## 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:

48:15

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

48/3 = 16

15/3 = 5

**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:

3 x 9 = 27

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:

58:5

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.