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In order to access this I need to be confident with:
Simplifying fractions
Adding and subtracting fractions
This topic is relevant for:
Algebraic Fractions
Here we will learn about algebraic fractions , including operations with fractions, and solving linear and quadratic equations written in the form of algebraic fractions.
There are also algebraic 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 are algebraic fractions?
Algebraic fractions are fractions that contain at least one variable.
The following algebraic expressions are examples of algebraic fractions:
x is the numerator: \quad \quad \quad \frac{x}{12} Both the numerator and the denominator contain an x term: \quad \quad \quad \frac{x+1}{2x}
An expression in terms of x is the denominator: \quad \quad \frac{3}{x+1}\ Both the numerator and the denominator contain an expression with x : \quad \quad \frac{3x+4}{2x-5}
The numerator is a multiple of x : \quad\quad \quad \quad \frac{2x}{15} The numerator and the denominator are quadratic expressions: \quad\quad \quad \quad \frac{(x+3)^{2}}{x^{2}-9}
The main aim of this lesson is to understand how to solve equations that include algebraic fractions.
All the examples above are expressions whereas the examples below are equations as we can find specific values for x for each example to solve the equation.
One step equation: \quad \quad \frac{x}{12}=4 A separate constant term: \quad \frac{x+1}{2x}+4=x
A quadratic equation: \quad\frac{3}{x+1}=x+5 A linear equation: \quad \frac{3x+4}{2x-5}=6
A second fraction: \quad \frac{2x}{15}=\frac{5x}{2} Double brackets, difference of two squares and simultaneous equations: \quad \frac{(x+3)^{2}}{x^{2}-9}=2x-1
It is important to be able to simplify algebraic fractions into their simplest form. If you need to practice this or need a quick refresher, see the lesson on simplifying algebraic fractions for further information.
Step-by-step guide: Simplifying algebraic fractions
How to solve equations including algebraic fractions
We need to be able to solve equations including algebraic fractions.
Let’s look at a simple example when \frac{8}{x}=2 .
Here, the denominator of the fraction contains the variable, so we first need to get the variable out of the denominator.
Rearranging the equation by multiplying both sides by x and then dividing by 2 , we get the value of x=4 .
We can substitute this into the original equation to prove that the answer is correct.
Here, \frac{8}{4}=2 so we have the correct answer.
We shall now consider more complicated cases when equations involve algebraic fractions.
In order to solve equations including algebraic fractions.
Convert each fraction so they all have a common denominator.
- Multiply the equation throughout by the common denominator.
- Solve the equation (linear or quadratic).
Explain how to solve equations including algebraic fractions
Algebraic fractions worksheet
Get your free algebraic fractions worksheet of 20+ questions and answers. Includes reasoning and applied questions.
Algebraic fractions examples
Example 1: equation with one fraction.
Solve the equation
Here, we only have one fraction and so we do not need to convert any other term into a fraction.
2 Multiply the equation throughout by the common denominator .
Multiplying the equation throughout by 3 (the denominator of the fractional term), we get
Make sure that you multiply every term in the equation by 3 .
3 Solve the equation (linear or quadratic) .
You can check your solution by substituting the value for x into the original equation and evaluating it.
Example 2: Equation with two fractions
Here, we have the two fractions with the denominators of 2 and 5 . The lowest common multiple of 2 and 5 is 10 and so we can convert the two fractions so that they have the same denominator.
Remember to use brackets to ensure that you multiply the entire numerator by 5 .
Remember to use brackets to ensure that you multiply the entire numerator by 2 .
We now have the equation
Multiply the equation throughout by the common denominator .
Multiplying the equation throughout by 10 (the denominator of the fractional terms), we get
Solve the equation (linear or quadratic) .
Example 3: Equation with x in the denominator
Here, we have one fraction so we do not need to find a common denominator.
Multiplying the equation throughout by x + 1 (the denominator of the fractional terms), we get
Example 4: Equation with three fractions
Here, we need to find the lowest common multiple of x, 2x, and 3x . As x is the highest common factor, x \times 1 \times 2 \times 3 is the lowest common multiple, which is equal to 6x. .
So by multiplying the numerator and denominator of each fraction by a constant, we can convert each fraction to have the common denominator of 6x: .
We now have an equation which we can immediately simplify.
Multiplying the equation throughout by 6x (the denominator of the fractional terms), we get
Example 5: Denominators are expressions in terms of x
Here, we need to find a common denominator for (x + 2) and (x − 4) .
The easiest way to do this is to multiply the two expressions together.
Top tip: do not expand the brackets too soon as you may be able to simplify the fraction before solving the equation.
By multiplying each fraction by the denominator of the other fraction, we get
Multiplying the equation throughout by (x+2)(x-4) (the denominator of the fractional term), we get
Example 6: Equation including a quadratic
Here, we have a single fraction and so we do not need to find a common denominator.
Multiplying the equation throughout by x (the denominator of the fractional terms), we get
Here, we have two possible solutions for x so we can check both:
Common misconceptions
- Multiplying the numerator by the denominator
Let us look at example 2 . When multiplying throughout by 10 to remove the denominator from each fraction, the numerator has also been multiplied by 10 . This means that the fractions have been multiplied by 100 , instead of 10 , leaving the next line of work to be incorrect.
- Ignoring the denominators
When given an equation including an algebraic fraction, if the denominators are ignored the question will be answered incorrectly.
- Not multiplying all terms by the denominator
When rearranging an equation, the denominator is moved to the other side of the equals sign, instead of each term being multiplied by it. You must remember to multiply throughout by any value, not just the opposite side of the equals sign.
- Simplifying fractions incorrectly (1)
When adding two fractions, the denominator must be the same. A common misconception for adding two fractions is to add the numerators and the denominators together because this method is emphasised when looking at multiplying fractions.
- Simplifying fractions incorrectly (2)
Seeing the same term on the numerator and denominator allow the misconception that they can both be cancelled.
Practice algebraic fractions questions
1. Solve the equation
2. Solve the equation
3. Solve the equation
4. Solve the equation
5. Solve the equation
6. Solve the equation
Algebraic fractions GCSE questions
1. Arron earns £40 per hour. One day, he receives a bonus of £5 . He shares this day’s earnings equally with his brother and sister. If he gives away £190 , how many hours did Arron work that day?
2. Use the quadratic formula to solve the equation
2x(x+2)+5x(x+3)=4(x+3)(x+2) or equivalent
x=3 or x=-\frac{8}{3}
3. Two shapes given below have the same area. Calculate the value for x .
Area of Triangle = \frac{2(x+8)}{2}=x+8
Area of Square = \left(\frac{3}{\sqrt{x}}\right)^{2}=\frac{9}{x}
x=1 or x=-9
Conclusion: x=1 only
Learning checklist
You have now learned how to:
- use algebraic methods to solve linear equations in 1 variable (including all forms that require rearrangement)
- simplify and manipulate algebraic fractions
The next lessons are
- Simultaneous equations
- Factorising
Still stuck?
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How to solve equations with fractions
What is a fractional equation?
In this lesson we’ll look at how to solve equations with numerical fractions as coefficients and terms.
Remember that multiplying a fraction by its reciprocal will always give you a value of ???1???.
For example ???4/5??? has a reciprocal of ???5/4??? because
???\frac{4}{5}\cdot\frac{5}{4}=1???
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To clear a fraction from an equation, multiply all of the terms on both sides of the equation by the fraction’s denominator.
For example, to clear the ???2??? from the fraction in ???5x+1/2=12???, multiply the equation by ???2??? on both sides.
???2\left(5x+\frac{1}{2}=12\right)???
???2(5x)+2\left(\frac{1}{2}\right)=2(12)???
???10x+1=24???
How to solve equations when there’s a fraction somewhere in the equation
Take the course
Want to learn more about algebra 2 i have a step-by-step course for that. :), clearing the fraction from the equation in order to solve for the variable.
Solve for the variable.
???\frac{4}{5}n=20???
To get rid of a fractional coefficient, we have to multiply both sides by its reciprocal, because that’ll make the fraction ???1???.
???\frac{5}{4}\cdot\frac{4}{5}n=\frac{5}{4}\cdot20???
???\frac{20}{20}n=\frac{100}{4}???
???1n=25???
If you have a fractional coefficient and another term, you can isolate the term with the variable and then multiply both sides by the reciprocal of the fractional coefficient.
???\frac{4}{7}x+14=22???
First isolate the fractional term.
???\frac{4}{7}x+14-14=22-14???
???\frac{4}{7}x=8???
Now get rid of the fractional coefficient by multiplying both sides of the equation by the reciprocal of ???4/7???.
???\frac{7}{4}\cdot\frac{4}{7}x=\frac{7}{4}\cdot8???
???\frac{28}{28}x=\frac{56}{4}???
???1x=14???
We could also do this same problem by first clearing the fraction. In order to get rid of the fraction, we have to multiply every term in the equation by its denominator.
???7\left(\frac{4}{7}x+14=22\right)???
???7\cdot\frac{4}{7}x+7\cdot14=7\cdot22???
???4x+98=154???
Now we can solve for the variable using inverse operations.
???4x+98-98=154-98???
???4x=56???
???\frac{4x}{4}=\frac{56}{4}???
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Solving Equations with Fractions
I know fractions are difficult, but with these easy step-by step instructions you'll be solving equations with fractions in no time.
Do you start to get nervous when you see fractions? Do you have to stop and review all the rules for adding, subtracting, multiplying and dividing fractions?
If so, you are just like almost every other math student out there! But... I am going to make your life so much easier when it comes to solving equations with fractions!
Our first step when solving these equations is to get rid of the fractions because they are not easy to work with!
Let see what happens with a typical two-step equation with the distributive property.
In this problem, we would typically distribute the 3/4 throughout the parenthesis and then solve. Let's see what happens:
Yuck! That just made this problem worse! Now we have two fractions to contend with and that means subtracting fractions and multiplying fractions.
So... let's stop here and say,
We DO NOT want to do this! DO NOT distribute fractions.
We are going to learn how to get rid of the fractions and make this much more simple!
So... what do we do? We are going to get rid of just the denominator in the fraction, so we will be left with the numerator, or just an integer!
I know, easier said than done! It's really not hard, but before I get into it, I want to go over one algebra definition.
We need to discuss the word term.
In Algebra, each term within an equation is separated by a plus (+) sign, minus (-) sign or an equals sign (=). Variable or quantities that are multiplied or divided are considered the same term.
That last example is the most important to remember. If a quantity is in parentheses, it it considered one term!
Let's look at a few examples of how to solve these crazy looking problems!
Example 1 - Equations with Fractions
Take a look at this example on video if you are feeling overwhelmed.
Hopefully you were able to follow that example. I know it's tough, but if you can get rid of the fraction, it will make these problems so much easier. Keep going, you'll get the hang of it!
In the next example, you will see two fractions. Since they have the same denominator, we will multiply by the denominator and get rid of both fractions.
Example 2 - Equations with Fractions with the Same Denominator
Did you notice how multiplying by 2 (the denominator of both fractions) allowed us to get rid of the fractions? This is the best way to deal with equations that contain fractions.
In the next example, you will see what happens when you have 2 fractions that have different denominators.
We still want to get rid of the fractions all in one step. Therefore, we need to multiply all terms by the least common multiple. Remember how to find the LCM? If not, check out the LCM lesson here .
Example 3 - Equations with Two Fractions with Different Denominators
Yes, the equations are getting harder, but if you take it step-by-step, you will arrive at the correct solution. Keep at it - I know you'll get it!
- Solving Equations
- Equations with Fractions
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Course: Algebra 1 > Unit 2
- Why we do the same thing to both sides: Variable on both sides
- Intro to equations with variables on both sides
- Equations with variables on both sides: 20-7x=6x-6
Equation with variables on both sides: fractions
- Equation with the variable in the denominator
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Video transcript
450+ Math Lessons written by Math Professors and Teachers
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Simplifying and Teaching Math for Over 23 Years
Tips, tricks, lessons, and tutoring to help reduce test anxiety and move to the top of the class.
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Chapter Contents ⊗
- Factoring and Fractions
- 1. Special Products
- 2. Common Factor and Difference of Squares
- 3. Factoring Trinomials
- 4. The Sum and Difference of Cubes
- 5. Equivalent Fractions
- 6. Multiplication and Division of Fractions
- 7. Addition and Subtraction of Fractions
8. Equations Involving Fractions
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In this section, we can find the solution easily by multiplying throughout by the lowest common denominator (LCD) . This will simplify our equation and make it easier to solve.
We need to remember to multiply all terms in the equation (both sides of the equal sign) by the LCD, otherwise the final answer won't be correct.
We start with some algebraic examples, then follow with some word problems involving fractions.
a. Algebraic types
Solve for x :
`x/5+3/10 = 1/2`
We first look at the denominators of the fractions and determine the lowest common denominator. In this case, it will be `10.`
We multiply throughout by `10` and the result is:
`10xx(x/5+3/10) = 10xx(1/2)` `2x+3=5`
Then, we just solve the simpler equation we've found.
Subtract `3` from both sides:
Divide boths sides by `2`, giving us the final answer:
`(2x)/3+2/5 = 8+x/2`
Once again, we look at the denominators of the fractions and determine the lowest common denominator. In this case, it will be `30.`
We multiply throughout by `30` and the result is:
`30xx((2x)/3+2/5) = 30xx(8+x/2)` `20x+12=240+15x`
Subtracting `12` from both sides:
`20x=228+15x`
Substract `15x` from both sides:
Divide both sides by `5`, giving us the final answer:
`x=228/5=45.6`
This next one has the variable in the denominator . We'll still need to find the LCD as before.
`4/x+1/3 = 7/(5x)`
In this case, the lowest common denominator is `15x.`
We multiply throughout by `15x` and the result is:
`15x xx(4/x+1/3) = 15x xx(7/(5x))` `60+5x=21`
Subtracting `60` from both sides:
`x=-39/5=-7.8`
Continues below ⇩
b. Word problems
An aquarium can be filled by one hose in 7 minutes and a second thinner hose in 10 minutes.
How long will it take to fill the tank if both hoses operate together?
Since each hose makes the filling time less , we have to add the reciprocals together and take the reciprocal of the result.
We need to use:
`1/T=1/T_1+1/T_2`
So we have:
`1/T=1/7+1/10`
`1/T=(10+7)/70=17/70`
`T=70/17=4.1176`
So it will take 4.1 minutes to fill the tank with both hoses operating together.
Related pages
You may find these useful:
- Division of Algebraic Expressions
- Solving equations
- Formulas and Literal Equations
- Applied verbal problems
For 2 resistors with resistances R 1 and R 2 in parallel, the combined resistance R is given by:
`1/R=1/R_1+1/R_2`
For a particular circuit, the combined resistance R was found to be 4 ohms (Ω), and R 1 = 10 Ω. Find R 2 .
`1/4=1/10+1/R_2`
We need to multiply throughout by the lowest common denominator: 20 R 2
`1/4=1/10+1/R_2` `(20R_2)/4=(20R_2)/10+(20R_2)/(R_2)` `5R_2=2R_2+20` `3R_2=20` `R_2=20/3=6 2/3Omega`
"`Omega`" is the symbol for "ohms", the unit of resistance.
A car averaged 30 km/h going from home to work and 40 km/h on the return journey. If the total time for the two journeys is 50 minutes, how far is it from home to work?
Let the length of the journey from home to work be x km.
Recall that
`text(speed) = text(distance)/text(time)`
`text(time) = text(distance)/text(speed)`
We must use the same time units throughout. We will use hours.
` 50\ text(minutes)=50/60=5/6text(hours)`
In the forward journey, the car's time was `x/30` hours.
For the return journey, the time was `x/40` hours.
The total time was `x/30+x/40=5/6\ text(hours)`
So `(4x+3x)/120=(7x)/120=5/6`
This gives us `7x=(5xx120)/6=100`
That is `x=100/7=14.286`
So the distance from home to work is 14.3 km.
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How To Solve Equations With Fractions
What is an equation.
An equation is a statement used in mathematics to show that two items are equal. In fractions, an equation is used to find the value of a fraction when one or more of its parts are unknown.
If you’re stuck on a math problem that involves fractions, don’t worry! In this article, we’ll show you how to solve equations with fractions step by step.
Step 1: Find the least common denominator
Firstly we need to find the least common denominator (LCD) of the fractions found in the equation, which is the smallest number that can be a common denominator for both of the fractions. For this equation the LCD is 12 as this is the lowest common multiple of 4 and 6.
Step 2: Multiply the least common denominator
Multiply the LCD to both sides of the equation.
Step 3: Simplify the equation
Simplify both sides of the equation and make sure we’re only working with whole numbers.
Step 4: Simplify until there's one term on both sides
Move all terms with the variable on one side and further simplify both sides of the Equations so we have one term on both sides.
Step 5: Divide the coefficient on both sides
Once the variable is isolated on one side, divide the coefficient on both sides to solve for the unknown variable.
Examples of how to solve equations with fractions
Q1) Find the value of x in:
First, let’s find the least common denominator (LCD) of the fractions:
6=2×3 15=3×5 LCD:2×3×5=30
Multiply 30 on both sides of the equation. Make sure to simplify after distributing 30.
Move 6x on on the left-hand side of the equation to isolate the term with the variable.
20x-6x=60 14x=60
Divide 14 on both sides of the equation to solve for x.
Q2) An unknown fraction is added to 1 and we divide the sum by 3. The result is equal to 3/4. What is the value of the unknown fraction?
We can let n be the unknown number. We can set up the equation to solve the problem.
First, look for the least common denominator (LCD). Since 3 and 4 don’t share any common factors, we find their product.
Multiplying 12 to both sides of the equation, we have:
Move 4 on the right-hand side of the equation.
4n=9-4 4n=5
Divide both sides by 4 to isolate n.
Hence, the unknown fraction is equal to 5/4.
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IMAGES
VIDEO
COMMENTS
How to solve equations involving fractions using the balance method from https://mr-mathematics.comThe full lesson includes a starter
How do we solve equations that involve fractions? Before we get started, I am assuming you already know how to solve equations by balancing:
View more at http://www.MathTutorDVD.com.In this lesson, you will learn how to solve fractional equations, which means that the variable may
How to solve equations including algebraic fractions · Convert each fraction so they all have a common denominator. · Multiply the equation throughout by the
To clear a fraction from an equation, multiply all of the terms on both sides of the equation by the fraction's denominator. For example, to
Our first step when solving these equations is to get rid of the fractions because they are not easy to work with! Let see what happens with a typical two-step
The general rule for solving equations with fractions — whether it be only on one side or both — is to try to get rid of all of them. The most common way to
We start with some algebraic examples, then follow with some word problems involving fractions. a. Algebraic types. Example 1. Solve for x:.
How To Solve Equations With Fractions · Step 1: Find the least common denominator · Step 2: Multiply the least common denominator · Step 3:
To clear all fractions from an equation, multiply both sides of the equation by the least common denominator of the fractions that appear in the