KS3 Maths

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Solving Equations with Fractions

KS3 Maths Algebra Solving Equations with Fractions

Fractions are just a way to represent division. We can add, subtract, multiply, and divide fractions, similar to whole numbers.

Note: When working with equations involving fractions, make sure to keep the denominators in check as they can affect your calculations.

One-Step Equations with Fractions

In one-step equations with fractions, you’re only a step away from finding the value of the variable. For example, if you have \frac{x}{2} = 3, you can multiply both sides by 2 to get x = 6.

The main aim here is to get the variable alone on one side of the equation, and in one-step equations with fractions, this usually means getting rid of the denominator.

Two or More step Equations with Fractions

Sometimes it takes two steps to isolate the variable. For example, consider 2 \times \frac{x}{3} + 1 = 3

  • Step 1: Eliminate the constant term by subtracting 1 from both sides, which gives you 2 \times \frac{x}{3} = 2
  • Step 2: Multiply both sides by 3 to get 2x = 6
  • Step 3: Then, divide both sides by 2 and you get x = 3

Clearing the Fractions

If your equation has fractions, you can get rid of them to make the equation easier to work with. For example:

\frac{x}{2} + \frac{3}{4} = 1

A quick way to get rid of the fractions is to multiply everything in the equation by a number that turns all the fractions into whole numbers. In this case, 4 will work. So, 4(\frac{x}{2}) + 4(\frac{3}{4}) = 4 \times 1.

After doing the multiplication, you’ll get 2x + 3 = 4

Remember: 4(\frac{x}{2}) = \frac{4x}{2}

Equations with Fractions on Both Sides

Sometimes you’ll find equations that have fractions on both sides. For example, let’s solve:

\frac{x}{2} = \frac{3}{4} + \frac{x}{4}

  • Multiply Everything by the Same Number: Here, 4 will also work. So, 4(\frac{x}{2}) = 4(\frac{3}{4}) + 4(\frac{x}{4})
  • The Fraction is Eliminated: After multiplying, you’ll have 2x = 3 + x

Next, isolate x on one side of the equation by subtracting x from both sides:

2x-x=3

x=3

Practice Questions

Question 1:

Solve \frac{x}{3} = 2

Multiply both sides by 3 to get x = 6

Question 2:

Solve 2 \times \frac{x}{4} + 1 = 3

First subtract 1 from both sides to get 2 \times \frac{x}{4} = 2

Then multiply both sides by 4 and divide by 2 to get x = 4

Question 3:

Solve \frac{x}{2} + 1 = \frac{x}{3} + 2

Multiply every term by 6 to get 3 \times x + 6 = 2 \times x + 12

Solve for x to get x = 6

Question 4:

Solve 2 \times (\frac{x}{3} + 1) = x

Expand the brackets to get 2 \times \frac{x}{3} + 2 = x

Multiply every term by 3 to clear the fraction, resulting in 2 \times x + 6 = 3x

Solve for x to get x = 6

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