🍰 Fraction Isles · Fractions & Proportion
Percents
Treat a percent as a fraction out of 100 and solve percent-of, percent-change and discount problems.
In short
- A percent is a fraction out of 100, so 40% is 0.4 — divide by 100 before you multiply by anything.
- Every percent problem is "part = percent x whole" with a different one of the three missing.
- Percent change is measured against the original amount, which is why a 20% rise and a 20% fall do not cancel.
- Building answers from 10%, 5% and 1% gives both a fast mental method and a way to check written work.
Per cent means per hundred
The word percent comes from *per centum*: "for every hundred". A percent is a fraction whose denominator is always 100, which is why percents are so easy to compare — they are all measured against the same whole.
- 40% = 40/100 = 0.4
- 7% = 7/100 = 0.07
- 125% = 125/100 = 1.25
Percents above 100 are perfectly legal; they simply describe something larger than the original whole. A crop that grows to 150% of last year's yield is half as much again.
To turn a percent into a decimal, divide by 100 — the point moves two places left. To go back the other way, multiply by 100 and the point moves two places right.
Finding a percent of a quantity
"Of" means multiply. 40% of 15 is 0.4 x 15 = 6.
The single most common error in this whole topic is multiplying by 40 instead of by 0.4. Ask yourself first: should the answer be bigger or smaller than 15? Since 40% is less than a whole, the answer must be smaller. 600 fails that check instantly.
Mental shortcuts are worth learning, because they double as a sanity check:
- 10% — move the decimal point one place left
- 1% — move it two places left
- 50% — halve it
- 25% — halve it twice
- 5% — half of 10%
From those you can build almost anything: 35% is 25% + 10%, and 18% is 10% + 5% + 1% + 1% + 1%.
The three percent questions
Every percent problem is one of three, and all three are the same equation read in different directions:
part = percent x whole
- *Find the part.* "What is 30% of 80?" — multiply: 0.3 x 80 = 24.
- *Find the percent.* "24 is what percent of 80?" — divide part by whole, then multiply by 100: 24/80 = 0.3 = 30%.
- *Find the whole.* "24 is 30% of what number?" — divide part by percent: 24 / 0.3 = 80.
Naming which quantity is missing before you start is worth more than memorising three separate recipes. If you are looking for the whole, your answer should be larger than the part you were given (as long as the percent is under 100).
Change, discount and tax
Percent change always compares the change with the original amount, never with the new one.
percent change = (change / original) x 100
If a toll rises from 40 gold to 50 gold, the change is 10 and the original is 40, so the rise is 10/40 = 25%.
For discounts and taxes there is a one-step shortcut worth having:
- A 20% discount means you pay 80% of the price: multiply by 0.8.
- A 15% tax means you pay 115% of the price: multiply by 1.15.
That shortcut also explains a result that surprises people: a 20% rise followed by a 20% fall does not return you to the start. 100 x 1.2 x 0.8 = 96, because the fall is measured against the larger amount.
Worked examples
Example 1
A cloak costs 60 gold. During the festival it is marked down by 35%. What is the sale price?
- The discount is a percent of the original price, so first turn 35% into a decimal: 35/100 = 0.35.
- Discount = 0.35 x 60 = 21 gold.
- Sale price = 60 - 21 = 39 gold.
- One-step check: paying after a 35% discount means paying 65% of the price, and 0.65 x 60 = 39 gold.
- Sense check: 39 is a bit under two-thirds of 60, which matches a discount of just over a third.
Example 2
18 is 24% of what number?
- The missing quantity is the whole, so the answer should be larger than 18.
- Write it as an equation: 0.24 x w = 18.
- Undo the multiplication by dividing: w = 18 / 0.24.
- w = 75.
- Check: 24% of 75 = 0.24 x 75 = 18.
Practice problems, with solutions
Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.
Problem 1
Difficulty 1 of 5Write 25% as a decimal.
Answer: 0.25
- 25% means 25 out of 100, so 25% = 25/100.
- 25 / 100 = 0.25
- Check: moving the point in 25.0 two places left gives 0.25.
Problem 2
Difficulty 3 of 5Jonas's caravan carries 10 feathers. 60% of them are marked with the guild seal. How many carry the seal?
Answer: 6 feathers
- 60% = 60/100 = 0.6
- "60% of 10" means 0.6 x 10.
- 0.6 x 10 = 6
- Sense check: 10% of 10 is 1, and 60% is about 6 times that.
Problem 3
Difficulty 4 of 5Kofi fired 1140 lanterns at the practice range and 969 of them struck the target. What percent struck the target?
Answer: 85 %
- Part = 969, whole = 1140.
- As a fraction: 969/1140 = 0.85
- Multiply by 100 to turn the decimal into a percent: 0.85 x 100 = 85
- So 969 is 85% of 1140.
Common mistakes
- Forgetting to divide by 100 and answering 600 for "40% of 15".
- Giving the discount instead of the price actually paid.
- Comparing the change with the new amount rather than with the original when finding a percent change.
- Dividing the whole by the part when asked "what percent of a is b", getting the reciprocal by mistake.
- Assuming a 20% increase followed by a 20% decrease returns to the starting value.
What you should be able to do
- Convert among percents, decimals and fractions.
- Find a percent of a number, mentally where possible.
- Find the whole given a part and a percent.
- Compute percent increase, decrease, discount and tax.
Where this fits in the curriculum
Common Core
- 6.RP.A.3.C
Grade 6 — Find a percent of a quantity, and find the whole given a part and the percent.
- 7.RP.A.3
Grade 7 — Solve multi-step percent problems: increase, decrease, discount, tax, tip and interest.
Ontario
SAT
- Problem Solving and Data Analysis
Percentages, percent change, discount, tax and interest.