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Daily Maths

Daily Maths

Percentage word problems: strategy and full-depth practice

Percentages- Formula Paper 1 + 2 10 min read

On this page
  1. In this note
  2. Formula bank
  3. A method for any percentage word problem
  4. Finding the 100% in the wording
  5. Ways they test it
  6. Mistakes to avoid
  7. Summary
  8. Practice
  9. Can I do this?
  10. Quiz

In this note

Syllabus 1.13: putting it all together
  • 1.13Read a long percentage problem and find the 100% for every step.
  • 1.13Decide what type each step is, and do it one step at a time.
  • 1.13Handle "more than / less than", percentages of percentages, and changing wholes.

This page uses all the earlier ones: fractions and decimals, increase and decrease, reverse percentages, profit and loss.

Formula bank

Percentage OFLearn itwhole \(\times\dfrac r{100}\)Use it when you know the whole and the %.
AS a percentageLearn it\(\dfrac{\text{part}}{\text{whole}}\times100\)Use it when you know the part and the whole.
Increase / decreaseLearn it\(\times\left(1\pm\dfrac r{100}\right)\)Use it when you know the start and the % change.
Percentage changeLearn it\(\dfrac{\text{change}}{\text{old}}\times100\)Use it when you know old and new.
ReverseLearn it\(\div\left(1\pm\dfrac r{100}\right)\)Use it when you know the amount AFTER the change.
Percentage of a percentageLearn it\(\dfrac{r_1}{100}\times\dfrac{r_2}{100}\)e.g. 15% of the 60% who are adults = 9% of everyone.Use it when the second % is of a smaller group.

A method for any percentage word problem

General methodHow to do any percentage word problem

  1. Read it twice: first the story, then underline each number and the words beside it (after, of, more than, including).
  2. For each percentage ask: a percentage of WHAT? That is the 100% for that step.
  3. Name the step's type with the table below, and use its formula.
  4. Write what each number means ("Ayesha's share = $600"). Keep full values; round only at the end.
  5. Answer what was asked (money to 2 d.p., the % sign, "increase" or "decrease").
You know…You want…Do this
a % and the wholethe partwhole × \(\tfrac{r}{100}\)
a part and the wholea %part ÷ whole × 100
an amount and a % changethe new amount× \((1\pm\tfrac{r}{100})\)
old and new amountsthe % changechange ÷ OLD × 100
the amount AFTER a changethe amount before÷ \((1\pm\tfrac{r}{100})\)
several changes in a rowfinal or overallmultiply the multipliers

Example In an exam, 60% of the students passed and 48 failed. How many passed?

The 48 who failed are the other 40%: find 100%, then the 60%.

  1. failed \(=100\%-60\%=40\%=48\)
  2. \(10\%=12\), so \(100\%=120\)
  3. passed \(=120-48=72\)

Answer 72 students

Try it yourself

1
After spending 30% of her money, Zara has $35 left. How much did she start with?
Hint
$35 is 70%.
Show answer

Answer $50

2
27 students are 45% of a year group. How many students are in the year group?
Show answer

Answer 60

Finding the 100% in the wording

"X is r% more (or less) than Y": the 100% is the thing after "than". So if A costs 25% more than B, B is 100% and A is 125%. Then B is cheaper than A by \(\tfrac{25}{125}=20\%\), not 25%.

General methodHow to handle a percentage of a percentage

  1. Notice the second percentage is of a SMALLER group ("15% of the adults").
  2. Multiply the two decimals: \(0.6\times0.15=0.09\), so 9% of everyone.
  3. Or take 100 people and work in numbers.

Example 60% of a club's members are adults. 15% of the adults are over 65. What percentage of all members are adults over 65?

Multiply the two percentages as decimals.

  1. \(0.6\times0.15=0.09\)

Answer 9%

Try it yourself

1
Ben is 25% taller than Ali. By what percentage is Ali shorter than Ben?
Hint
Ali = 100, Ben = 125.
Show answer

Answer 20%

2
50% of students walk; 30% of the walkers bring lunch. What % of ALL students are walkers who bring lunch?
Show answer

Answer 15%

Ways they test it

Type 1 Hidden reverse

"60% passed; 48 failed. How many passed?"

Spot it you are given the 'other' part.

  1. The given number is (100 − r)%.
  2. Find 100%, then the part asked for.

Type 2 Share, then percentages, then a fraction

"Share $960 in the ratio 5 : 3. Ayesha spends 35% of her share and gives ⅕ of what is left to charity..." (6 marks)

Spot it several steps, each with its own whole.

  1. Do each step on the amount it is OF.
  2. Write what each number is.

For example Ayesha's $600; spends 35%; gives ⅕ of the rest

  1. \(600\times0.65=390\) left
  2. keeps \(\tfrac45\) of 390 = 312

Answer $312

Type 3 More than 100%

"Visitors in 2025 were 240% of 2020. 2025: 4320. Find 2020 and the % increase."

Spot it a percentage over 100.

  1. Reverse: ÷ 2.4.
  2. Increase = 240% − 100% = 140%.

Type 4 Two groups, different rates

"40% of a class of 40 are girls. 75% of girls and ⅝ of boys pass. What % passed?" (3 marks)

Spot it different rates for different groups.

  1. Work in NUMBERS for each group.
  2. Add, then ÷ the total.

For example 16 girls, 24 boys

  1. \(12+15=27\) passed
  2. \(\tfrac{27}{40}\times100=67.5\)

Answer 67.5% (not the average 68.75%)

Type 5 First year over a target

"$20 000 rises 5% a year. After how many years is it first over $25 000?"

Spot it "first", "exceed".

  1. Try \(n\) years: \(20\,000\times1.05^{n}\).
  2. Show the year below AND the first year above.

For example \(20\,000\times1.05^{n}>25\,000\)

  1. \(n=4\): 24 310.13
  2. \(n=5\): 25 525.63

Answer after 5 years

Mistakes to avoid

  • ✗ Adding percentages of different groups✓ work in numbers of peopleEach % has its own 100%.
  • ✗ "A is 25% more than B, so B is 25% less than A"✓ B is 20% less than AThe 100% changes to A.
  • ✗ Averaging two rates (75% and 62.5%)✓ numbers per group, then ÷ totalThe groups are different sizes.
  • ✗ Rounding in the middle✓ round only the final answerEarly rounding can lose the accuracy mark.

Summary

  • For every percentage ask "of what?": that is its 100%.
  • "More / less than Y": Y is the 100%.
  • Percentage of a percentage: multiply the decimals.
  • Different groups: work in numbers, not percentages.
  • One step at a time, with words beside the numbers.

If you see …

  • "of the ... who"a new, smaller 100%.
  • "more than" / "less than"the thing after "than" is 100%.
  • "first year"try whole years, show both sides of the target.

Practice

Practice: easy, medium, hard

Easy

1
A price rises 20%, then another 20%. Total percentage increase?
Show working
  1. \(1.2^{2}=1.44\)

Answer 44%

2
In a test, 30 of 70 candidates got an A. 12 of these were girls. What percentage of the A grades went to girls?
Show working
  1. \(\tfrac{12}{30}\times100=40\)

Answer 40%

3
Same test: what percentage of ALL candidates were girls with an A? (3 s.f.)
Show working
  1. \(\tfrac{12}{70}\times100=17.1\)

Answer 17.1%

Medium

1
Visitors in 2025 were 240% of the 2020 number. In 2025 there were 4320. How many in 2020?
Guide me

What is it asking? the 2020 number

What do we know? 4320 = 240%

Plan ÷ 2.4

Show working
  1. \(4320\div2.4=1800\)

Answer 1800

2
A bike costs $5400 in Shop P, 35% more than in Shop Q. Find Shop Q's price.
Hint
Q is the 100%.
Guide me

What is it asking? Q's price

What do we know? P = 135% of Q

Plan ÷ 1.35

Show working
  1. \(5400\div1.35=4000\)

Answer $4000

3
Rabia's savings rose 5% a year for 2 years to $6615. How much did she start with?
Guide me

What is it asking? the starting amount

What do we know? two 5% rises

Plan ÷ \(1.05^{2}\)

Show working
  1. \(6615\div1.05^{2}=6000\)

Answer $6000

Hard

1
A theatre has 1500 seats: 60% in the stalls, the rest in the balcony. 25% of stalls seats and half the balcony seats are empty. What percentage of all seats are empty?
Hint
Work in numbers of seats.
Guide me

What is it asking? % of all seats empty

What do we know? two groups with different rates

Plan seats in each group, empty in each, add, ÷ 1500

Show working
  1. stalls 900, balcony 600
  2. empty \(=225+300=525\)
  3. \(\tfrac{525}{1500}\times100=35\)

Answer 35%

PartAnswer / workingMarksGuidance
900 and 600B1
\(0.25\times900+0.5\times600\)M1
35A1not 37.5
Total: 3 marks
2
Ayesha and Bilal share $960 in the ratio 5 : 3. Ayesha spends 35% of her share, then gives ⅕ of what is left to charity. (a) How much does she keep? (b) As a % of $960?
Guide me

What is it asking? what she keeps, and as a %

What do we know? $960 in 5 : 3

Plan her share, × 0.65, × 0.8, then ÷ 960

Show working
  1. share \(=600\)
  2. left \(=600\times0.65=390\)
  3. keeps \(=390\times0.8=312\)
  4. \(\tfrac{312}{960}\times100=32.5\)

Answer (a) $312 (b) 32.5%

PartAnswer / workingMarksGuidance
(a)600B1
390 and 78M1
312A1
(b)\(\tfrac{312}{960}\times100\)M1
32.5A1
Total: 5 marks
3
A salary of $20 000 rises 5% every year. After how many whole years is it first more than $25 000?
Guide me

What is it asking? the first whole year over 25 000

What do we know? \(20\,000\times1.05^{n}\)

Plan try n = 4 and n = 5

Show working
  1. \(n=4\): 24 310.13
  2. \(n=5\): 25 525.63

Answer 5 years

PartAnswer / workingMarksGuidance
\(20\,000\times1.05^{n}\) for some nM1
both 24 310 and 25 526 seenM1
5A1
Total: 3 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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