Numerical Reasoning: Percentages and Ratios Made Simple

Numerical reasoning percentages and ratios explained: the four formulas you need, percentage points versus percent, reverse percentages, and two fully worked test questions.

Percentages and ratios carry most of the marks in a numerical reasoning test, and four formulas cover nearly all of them. Percentage of a total is the value divided by the total, times 100. Percentage change is the new value minus the original, divided by the original, times 100. A value after a change is the original multiplied by one plus or minus the change as a decimal. A reverse percentage is the new value divided by one plus or minus the change. For ratios, find the value of one part first, then scale. Everything else is a variation. The marks are lost not on the arithmetic but on four specific confusions: the wrong base, percentage points against percent, sequential changes, and ratio totals. This guide fixes each one, with two fully worked questions.

TL;DR

Percentage change always divides by the original value, never the new one.
A move from 12 percent to 15 percent is 3 percentage points and a 25 percent relative rise. Both are correct answers to different questions.
Never add sequential percentages. Multiply the decimal factors: a 10 percent rise then a 10 percent fall gives 0.99, a net 1 percent fall.
Reverse percentages divide by the factor. A price of 51 after 40 percent off means 51 divided by 0.6.
For ratios, divide the total by the sum of the parts to get one part, then multiply.
Round only at the end, and only to the precision the question requests.

Which formulas do you actually need?

Task	Formula	Example
Percentage of a total	value divided by total, times 100	45 out of 180 is 25 percent
Percentage change	(new minus original) divided by original, times 100	150 to 174 is a 16 percent rise
Value after a change	original times (1 plus or minus change)	800 up 12 percent is 800 times 1.12, or 896
Reverse percentage	new divided by (1 plus or minus change)	96 after a 20 percent rise came from 96 divided by 1.2, or 80
Share from a ratio	(part divided by sum of parts) times total	3 to 5 of 640 gives 240 and 400
Weighted average	sum of (value times weight) divided by total weight	see the second worked example below

If your test also includes rates, currency conversion or unit changes, the same discipline applies. Identify the base, then convert once.

You can pressure-test these on five free timed numerical questions before you read the worked solutions.

Why does the base cause so many errors?

Every percentage is a percentage of something. Test writers exploit the moment you forget what that something is.

Take revenue rising from 150 to 174. The change is 24. As a percentage of the original 150, that is 16 percent. As a percentage of the new 174, it is 13.8 percent. Both appear in the answer options. Only the first answers “by what percentage did revenue increase”.

The fix is a spoken habit. Before dividing, say “as a percentage of what?” and write that number down first. For an increase or decrease question, the answer is always the earlier value. For a share question, it is the total.

The second base trap is sequential change. When a value changes twice, the second change applies to the already-changed figure. That is why percentages never add. A 10 percent rise followed by a 10 percent fall gives 1.10 times 0.90, which is 0.99, a net fall of 1 percent, not a return to the start.

What is the difference between percentage points and percent?

This distinction appears constantly in tests built around market share, margins, interest rates and survey results.

A percentage point is the plain arithmetic difference between two percentages. If a margin moves from 12 percent to 15 percent, that is 3 percentage points.

A percent change is relative. The same move is 3 divided by 12, which is 0.25, so a 25 percent increase in the margin.

Both statements describe the same event. Read which one the question asks for. “By how many percentage points” means subtract. “By what percentage” means divide by the starting figure. A useful worked illustration: if a rate rises from 2 percent to 3 percent, that is 1 percentage point and a 50 percent relative rise.

Worked example 1: reverse percentage with a discount

Data. A supplier lists a maintenance contract at a headline price. A customer receives a 40 percent discount and pays 51,000 GBP. A second customer receives a 15 percent discount on the same headline price.

Question. How much does the second customer pay?

A) 58,650 GBP B) 72,250 GBP C) 85,000 GBP D) 43,350 GBP

Worked solution.

Step 1. Find the headline price. A 40 percent discount means the customer pays 60 percent, a factor of 0.6. So headline equals 51,000 divided by 0.6, which is 85,000 GBP. Check: 85,000 times 0.6 equals 51,000. Correct.

Step 2. Apply the second discount. A 15 percent discount is a factor of 0.85. So 85,000 times 0.85 equals 72,250 GBP. The answer is B.

Why the distractors work. Option C is the headline price, which is a true intermediate value and therefore tempting. Option A applies 15 percent off the discounted 51,000 rather than the headline. Option D applies 40 percent plus 15 percent as a combined 55 percent discount, the classic additive error. Multiplying the factors gives 0.6 times 0.85, which is 0.51, so the true combined discount would be 49 percent, not 55.

Worked example 2: three-part ratio and a weighted average

Data. A support desk of 48 staff is split across first line, second line and escalations in the ratio 5 to 4 to 3. Average tickets closed per person per day are 22 for first line, 14 for second line, and 6 for escalations.

Question. What is the average number of tickets closed per person per day across the whole desk, to one decimal place?

A) 14.0 B) 15.3 C) 15.5 D) 16.0

Worked solution.

Step 1. Find one part. The parts sum to 5 plus 4 plus 3, which is 12. One part equals 48 divided by 12, which is 4 people.

Step 2. Size each group. First line is 5 parts, so 20 people. Second line is 4 parts, so 16 people. Escalations is 3 parts, so 12 people. Check: 20 plus 16 plus 12 equals 48. Correct.

Step 3. Total tickets. First line closes 20 times 22, which is 440. Second line closes 16 times 14, which is 224. Escalations close 12 times 6, which is 72. The total is 440 plus 224 plus 72, which is 736.

Step 4. Divide by headcount. 736 divided by 48 equals 15.333, so 15.3 per person per day. The answer is B.

Why the distractors work. Option A is the plain average of 22, 14 and 6, which is 14.0. That ignores group size and is the single most common error in this question type. An average of averages is only valid when the groups are the same size, and here they are 20, 16 and 12 people. Options C and D come from mis-sizing the groups, usually by splitting 48 into three equal teams.

To keep the arithmetic honest, always write down the check line in step 2. If your parts do not sum back to the stated total, stop and re-read the ratio before doing anything else.

How do you speed this up under time pressure?

Convert percentages to factors immediately. Write 0.92 rather than “8 percent decrease”. One multiplication replaces two operations.
Learn the common fractions. 12.5 percent is one eighth, 33.3 percent is one third, 16.7 percent is one sixth. Recognising these turns a calculator step into a mental one.
Estimate before calculating. If you expect roughly 70,000 and the calculator says 7,225, you have found a decimal slip.
Write intermediate values down. Recalculating a number you already had is the most common invisible time loss.
Round last. Rounding at step one and again at step three compounds the error past the option spacing.

These techniques sit inside the broader timing method covered in our numerical reasoning test tips. If you are sitting a publisher-branded test, see the SHL numerical reasoning practice guide, and for a full multi-stage assessment day, how to pass online assessment tests covers the sitting itself.

Frequently Asked Questions

What is the formula for percentage change?

New value minus original value, divided by the original value, multiplied by 100. If the result is negative, it is a decrease.

How do I work out a reverse percentage?

Divide the new value by the factor. For a 25 percent increase, divide by 1.25. For a 25 percent decrease, divide by 0.75.

Can I add two percentage changes together?

No. Multiply the factors instead. A 20 percent rise followed by a 10 percent rise is 1.2 times 1.1, which is 1.32, so a 32 percent rise rather than 30 percent.

How do I split a total by a ratio?

Add the parts, divide the total by that sum to get the value of one part, then multiply each part by it. Always check that your results sum back to the original total.

When is an average of averages wrong?

Whenever the groups being averaged are different sizes. Use a weighted average: multiply each value by its group size, sum, then divide by the total size.

Do I need to memorise these formulas?

You need four of them fluently: percentage of a total, percentage change, value after a change, and reverse percentage. Ratios need a method rather than a formula. Everything else in a graduate test is a combination of those.