Percentage questions: methods and worked examples

Percentage of a number

Multiply by the percentage as a decimal: 35% of 240 is 0.35 x 240 = 84.

Mental shortcut: build it from 10% and 5%. 10% of 240 is 24, so 30% is 72, 5% is 12, and 35% is 84. This is fast enough for tests without a calculator.

Share of a total: "What percentage of 80 is 18?" Divide, then multiply by 100: 18 / 80 = 0.225, or 22.5%.

Percentage change

Percentage change = (new - old) / old x 100.

Sales rose from 400 to 460: (460 - 400) / 400 = 60 / 400 = 15%.

Costs fell from 250 to 200: (200 - 250) / 250 = -50 / 250 = -20%, a 20% decrease.

The trap: dividing by the new value. 60 / 460 is 13%, a classic wrong option. Always divide by where you started.

Reverse percentages

When a question gives you the figure after a change, divide by the multiplier.

A price after a 25% increase is $150. The original is $150 / 1.25 = $120.

A price after a 20% discount is $64. The original is $64 / 0.8 = $80.

The trap: taking 25% off $150 ($112.50) or adding 20% to $64 ($76.80). Neither gets you back to the start, because the percentage applied to a different base.

Percentage points

When a rate itself changes, there are two ways to describe it.

Interest rises from 4% to 5%. That is a rise of 1 percentage point, and a 25% increase in the rate (1 / 4).

Tests use this to set traps: read whether the question asks for "percentage points" or "percent".

Changes in a row

Successive percentage changes multiply; they do not add.

A price rises 10%, then falls 10%. Multiply: 1.1 x 0.9 = 0.99, a 1% fall overall, not zero.

Sales grow 20% two years running: 1.2 x 1.2 = 1.44, a 44% increase, not 40%.

Percentages in numerical reasoning tables

In numerical reasoning tests, percentage questions rarely come as bare sums. They come from a table or chart, and the hard part is picking the right two numbers. Take this small table of sales (in units):

• North: 240 in Q1, 270 in Q2
• South: 180 in Q1, 171 in Q2
• West: 150 in Q1, 159 in Q2

"By what percentage did North's sales grow from Q1 to Q2?" (270 - 240) / 240 = 30 / 240 = 12.5%.

"What was South's percentage change?" (171 - 180) / 180 = -9 / 180 = -5%, a 5% fall.

"What share of Q2 sales came from North?" Q2 total is 270 + 171 + 159 = 600. North's share is 270 / 600 = 45%.

"What was the overall percentage growth?" Q1 total is 570, Q2 total is 600, so 30 / 570 = about 5.3%.

That last one has a trap. Averaging the three regional changes (12.5%, -5% and 6%) gives 4.5%, which is wrong, because the regions are different sizes. Always go back to the totals.

Read the units and the time period. Tables often mix thousands and millions, or monthly and yearly figures. Many wrong options come from using the right method on the wrong row or column.

Fractions to know by heart

Fast cognitive tests usually ban calculators. Wonderlic says calculators are not allowed on its test (Wonderlic FAQ), and Criteria says the same for the CCAT (Criteria). Knowing these conversions turns many percentage questions into one step:

• 1/2 = 50%, 1/4 = 25%, 3/4 = 75%
• 1/5 = 20%, 2/5 = 40%, 1/10 = 10%
• 1/8 = 12.5%, 3/8 = 37.5%, 5/8 = 62.5%, 7/8 = 87.5%
• 1/3 = about 33.3%, 2/3 = about 66.7%, 1/6 = about 16.7%

So 37.5% of 64 is 3/8 of 64: 64 / 8 = 8, and 8 x 3 = 24. And "what percentage is 15 of 40?" becomes 15/40 = 3/8 = 37.5%.

The swap trick. x% of y equals y% of x. So 8% of 25 is the same as 25% of 8, which is 2. Use it whenever one of the two numbers is friendlier as a percentage.

Profit, markup, margin and tax

Business-style questions use the same methods, but the wording decides which number you divide by.

Markup is profit as a percentage of cost. An item costs $80 and sells for $100. Profit is $20, so the markup is 20 / 80 = 25%.

Margin is profit as a percentage of the selling price. Same item: 20 / 100 = 20%. One sale, two different percentages, and both usually appear as options.

Adding tax. $50 plus 8% tax: 50 x 1.08 = $54.

Removing tax. A price of $54 includes 8% tax. The pre-tax price is 54 / 1.08 = $50. Taking 8% off $54 gives $49.68, which is the reverse percentage trap again.

A percentage of a percentage. If 40% of staff work in sales and 25% of the sales staff are managers, sales managers are 0.4 x 0.25 = 10% of all staff.

Three multi-step examples

Harder percentage questions chain two of the methods above. Break them into steps.

A team grew 20% to 36 people. How many joined? This is a reverse percentage first: 36 / 1.2 = 30 people before. Then subtract: 36 - 30 = 6 joined. Taking 20% of 36 (7.2) is the trap.

A $200 price is cut 15%, then a further 10%. What is the final price? Multiply the multipliers: 200 x 0.85 = 170, then 170 x 0.9 = $153. The total discount is 23.5%, not 25%.

In year one, 360 of 480 people surveyed agreed. In year two, 378 of 540 agreed. What happened to the agreement rate? Year one: 360 / 480 = 75%. Year two: 378 / 540 = 70%. The rate fell 5 percentage points, even though more people agreed. Questions like this test whether you compare rates or raw counts.

The traps, in one list

Most wrong answers on percentage questions come from a short list of mistakes. Check for these before you choose:

• Dividing by the new value in a percentage change. Divide by the old value.
• Taking the percentage off the final figure in a reverse percentage. Divide by the multiplier instead.
• Adding successive changes. Multiply the multipliers.
• Confusing percent with percentage points when a rate changes.
• Averaging percentages across groups of different sizes. Use the totals.
• Mixing up markup and margin. Check whether the question compares profit with cost or with price.
• Using the wrong row or year in a table. Point at the two numbers before you calculate.

A speed method for timed tests

1. Name the type. Is it percentage of, change, reverse, points, or changes in a row? Each has one method, so naming it picks the method.
2. Estimate before you calculate. 19% of 412 is close to 20% of 400, which is 80. The exact answer is 78.28. If the options are far apart, the estimate alone may be enough.
3. Work in multipliers. A 15% rise is x1.15, a 15% fall is x0.85. Multipliers make reverse and successive questions one line each.
4. Check the answer against the question. If the question asks for a decrease and your answer is an increase, you divided the wrong way round.

On tests with a calculator, the risk is typing, not arithmetic. Estimate first so you notice when a result is out by a factor of 10.

Percentages often appear beside ratios and rates. If you want to practice those next, see our guides to ratio questions and speed, distance and time.