Example 3 - Finding the difference
\(\frac{2}{3}\) of A is equal to B.
30% of B is equal to C.
The ratio of C:D is 3:5.
If B is equal to 16, find the difference between A and D.
1. What do I have to do?
Read the question through twice.
Highlight or underline the important pieces of information in the question.
2. What information do I need?
The most important parts of this question are:
- two thirds of A equals B
- 30% of B equals C
- C:D ratio is 3:5
- B is 16
The fact that B is given is a way into the question later.
The question asks the difference between A and D.
A and D are two numbers. It is not clear if they are whole numbers, decimals or fractions.
Finding a difference will involve subtracting two numbers.
From the question, A must be bigger than B (as B is two thirds of A) and B must be bigger than C (as C is 30% of B). This means A must be bigger than C.
Also D must be bigger than C as the ratio of C:D is 3:5.
3. What information don’t I need?
Everything given in the question needs to be used at some point.
The final calculation does not need to involve B and C, although these will be needed to work out the values of A and D.
4. What maths can I do?
B is given and so this is the best way into the question.
The ratio will be found later.
Since 30% of B is equal to C and two thirds of A is B, this information can be used to find A and C.
Step A
Use the information that B is equal to 16 to work out the value of A.
\(\frac{2}{3}A = 16\)
Dividing both sides by 2 gives:
\(\frac{1}{3}A = 8\)
Multiplying both sides by 3 gives:
\(A = 24\)
Step B
Use the information that B is equal to 16 to work out the value of C.
C is 30% of 16.
To work out 30% of 16, find 10% then multiply your answer by 3.
To find 10% divide 16 by 10.
\(16 \div 10 = 1.6\)
\(1.6 \times 3 = 4.8\)
Therefore the value of C is 4.8.
Step C
Use the information that C is equal to 4.8 to work out the value of D.
The ratio of C:D is 3:5 therefore C takes up 3 parts of the ratio.
If 4.8 is equal to 3 parts then divide 4.8 by 3 to find out what 1 part of the ratio is worth.
\(4.8 \div 3 = 1.6\)
Use the information that 1 part of the ratio is worth 1.6 to work out what 5 parts of the ratio are worth. Do this by multiplying 1.6 by 5.
\(1.6 \times 5 = 8\)
Therefore the value of D is 8.
Step D
Find the difference between A and D by taking them away from each other.
\(24 - 8 = 16\)
Therefore the difference between A and D is 16.
5. Is my solution correct?
It is important to check any calculations at the end, even if a calculator was used.
It was given that A must be bigger than B and it is.
B must be bigger than C and it is.
Also D must be bigger than C which it is.
6. Have I completed everything?
The answer is supposed to be a number difference, which it is.
A subtraction was carried out at the very end, which is what must happen to find a difference.
Nothing else was asked for.
