Ratio and Proportion for the ESAT

Q: Can I divide a quantity into more than two parts using this method?

Yes. If the ratio is $x : y : z$, you add all three terms together to find the total number of parts, divide the quantity by that sum, and then multiply by $x$, $y$, and $z$ respectively.

Updated July 2026

This guide covers the fundamental methods for dividing quantities into multiple parts using ratios and expressing existing divisions as ratios. Mastering these techniques is essential for ESAT Mathematics 1, as they form the basis for solving problems involving proportions, scale, and unit conversions.

Core concept

To divide a quantity $Q$ in the ratio $x : y$ , you must first determine the value of a single part by dividing $Q$ by the total number of parts, $x + y$ .

Dividing in a given ratio

When you need to divide a total quantity, $Q$ , into two or more parts based on a specific ratio, such as $x : y$ , you follow a structured three step process. This method ensures that the proportions are maintained correctly across all segments of the total.

Find the total number of parts by adding the terms of the ratio together: $x + y$ .
Divide the total quantity $Q$ by the total number of parts to find the value of exactly one part. This is calculated as $Q \div (x + y)$ .
Multiply the value of this single part by $x$ to find the value of the first portion, and then multiply the value of the single part by $y$ to find the value of the second portion.

It is always advisable to perform a final check by adding the resulting values together. The sum of the parts must equal the original quantity $Q$ . If they do not, an error has occurred in the division or multiplication steps.

Worked Example: Dividing Currency

Question: Divide £450 in the ratio $11 : 7$ .

Step 1: Calculate the total number of parts. $11 + 7 = 18$ parts.

Step 2: Find the value of one part. $1$ part is $£450 \div 18 = £25$ .

Step 3: Calculate the value of each side of the ratio. $11$ parts are $£25 \times 11 = £275$ . $7$ parts are $£25 \times 7 = £175$ .

Check: Verify the total. $£275 + £175 = £450$ .

Expressing a division into parts as a ratio

In some instances, you will be given the sizes of different parts and asked to express their relationship as a ratio. The most critical step in this process is ensuring that all parts are expressed in the same units of measurement before you attempt to write the ratio.

Once the units are consistent, you use ratio notation (the colon symbol) to relate the parts. You should then simplify the ratio by dividing all terms by their highest common factor, similar to how you would simplify a fraction.

Worked Example: Comparing Lengths

Question: A piece of ribbon is divided into $2$ pieces, A and B. Piece A is $125$ cm long and piece B is $2.75$ m long. What is the ratio of the lengths of A and B?

Step 1: Put the lengths of A and B into the same units. Piece A is $125$ cm. Piece B is $2.75$ m, which is equivalent to $275$ cm.

Step 2: Express the relationship using ratio notation. The ratio of the lengths A : B is $125 : 275$ .

Step 3: Simplify the ratio. Dividing both sides of the ratio by their common factor of $25$ : $125 \div 25 = 5$ $275 \div 25 = 11$

The simplified ratio is $5 : 11$ .

Key takeaways

To find the value of one part, divide the total quantity by the sum of all numbers in the ratio.
Always convert all measurements to the same units before expressing them as a ratio.
Verify your final answers by ensuring the sum of the calculated parts equals the original total quantity.
Ratios should be simplified by dividing by the highest common factor to reach their simplest form.

Tips

When simplifying ratios, look for common divisibility rules. If both numbers end in $25, 50, 75,$ or $00$ , they are divisible by $25$ . This can save significant time during the exam.

Cautions

A common error is dividing the total quantity by the ratio terms individually rather than by their sum. Always remember that the sum of the ratio terms represents the 'whole' of the quantity.

Insight

Ratios are closely linked to fractions. If a quantity is divided in the ratio $a : b$ , then the first part is the fraction $a / (a + b)$ of the whole, and the second part is $b / (a + b)$ of the whole.

Frequently asked questions

Can I divide a quantity into more than two parts using this method?

Yes. If the ratio is $x : y : z$ , you add all three terms together to find the total number of parts, divide the quantity by that sum, and then multiply by $x$ , $y$ , and $z$ respectively.

Does the order of the numbers in a ratio matter?

Yes. The order is specific to the parts described. In the ribbon example, the ratio A : B is $5 : 11$ , whereas the ratio B : A would be $11 : 5$ .

What if the division of the total by the number of parts results in a decimal?

In many ESAT questions, the numbers are chosen to work out cleanly. However, if you get a decimal, continue the calculation with the decimal value, as the final parts may still be integers or required in decimal form.

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