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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

## What are algebraic fractions?

Algebraic fractions are fractions that contain at least one variable.

The following algebraic expressions are examples of algebraic fractions:

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

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 .

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

## Algebraic fractions examples

Example 1: equation with one fraction.

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) .

## Example 2: Equation with two fractions

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 .

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

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.

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

## Practice algebraic fractions questions

## Algebraic fractions GCSE questions

2. Use the quadratic formula to solve the equation

2x(x+2)+5x(x+3)=4(x+3)(x+2) or equivalent

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}

## Learning checklist

- 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

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## Privacy Overview

## How to solve equations with fractions

## What is a fractional equation?

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???

I create online courses to help you rock your math class. Read more.

???2\left(5x+\frac{1}{2}=12\right)???

???2(5x)+2\left(\frac{1}{2}\right)=2(12)???

## How to solve equations when there’s a fraction somewhere in the equation

## Take the course

???\frac{5}{4}\cdot\frac{4}{5}n=\frac{5}{4}\cdot20???

???\frac{20}{20}n=\frac{100}{4}???

First isolate the fractional term.

???\frac{4}{7}x+14-14=22-14???

???\frac{7}{4}\cdot\frac{4}{7}x=\frac{7}{4}\cdot8???

???\frac{28}{28}x=\frac{56}{4}???

???7\left(\frac{4}{7}x+14=22\right)???

???7\cdot\frac{4}{7}x+7\cdot14=7\cdot22???

Now we can solve for the variable using inverse operations.

???\frac{4x}{4}=\frac{56}{4}???

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## Solving Equations with Fractions

Let see what happens with a typical two-step equation with the distributive property.

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!

We need to discuss the word 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.

## Example 2 - Equations with Fractions with the Same Denominator

## Example 3 - Equations with Two Fractions with Different Denominators

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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

## Want to join the conversation?

## Video transcript

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

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.

- 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

Need help? Chat with a tutor anytime, 24/7 .

Solve your algebra problem step by step!

Get help with your math queries:

We start with some algebraic examples, then follow with some word problems involving fractions.

## a. Algebraic types

`x/5+3/10 = 1/2`

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.

Divide boths sides by `2`, giving us the final answer:

`(2x)/3+2/5 = 8+x/2`

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`

## 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?

`1/T=1/T_1+1/T_2`

`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

- 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`

`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.

Let the length of the journey from home to work be x km.

`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)`

This gives us `7x=(5xx120)/6=100`

So the distance from home to work is 14.3 km.

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## How To Solve Equations With Fractions

## Step 1: Find the least common denominator

## 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

## Step 5: Divide the coefficient on both sides

## Examples of how to solve equations with fractions

First, let’s find the least common denominator (LCD) of the fractions:

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.

Divide 14 on both sides of the equation to solve for x.

We can let n be the unknown number. We can set up the equation to solve the problem.

Multiplying 12 to both sides of the equation, we have:

Move 4 on the right-hand side of the equation.

Divide both sides by 4 to isolate n.

Hence, the unknown fraction is equal to 5/4.

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## COMMENTS

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