}~\times~\frac{2}{5}\))

= (\(\frac{1~\times~2}{6~\times~5}\))                                       [Multiplying numerator of fractions and multiplying denominator of fractions]

=  \(\frac{2}{30}\)

(\(\frac{2}{5}~÷~\frac{1}{6}\))

= (\(\frac{2 ~\times~ 5}{6~\times~ 1}\))                                     [Multiplying dividend with the reciprocal of divisor]

= (\(\frac{2 ~\times~ 6}{5 ~\times~ 1}\))

= \(\frac{12}{5}\)

Example 1: Solve \(\frac{7}{8}\) + \(\frac{2}{3}\)

Let’s add \(\frac{7}{8}\)  and  \(\frac{2}{3}\)   using equivalent fractions. For this we need to find a common denominator or a common multiple of the two denominators 8 and 3, which is, 24.

\(\frac{7}{8}\) + \(\frac{2}{3}\)

= \(\frac{21~+~16}{24}\)    

= \(\frac{37}{24}\)

Example 2: Solve \(\frac{11}{13}\) – \(\frac{12}{17}\)

Solution:  

Let’s subtract  \(\frac{12}{17}\) from \(\frac{11}{13}\)   using equivalent fractions. For this we need a common denominator or a common multiple of the two denominators 13 and 17, that is, 221.

\(\frac{11}{13}\) – \(\frac{12}{17}\)

= \(\frac{187~-~156}{221}\)

= \(\frac{31}{221}\)

Example 3: Solve \(\frac{15}{13} ~\times~\frac{18}{17}\)

Multiply the numerators and multiply the denominators of the 2 fractions.

\(\frac{15}{13}~\times~\frac{18}{17}\)

= \(\frac{15~~\times~18}{13~~\times~~17}\)

= \(\frac{270}{221}\)

Example 4: Solve \(\frac{25}{33}~\div~\frac{41}{45}\)

Divide by multiplying the dividend with the reciprocal of the divisor.

\(\frac{25}{33}~\div~\frac{41}{45}\)

= \(\frac{25}{33}~\times~\frac{41}{45}\)                            [Multiply with reciprocal of the divisor \(\frac{41}{45}\) , that is, \(\frac{45}{41}\)  ]

= \(\frac{25~\times~45}{33~\times~41}\)

= \(\frac{1125}{1353}\)

Example 5: 

Sam was left with   \(\frac{7}{8}\)  slices of chocolate cake and    \(\frac{3}{7}\)  slices of vanilla cake after he shared the rest with his friends. Find out the total number of slices of cake he had with him. Sam shared   \(\frac{10}{11}\)  slices from the total number he had with his parents. What is the number of slices he has remaining?

To find the total number of slices of cake he had after sharing we need to add the slices of each cake he had,

=   \(\frac{7}{8}\) +   \(\frac{3}{7}\)   

=   \(\frac{49~+~24}{56}\)

=   \(\frac{73}{56}\)

To find out the remaining number of slices Sam has   \(\frac{10}{11}\)  slices need to be deducted from the total number,

= \(\frac{73}{56}~-~\frac{10}{11}\)

=   \(\frac{803~-~560}{616}\)

=   \(\frac{243}{616}\)

Hence, after sharing the cake with his friends, Sam has  \(\frac{73}{56}\) slices of cake, and after sharing with his parents he had  \(\frac{243}{616}\)  slices of cake left with him.

Example 6: Tiffany squeezed oranges to make orange juice for her juice stand. She was able to get 25 ml from one orange. How many oranges does she need to squeeze to fill a jar of   \(\frac{15}{8}\) liters? Each cup that she sells carries 200 ml and she sells each cup for 64 cents. How much money does she make at her juice stand?

First  \(\frac{15}{8}\) l needs to be converted to milliliters.

\(\frac{15}{8}\)l into milliliters =  \(\frac{15}{8}\) x 1000 = 1875 ml

To find the number of oranges, divide the total required quantity by the quantity of juice that one orange can give.

The number of oranges required for 1875 m l of juice =  \(\frac{1875}{25}\) ml = 75 oranges

To find the number of cups she sells, the total quantity of juice is to be divided by the quantity of juice that 1 cup has

=  \(\frac{1875}{200}~=~9\frac{3}{8}\) cups

We know that, the number of cups cannot be a fraction, it has to be a whole number. Also each cup must have 200ml. Hence with the quantity of juice she has she can sell 9 cups,   \(\frac{3}{8}\) th  of a cup cannot be sold alone.

Money made on selling 9 cups = 9 x 64 = 576 cents

Hence she makes 576 cents from her juice stand.

What is a mixed fraction?

A mixed fraction is a number that has a whole number and a fractional part. It is used to represent values between whole numbers.

How will you add fractions with unlike denominators?

When adding fractions with unlike denominators, take the common multiple of the denominators of both the fractions and then convert them into equivalent fractions. 

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Equivalent Fractions 3rd Grade Resources, Worksheets and Activities

Equivalent Fractions 3rd Grade Resources, Worksheets and Activities

February 28, 2023 by Anthony Persico

5 Free Equivalent Fractions 3rd Grade Resources

The best equivalent fractions for 3rd graders resources includes worksheets, activities, and games.

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Equivalent fractions for 3rd graders: This guide shares several free activities and worksheets for exploring equivalent fractions.

One of the trickiest topics for 3rd graders to grasp is the concept of equivalent fractions.

To be successful, students need both a procedural understanding of working with equivalent fractions as well as a deep conceptual understanding that will allow them to apply their understanding to more challenging topics involving fractions.

The free Equivalent Fractions 3rd Grade Resources shared in this post will help you to give your students opportunities to develop both procedural fluency and conceptual understanding of equivalent fractions.

Below, you will find links to access 5 equivalent fractions for 3rd graders activities (including equivalent fractions worksheets with answer keys). You can pick-and-choose which resources you want to use to supplement your lesson plans or you can utilize all 5 equivalent fractions for 3rd graders activities in chronological order over the course of several consecutive lessons to form a complete unit on equivalent fractions.

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This post includes a link to download 3 Equivalent Fractions Worksheet 3rd Grade Versions with answer kets.

You can use any of the links below to jump to a specific equivalent fractions for 3rd graders worksheet, or you can scroll through the entire post to access all of the free resources.

Using Fraction Strips to Explore and Understand Equivalent Fractions (Conceptual)

Equivalent Fractions Explained - Step-by-Step Guide for Students (Procedural)

Exploring Equivalent Fractions Using Legos (Conceptual and Procedural)

Using Fraction Circles to Solve Equivalent Fractions (Conceptual and Procedural)

Equivalent Fractions Worksheet 3rd Grade Level (Procedural/Review) - x3

Again, we recommend starting with a conceptual exploration of the topic before you teach procedure. Then, you can utilize activities that combine both skills before finishing with practice or review activities that assess how well your students understand equivalent fractions.

Do you want more free 3rd Grade math resources in your inbox every week? Click here to sign up for our math education email newsletter

Equivalent Fractions 3rd Grade Resources

1.) fraction strips.

When first introducing your 3rd grades to the concept of fractions and equivalent fractions, we recommend giving them an opportunity to conceptually explore the topic before introducing them to any procedure (such as finding common denominators).

This way, students can develop a visual understanding of the concept that they can utilize later on when procedure comes into play.

One of the best hands-on activities for exploring fractions is the use of Fraction Strips —rectangular sheet of colored paper that represent fractions (although fraction strips can also be digital or made out of plastic).

ratio and fractions problem solving

Equivalent Fraction Games for 3rd Grade: Fraction Strips are a fun way to visually explore the concept of equivalent fractions before learning/memorizing procedures.

For example, the image above shows how a 3rd grade student could use fraction strips to visualize why 4/8, 2.4, and 1/2 are all equivalent fractions before ever learning any procedure.

You can learn more about using fraction strips in your classroom by accessing our Free Guide to Fraction Strips , which will show you how to have students make their own fraction strips by hand in addition to a printable fraction strip activity and virtual fraction strip resources.

2.) Equivalent Fractions Explained - Free Student Guide

After 3rd graders have developed an initial conceptual understanding of equivalent fractions are, they are ready to learn the procedure of determining whether or not two given fractions are equivalent.

Our free step-by-step guide to equivalent fractions builds upon and extends your students’ understanding of equivalent fractions to include procedure. As a teacher, you can use this free guide and included examples and visual aides to structure your equivalent fractions 3rd grade lesson.

ratio and fractions problem solving

Our free equivalent fractions 3rd grade guide combines procedural skills with visual representations.

The free guide is also an excellent resource for students who have missed class time or need an in-depth review of equivalent fractions.

3.) Solving Equivalent Fractions Using Legos

After students have begun to build conceptual understanding and procedural fluency, the next step can be participating in activities that combine both skills. Our next equivalent fractions for 3rd graders resource is a suggestion to use Legos as a hands-on tool and visual aide for exploring and solving equivalent fractions.

The image below shows how you can use lego bricks to represent one whole and the fractions 1/2, 1/4, 1/8, and 3/4. Using legos, students have a side-by-side visual aid that corresponds with their mathematical work.

ratio and fractions problem solving

Equivalent Fraction Games for 3rd Grade: Legos are a great hands-on tool and visual aide for exploring equivalent fractions.

For example, like Fraction Strips , 3rd grade students can use legos to visualize why 4/8 and 2/4 are equivalent fractions and why they both can be simplified down to 1/2. This simple yet powerful hands-on activity will help your 3rd grade students to develop both conceptual understanding and procedural fluency.

ratio and fractions problem solving

Example of using Legos to model an equivalent fractions scenario.

4.) Solving Equivalent Fractions Using Fraction Circles

Yet another effective equivalent fractions 3rd grade activity that helps students to develop both conceptual understanding and procedural fluency is the use of fraction circles as a visual aide for solving problems.

Circle diagrams are also commonly used to represent fractions, so it is important for 3rd graders to gain experience working with them.

For example, the image below shows how your 3rd grade students could use fraction circles to determine that 6/10 and 3/5 are equivalent fractions, and that 3/4 and 6/9 are not equivalent fractions.

ratio and fractions problem solving

Fraction Circles can be used to help students with determining whether two given fractions are equivalent or not.

If you are looking for an awesome (and free) virtual resource for using fraction circles, we recommend the fraction circle’s feature on Mathigon’s Polypad app .

ratio and fractions problem solving

Mathigon’s Polypad app is a great free virtual resource for exploring equivalent fractions 3rd grade.

5.) Equivalent Fractions Worksheets 3rd Grade

Finally, once your students have had plenty of experience exploring equivalent fractions and gaining strong conceptual understanding and procedural fluency, they are ready to apply their skills to solving problems that resemble what they will likely see on assessments and exams.

You can use the links below to download 3 different Equivalent Fractions Worksheets 3rd Grade PDF files with complete answer keys. All three worksheets are samples from the 3rd Grade Worksheet Libraries available on our membership website.

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Equivalent Fractions Worksheet 3rd Grade Preview

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▶ Equivalent Fractions Worksheet 3rd Grade: Coloring in Equivalent Fractions

▶ Equivalent Fractions Worksheet 3rd Grade: Equivalent Fraction Statements: True or False?

▶ Equivalent Fractions Worksheet 3rd Grade: Fill in the Missing Values to Make the Fractions Equivalent

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✓ 19 Fractions Critiquing Error Analysis Word Problems in 4 Formats:

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Equivalent Fraction Families – Reasoning and Problem Solving

Equivalent Fraction Families - Reasoning and Problem Solving

This worksheet includes a range of reasoning and problem solving questions for pupils to practise the main skill of finding equivalent fraction families.

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Mathematics Year 4: (4F2) Recognise and show, using diagrams, families of common equivalent fractions

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  2. Solving Ratio Problems with Part and Whole

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  4. Additions of Fraction Math Hack #shorts #youtubeshorts #trick #hack

  5. How to Make an ORIGAMI SWAN

  6. Simple ratio short trick... #education #mathstricks #mathshorts #learning #mathematics #mathskills

COMMENTS

  1. Rates with fractions (video)

    The reciprocal of a fraction is it flipped. For example, the reciprocal of 1/5 is 5/1 (which is 5.) The reciprocal of an integer, is 1 divided by the integer. For example, the reciprocal of 5 is 1/5 and the reciprocal of 98 is 1/98. Comment on Admiral Betasin's post "The reciprocal of a fract...". Show more...

  2. Ratios and rates

    Unit: Ratios and rates 1,700 Possible mastery points Skill Summary Intro to ratios Visualize ratios Quiz 1: 5 questions Practice what you've learned, and level up on the above skills Equivalent ratios Quiz 2: 5 questions Practice what you've learned, and level up on the above skills Ratio application

  3. A 4-Step Guide To Simplify Ratios With Fractions

    A ratio containing both mixed and proper fractions can be intimidating, causing you to draw a blank and eating into your time allowance. This is easily avoidable if you're confident in converting mixed fractions to improper fractions. All you need to remember are two simple rules: multiply the whole by the denominator, and add to the numerator.

  4. Ratio Problem Solving

    Ratio problem solving is a collection of word problems that link together aspects of ratio and proportion into more real life questions. This requires you to be able to take key information from a question and use your knowledge of ratios (and other areas of the curriculum) to solve the problem.

  5. Ratio Word Problems

    Ratio Word Problems Here you will find a range of problem solving worksheets about ratio. The sheets involve using and applying knowledge to ratios to solve problems. The sheets have been put in order of difficulty, with the easiest first. Each problem sheet comes complete with an answer sheet. Using these sheets will help your child to:

  6. Complete Guide to Fractions and Ratios in ACT Math

    Because you can reduce ratios, you can also do the opposite and increase them. In order to do so, you must multiply each piece of the ratio by the same amount (just as you had to divide by the same amount on each side to reduce the ratio). So the ratio of 3: 2 can also be. 3 ( 2): 2 ( 2) = 6: 4. 3 ( 3): 2 ( 3) = 9: 6.

  7. Problem Solving with Ratio and Fractions

    Problem Solving with Ratio and Fractions November 5, 2020 Students are challenged to solve a range of problems involving arithmetic with fractions. There are five problems that link to ratio, probability, mean averages and money. Begin Lesson Download Worksheet Fractions | Foundation GCSE Maths | Higher GCSE Maths

  8. Ratios as Fractions: Properties & Examples

    Ratios as Fractions Calculus Absolute Maxima and Minima Absolute and Conditional Convergence Accumulation Function Accumulation Problems Algebraic Functions Alternating Series Antiderivatives Application of Derivatives Approximating Areas Arc Length of a Curve Area Between Two Curves Arithmetic Series Average Value of a Function

  9. 3 Ways to Solve Fraction Questions in Math

    To add fractions, they must have the same denominator. If they do, simply add the numerators together. [2] For instance, to solve 5/9 + 1/9, just add 5 + 1, which equals 6. The answer, then, is 6/9 which can be reduced to 2/3. 2. Subtract fractions with the same denominator by subtracting the numerators.

  10. Ratio To Fraction

    In order to find a ratio given a fraction: Subtract the numerator from the denominator of the fraction. State the parts of the ratio in the correct order. Explain how to find a ratio given a fraction Fraction to ratio examples Example 5: standard question \frac {3} {4} 43 of a school of fish are male. The rest are female.

  11. Ratio Calculator

    The ratio calculator performs three types of operations and shows the steps to solve: Simplify ratios or create an equivalent ratio when one side of the ratio is empty. Solve ratios for the one missing value when comparing ratios or proportions. Compare ratios and evaluate as true or false to answer whether ratios or fractions are equivalent.

  12. How to solve ratio problems with fractions

    Ratio & Fractions problem. Solving Proportions with an Unknown Ratio To check the accuracy of our answer, simply divide the two sides of the equation and compare the Get arithmetic support online. The best way to download full math explanation, it's download answer here. ...

  13. Problem Solving using Fractions (Definition, Types and Examples)

    Fractions can be Multiplied by Using: 1. Tape diagrams 2. Area models 3. Repeated addition 4. Unit fractions 5. Multiplication of numerators, and multiplication of denominators of the two fractions. Division operations on fractions can be performed using a tape diagram and area model.

  14. Solving ratio problems

    To solve a problem involving ratios and fractions, you may be given the ratio or the fraction . When given the ratio: Add the ratio parts together to find the denominator of the...

  15. Fractions

    Learn about fractions using our free math solver with step-by-step solutions. Skip to main content. Microsoft Math Solver. Solve Practice Download. Solve Practice. Topics ... Type a math problem. Type a math problem. Solve. Examples \frac{ 4 }{ 12 } - \frac{ 9 }{ 7 } \frac{ 4 }{ 12 } \times \frac{ 9 }{ 8 } ...

  16. Fraction Calculator & Problem Solver

    Understand the how and why See how to tackle your equations and why to use a particular method to solve it — making it easier for you to learn.; Learn from detailed step-by-step explanations Get walked through each step of the solution to know exactly what path gets you to the right answer.; Dig deeper into specific steps Our solver does what a calculator won't: breaking down key steps ...

  17. Ratios & Proportions Calculator

    Ratios & Proportions Calculator Compare ratios, convert ratios to fractions and find unknowns step-by-step full pad » Examples Related Symbolab blog posts Practice, practice, practice Math can be an intimidating subject. Each new topic we learn has symbols and problems we have never seen. The unknowing... Read More

  18. Fraction division problems 6th grade

    Fraction division problems 6th grade. Explore all of our fractions worksheets, from dividing shapes into equal parts to multiplying and dividing improper fractions and mixed numbers. Get mathematics support online. Enhance your educational performance. Writing Versatility.

  19. Equivalent Fractions 3rd Grade Resources, Worksheets and Activities

    Equivalent Fraction Games for 3rd Grade: Legos are a great hands-on tool and visual aide for exploring equivalent fractions. For example, like Fraction Strips, 3rd grade students can use legos to visualize why 4/8 and 2/4 are equivalent fractions and why they both can be simplified down to 1/2. This simple yet powerful hands-on activity will ...

  20. Fractions Calculator

    A mixed number is a combination of a whole number and a fraction. How can I compare two fractions? To compare two fractions, first find a common denominator, then compare the numerators.Alternatively, compare the fractions by converting them to decimals. ... Math notebooks have been around for hundreds of years. You write down problems ...

  21. Ratios as Fractions

    A video revising the techniques and strategies for solving problems with ratios as fractions (Higher Only).This video is part of the Ratio & Proportion modul...

  22. Fractions, percentages and ratio

    Divide fractions by integers Multiply decimals by 10, 100, 1000 Subtracting mixed numbers Multiply mixed numbers by an integer Tenths Fractions on a number line Order simple fractions...

  23. 4th Grade Fractions Activities with Equivalent Comparing Problem Solving

    CCSS 4.NF.A.2. Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or ...

  24. Adding and subtracting mixed fractions interactive worksheet

    Adding and subtracting mixed fractions Adding and subtracting mixed numbers with like and unlike denominators ID: 3363717 ... English School subject: Math Grade/level: 4 Age: 7-10 Main content: Fraction Word Problems Other contents: Word Problems Add to my workbooks (0) Embed in my website or blog Add to Google Classroom

  25. Equivalent Fraction Families

    Equivalent Fraction Families - Reasoning and Problem Solving. This worksheet includes a range of reasoning and problem solving questions for pupils to practise the main skill of finding equivalent fraction families. (0 votes, average: 0.00 out of 5) You need to be a registered member to rate this.

  26. Fractions Questions: KS2 Fraction Problem-Solving Resources

    A Fantastic Variety of Fractions Questions for KS2. There are many different ways that we can use fractions in KS2 maths, and this collection of resources includes questions on all of these methods. Pupils will need to find the answer to each fraction problem by solving the equations. From adding and subtracting fractions to fractions word ...