Compare Mixed Numbers and Improper Fractions | KS3 Maths | MME

## What you need to know

Things to remember:

• To compare mixed numbers and improper fractions, we need to convert them into the same form.
• To convert a mixed number into an improper fraction, multiply the whole number by denominator (the bottom of the fraction) and add it to the numerator (the top of the fraction)
• To convert an improper fraction into a mixed number, divide the numerator (top of the fraction) by the denominator (the bottom of the fraction). The whole number part of the division will be the whole number part of the mixed number, and the remained will be the new numerator.

Which is bigger out of the following?

$4\dfrac{2}{5}$ or $\dfrac{9}{5}$

To compare these, we first need to make them both into either mixed numbers or top-heavy fractions. Let’s start with converting to mixed top-heavy fractions, which we do in three steps:

Step 1: Multiply the whole number by the denominator (bottom) of the fraction.

$$4\times5=20$$

Step 2: Add the answer in Step 1 to the top numerator (top) of the fraction.

$$2+20=22$$

Step 3: Put the answer to Step 2 over the denominator of the fraction.

$$\dfrac{22}{5}$$

$$4\dfrac{2}{5}=\dfrac{22}{5}$$

And now we can compare them.

$\dfrac{22}{5}$ or $\dfrac{9}{5}$

22 is bigger than 9, so $4\frac{2}{5}$ must be bigger than $\frac{9}{5}$.

What happens if we wanted to make them both into mixed numbers? We can convert a top-heavy fraction into a mixed number in two steps.

Step 1: Divide the fraction to find the whole number and remained.

$$9\div5=2\text{ remainder }4$$

Whole number part = 1

Remainder = 4

Step 2: Turn into a mixed number. The whole number part is the number before the fraction, and the remainder goes on top of the original fraction.

$$\frac{9}{5}=1\frac{4}{5}$$

Now we can compare them. To compare mixed numbers, we possibly need to do two steps.

$4\dfrac{2}{5}$ or $1\dfrac{4}{5}$

Step 1: Compare the whole numbers.

4 is bigger than 1, so $4\dfrac{2}{5}$ is bigger than $1\dfrac{4}{5}$. This is the same answer as before.

However, if the whole number parts had been the same, then we’d need to compare the fractions.

Which is bigger out of the following?

$1\dfrac{2}{5}$ or $1\dfrac{4}{5}$

Step 2: Because the whole numbers are the same, compare the fractions.

$1\dfrac{4}{5}$ is bigger than $\dfrac{2}{5}$, so $1\dfrac{4}{5}$ is bigger than $1\dfrac{2}{5}$

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## Example Questions

Step 1: Multiply the whole number by the denominator (bottom) of the fraction.

$$7\times9=63$$

Step 2: Add the answer in Step 1 to the top numerator (top) of the fraction.

$$4+63=67$$

Step 3: Put the answer to Step 2 over the denominator of the fraction.

$$\frac{67}{9}$$

$$7\frac{4}{9}=\frac{63}{9}$$

And now we can compare them.

$\frac{63}{9}$ or $\frac{47}{9}$

47 is smaller than 63, so $\dfrac{47}{9}$ must be smaller than $7\dfrac{4}{9}$.

Step 1: Divide the fraction to find the whole number and remained.

$$13\div7=1\text{ remainder }6$$

Whole number part = 1

Remainder = 6

Step 2: Turn into a mixed number. The whole number part is the number before the fraction, and the remainder goes on top of the original fraction.

$$\frac{13}{7}=1\frac{6}{7}$$

Now we can compare them. To compare mixed numbers, we possibly need to do two steps.

$1\dfrac{5}{7}$ or $1\dfrac{6}{7}$

Step 1: Compare the whole numbers

The whole numbers are the same, so move on to Step 2.

Step 2: Compare the fractions.

6 is biggest than 7, so $1\dfrac{6}{7}$ must be bigger than $1\dfrac{5}{7}$.

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