About Arithmetical Reasoning
Arithmetical reasoning covers word problems solved by setting up a small equation rather than by applying a formula for a named topic. The wording is the only difficulty: once the sentence is written as algebra the rest is ordinary arithmetic. The skill worth practising is translation.
What you need to understand
- Name the unknown quantity and write what the sentence says as an equation.
- A sum and a difference together fix both numbers: the larger is half the sum plus half the difference.
- A ratio says how the total divides, so the ratio terms should be added to find the number of parts.
- Each part of a ratio is the total divided by the number of parts.
- Checking by substituting the answer back into the original sentence catches translation errors immediately.
How to work through these questions
- Read the problem and decide what quantity it is really asking for.
- Give that quantity, and any other unknown, a short name.
- Write each sentence as an equation using those names.
- Solve the equations, working with the simplest one first.
- Put the answer back into the original wording to confirm it fits.
Mistakes that cost marks
- Answering a related quantity rather than the one asked for, such as giving the ratio term instead of the value.
- Forgetting to divide the total by the number of parts before scaling.
- Mixing up the two unknowns when the question asks about one of them specifically.
- Treating a ratio as a pair of actual values rather than a proportion.
- Stopping at the equation instead of solving it, or solving it and forgetting the units.
Worked example
Two numbers are in the ratio 3 : 4 and their sum is 140. What is the larger number?
- The ratio 3 : 4 means the numbers divide the total into 3 plus 4, which is 7 parts.
- Each part is the total divided by 7, so 140 divided by 7 is 20.
- The smaller number is 3 parts, which is 3 times 20 and equals 60.
- The larger number is 4 parts, which is 4 times 20 and equals 80.
- Checking: 60 plus 80 is 140, so the numbers are right.
Answer: 80
Practice questions with answers
A few Arithmetical Reasoning questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
The sum of two numbers is 122 and their difference is 34.
What is the larger number?
Answer: Option B — with explanation
The larger number is half the sum plus half the difference.
Half of 122 is 61.0 and half of 34 is 17.0, so the larger number is 78.
Checking: the other number is 44, and 78 + 44 = 122 while 78 - 44 = 34.
Common mistakes
- gave the smaller number instead of the larger one: 44
- gave the sum instead of a number: 122
- gave the difference instead of a number: 34
Question 2
Two numbers are in the ratio 5 : 9, and their sum is 126.
What is the first number?
Answer: Option C — with explanation
The ratio 5 : 9 has 14 parts in total.
Each part is 126 divided by 14, which is 9.
The first number is 5 x 9 = 45 and the second is 9 x 9 = 81, so the first number is 45.
Common mistakes
- gave the ratio term instead of the value: 5
- reported the other number in the pair: 81
- gave the value of one part instead of the whole number: 9
Question 3
The sum of two numbers is 161 and their difference is 19.
What is the larger number?
Answer: Option D — with explanation
The larger number is half the sum plus half the difference.
Half of 161 is 80.5 and half of 19 is 9.5, so the larger number is 90.
Checking: the other number is 71, and 90 + 71 = 161 while 90 - 71 = 19.
Common mistakes
- gave the sum instead of a number: 161
- gave the smaller number instead of the larger one: 71
- gave the difference instead of a number: 19
Question 4
Two numbers are in the ratio 2 : 5, and their sum is 112.
What is the second number?
Answer: Option A — with explanation
The ratio 2 : 5 has 7 parts in total.
Each part is 112 divided by 7, which is 16.
The first number is 2 x 16 = 32 and the second is 5 x 16 = 80, so the second number is 80.
Common mistakes
- gave the ratio term instead of the value: 5
- gave the value of one part instead of the whole number: 16
- reported the other number in the pair: 32
Question 5
The sum of two numbers is 199 and their difference is 29.
What is the larger number?
Answer: Option D — with explanation
The larger number is half the sum plus half the difference.
Half of 199 is 99.5 and half of 29 is 14.5, so the larger number is 114.
Checking: the other number is 85, and 114 + 85 = 199 while 114 - 85 = 29.
Common mistakes
- gave the smaller number instead of the larger one: 85
- gave the difference instead of a number: 29
- gave the sum instead of a number: 199
Frequently asked questions
How do I start an arithmetical reasoning problem?
Decide what the question is asking for, give that quantity a name and write the sentence as an equation in that name. The wording is the only barrier; the arithmetic is usually simple.
How do ratios work in these problems?
A ratio of a to b divides the total into a plus b equal parts. One part is the total divided by that number, and each quantity is its ratio term multiplied by the size of a part.
How do I find two numbers from their sum and difference?
The larger number is half the sum plus half the difference, and the smaller is half the sum minus half the difference. Adding the two statements shows why: the difference cancels one way and doubles the other.
How do I avoid answering the wrong quantity?
Re-read the last line of the question just before choosing an option, and check that the number you have found is the one being asked for rather than its partner.
Is checking worth the time?
Yes. Substituting your answer back into the original sentence takes a few seconds and catches almost every translation slip, which is where nearly all the marks are lost.
