Percentage word problems

Converting to a decimal comes first#

You do not add or multiply percentages directly: first you turn them into a decimal by dividing by 100. $18\% = 0.18$, $7\% = 0.07$, $120\% = 1.2$, $0.5\% = 0.005$. Going the other way means multiplying by 100: $0.35 = 35\%$.

Dividing by 100 moves the decimal point two places to the left. If that move is not yet automatic, go through operations with decimals first: every "ten times off" mistake grows from here.

The second thing to settle before you calculate is what the whole is — the 100%. The whole is the thing you take the percentage OF: the price before the discount, the whole class, the whole journey, the whole salary.

Type 1: finding a percent of a number#

The problem: a workbook has 250 problems, and a student has solved 18% of them. How many problems are solved?

Step 1. The whole is stated directly: 250 problems, which is 100%.

Step 2. Convert the percent to a decimal: $18\% = 0.18$.

Step 3. Multiply: $250 \cdot 0.18 = 45$.

Answer: 45 problems.

The same calculation in your head, through one percent: $250 \div 100 = 2.5$ is 1%, so 18% is $2.5 \cdot 18 = 45$. The "one percent first" method is handy when decimals will not come in your head, and it gives the same number.

Type 2: finding the whole from a percent#

The problem: 12% of a number is 42. Find the number.

Step 1. Here the unknown is the whole itself — it is the answer.

Step 2. Write down the relationship: $0.12 \cdot x = 42$. This is an ordinary equation, and the rules from how to solve linear equations apply.

Step 3. Divide both sides by $0.12$: $x = 42 \div 0.12 = 350$.

Step 4. Check: $350 \cdot 0.12 = 42$. It matches.

Type 2 differs from type 1 by one operation — dividing instead of multiplying. They are easy to mix up, so estimate the answer in advance: 42 is only 12% of the whole, so the whole must be much larger than 42. The 5.04 you get by multiplying fails that estimate.

Type 3: what percent one number is of another#

The problem: 27 of 45 exercises are done. What percentage is that?

Step 1. The whole is 45: that is what you measure the share against.

Step 2. Divide the part by the whole: $27 \div 45 = 0.6$.

Step 3. Turn the decimal into a percentage by multiplying by 100: $0.6 = 60\%$.

Answer: 60%.

The main trap in type 3 is dividing the other way round, $45 \div 27$. You get 1.67, or 167%, and a part cannot be larger than the whole. The rule is simple: whatever counts as 100% always goes at the bottom.

Price increases and decreases#

The problem: a price of 800 dollars went up by 25%, and a month later the new price went down by 25%. What does the item cost now?

Step 1. The increase: $800 \cdot 0.25 = 200$, so the new price is $800 + 200 = 1000$ dollars.

Step 2. The decrease is now taken from 1000, not from 800: $1000 \cdot 0.25 = 250$.

Step 3. The result: $1000 - 250 = 750$ dollars — less than at the start.

This is not a paradox and not a mistake: the percentages were taken of different bases. The same calculation is shorter with multipliers: a 25% increase means multiplying by 1.25, a 25% decrease means multiplying by 0.75, so $800 \cdot 1.25 \cdot 0.75 = 750$.

A discount works the same way: an item priced at 1500 dollars with 30% off costs $1500 \cdot 0.7 = 1050$ dollars. Multiplying by 0.7 straight away is quicker than working out the discount and subtracting it.

Mistakes that come back every year#

Taking the percentage of the wrong number. After reading the problem, underline the word that follows "of": that is what the percentage is taken from.

Adding percentages of different amounts: "the price rose 10%, then another 10%" is not 20% but a multiplication by $1.1 \cdot 1.1 = 1.21$, that is 21%.

Confusing "20% more than" with "is 20% of". The first is an increase on the whole, the second is a fifth of the whole.

Forgetting to turn the answer back into a percentage in type 3 problems and writing 0.6 instead of 60%.

How to check the answer#

Estimate with round percentages. A half is 50%, a quarter is 25%, a tenth is 10%. You can estimate 18% of 250 like this: 10% is 25, 20% is 50, so the answer is a bit under 50. The 45 you got falls inside that range. If the answer falls outside, the mistake is not in the arithmetic but in the choice of operation.

Step-by-step plan

1. Step 1 — convertingConvert percentages to decimals and back in your head: 5%, 40%, 125%, 0.2%, 0.45, 1.3.
2. Step 2 — percent of a numberTen type 1 problems, each solved two ways — by multiplying and through one percent.
3. Step 3 — the whole from a percentType 2 problems through an equation, always checking by substitution.
4. Step 4 — share as a percentageType 3 problems: decide what goes at the bottom before you calculate.
5. Step 5 — prices and discountsIncreases and decreases with the multipliers 1.25 and 0.75; problems with two changes in a row.

Check yourself

Sources

• Khan Academy — Pre-algebraPercentage problems with step-by-step explanations and checking
  free
• OpenStax Prealgebra 2eFree textbook; chapter 6 covers percents, discounts and markups
  free

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