Percent Problems

A percent is a ratio out of $100$ . Percents are everywhere: sales and discounts, sales tax, test scores, survey results, and news stories about prices going up. In this lesson you’ll solve the three basic kinds of percent problems, work out percent increases and decreases, and see why a $20\%$ increase followed by a $20\%$ decrease does not bring you back to where you started.

Key ideas

Percent means “per hundred”

$35\%$ means $35$ out of every $100$ :

35\% = \frac{35}{100} = 0.35

To change a percent to a decimal, divide by $100$ (move the decimal point two places left). Watch the small ones: $5\% = 0.05$ and $0.5\% = 0.005$ . To change a fraction or decimal to a percent, multiply by $100\%$ : $\dfrac{3}{8} = 0.375 = 37.5\%$ .

Three kinds of percent problems

Every basic percent problem uses the same relationship:

\text{part} = \text{percent} \times \text{whole}

You know	You want	What to do	Example
percent and whole	the part	multiply	$15\%$ of $240 = 0.15 \times 240 = 36$
part and whole	the percent	divide part by whole	$18$ out of $72$ : $18 \div 72 = 0.25 = 25\%$
part and percent	the whole	divide part by percent	$30\%$ of what is $27$ ? $\ 27 \div 0.30 = 90$

You can also set each one up as a proportion, like $\dfrac{18}{72} = \dfrac{x}{100}$ , as in ratios, rates and proportions.

Percent increase and decrease

To find a percent change, compare the change to the original amount:

\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%

A positive answer is an increase; a negative answer is a decrease.

To apply a percent change, use a multiplier:

An increase of $r\%$ means multiplying by $1 + \dfrac{r}{100}$ . A $13\%$ increase multiplies by $1.13$ .
A decrease of $r\%$ means multiplying by $1 - \dfrac{r}{100}$ . A $25\%$ decrease multiplies by $0.75$ .

The multiplier does the whole calculation in one step: you keep $75\%$ of the price after a $25\%$ discount.

Sales tax and discounts

In Ontario, most purchases have $13\%$ HST (harmonized sales tax) added at the till. To find the total, multiply by $1.13$ . For a discount, multiply by $1$ minus the discount. Prices are rounded to the nearest cent at the end.

It doesn’t matter whether you take the discount first or add the tax first: multiplying by $0.75$ and then $1.13$ gives the same answer as multiplying by $1.13$ and then $0.75$ .

Successive percents don’t just add

Suppose a price of $100 goes up $20\%$ , then down $20\%$ :

100 \times 1.20 = 120 \qquad\qquad 120 \times 0.80 = 96

The final price is $96, not $100. The decrease is $20\%$ of a bigger number ( $120$ ), so it takes away more than the increase added. Using multipliers: $1.20 \times 0.80 = 0.96$ , which is a $4\%$ decrease overall.

Percents in data

Surveys and reports often give results as percents. To turn a percent back into a count, multiply by the total. If $38\%$ of $250$ students walk to school, then $0.38 \times 250 = 95$ students walk. The percents for all the categories of a survey (where each person picks one answer) add up to $100\%$ .

Worked examples

Example 1: The three kinds

(a) Find $15\%$ of $240$ .
(b) $18$ is what percent of $72$ ?
(c) $30\%$ of a number is $27$ . What is the number?

Solution.

(a) Part $=$ percent $\times$ whole: $0.15 \times 240 = 36$ .

(b) Percent $=$ part $\div$ whole: $18 \div 72 = 0.25 = 25\%$ .

27 \div 0.30 = 90

Check: $30\%$ of $90$ is $0.30 \times 90 = 27$ . ✓

Example 2: Discount and HST

A hoodie has a regular price of $60.00. It’s on sale for $30\%$ off, and $13\%$ HST is added to the sale price. What is the total?

Solution. After a $30\%$ discount you pay $70\%$ of the price:

60.00 \times 0.70 = 42.00

Then add $13\%$ HST by multiplying by $1.13$ :

42.00 \times 1.13 = 47.46

The total is $47.46. (The tax itself is $42.00 \times 0.13 = 5.46$ dollars.)

Example 3: Percent change

(a) The price of a monthly bus pass rose from $128 to $156. Find the percent increase, to one decimal place.
(b) The population of a town fell from $12\,500$ to $11\,800$ . Find the percent change.

Solution.

(a)

\frac{156 - 128}{128} \times 100\% = \frac{28}{128} \times 100\% = 21.875\% \approx 21.9\%

(b)

\frac{11\,800 - 12\,500}{12\,500} \times 100\% = \frac{-700}{12\,500} \times 100\% = -5.6\%

The population decreased by $5.6\%$ . Notice that we divide by the original value each time.

Example 4: Up 20%, then down 20%

A bike costs $400. The store raises the price by $20\%$ , then later puts it on sale for $20\%$ off.

(a) What is the sale price?
(b) What is the overall percent change?
(c) What percent discount would have brought the price back to exactly $400?

Solution.

(a) $400 \times 1.20 = 480$ , then $480 \times 0.80 = 384$ . The sale price is $384.

(b) $\dfrac{384 - 400}{400} \times 100\% = \dfrac{-16}{400} \times 100\% = -4\%$ , a $4\%$ decrease. (The multipliers agree: $1.20 \times 0.80 = 0.96$ .)

\frac{80}{480} \times 100\% = 16.\overline{6}\% \approx 16.7\%

A discount of about $16.7\%$ (not $20\%$ ) would undo a $20\%$ increase.

Common mistakes

Dividing by the new value to find percent change. Always divide by the original amount. From $128$ to $156$ is a change of $28$ , and $\dfrac{28}{128} \approx 21.9\%$ , not $\dfrac{28}{156} \approx 17.9\%$ .

Thinking $+20\%$ then $-20\%$ cancels out. The two percents are of different amounts. Multiply the multipliers instead: $1.2 \times 0.8 = 0.96$ , a $4\%$ loss.

Combining a discount and tax by subtracting the percents. $30\%$ off with $13\%$ tax is not $17\%$ off. The combined multiplier is $0.70 \times 1.13 = 0.791$ , so you pay $79.1\%$ of the regular price, which is $20.9\%$ off.

Multiplying when you should divide to find the whole. If $30\%$ of a number is $27$ , the number must be bigger than $27$ . Dividing, $27 \div 0.30 = 90$ , makes sense; multiplying, $0.30 \times 27 = 8.1$ , doesn’t.

Putting the decimal point in the wrong place. $5\% = 0.05$ , not $0.5$ . And $0.5\% = 0.005$ . Check your answer for reasonableness: $5\%$ of $80 should be small ($4).

Undoing tax by subtracting $13\%$ of the total. If a price with tax is $56.50, the price before tax is $56.50 \div 1.13 = 50$ dollars. Subtracting $13\%$ of $56.50$ gives $49.155$ , which is wrong, because the tax was $13\%$ of the smaller, pre-tax price.

Practice

1. (Warm-up)

(a) Write $45\%$ as a decimal and as a fraction in lowest terms.
(b) Write $0.8\%$ as a decimal.
(c) Write $\dfrac{3}{8}$ as a percent.

Solution

(a) $45\% = 0.45 = \dfrac{45}{100} = \dfrac{9}{20}$ .

(b) $0.8\% = 0.8 \div 100 = 0.008$ .

2. (Warm-up)

(a) Find $20\%$ of $85$ .
(b) You got $12$ out of $16$ on a quiz. What percent is that?
(c) Find $150\%$ of $60$ .

Solution

(a) $0.20 \times 85 = 17$ .

(b) $12 \div 16 = 0.75 = 75\%$ .

3. (Warm-up) $40\%$ of a number is $30$ . Find the number.

Solution

30 \div 0.40 = 75

Check: $0.40 \times 75 = 30$ . ✓

4. (Core) A pizza costs $24.00 before tax.

(a) Find the HST and the total with tax.
(b) You also leave a $15\%$ tip, calculated on the price before tax. What do you pay in all?

Solution

(a) HST: $0.13 \times 24.00 = 3.12$ , so the HST is $3.12. Total: $24.00 + 3.12 = 27.12$ , or $27.12. (Or in one step: $24.00 \times 1.13 = 27.12$ .)

(b) Tip: $0.15 \times 24.00 = 3.60$ . Total paid: $27.12 + 3.60 = 30.72$ , so you pay $30.72.

5. (Core) Find each percent change.

(a) A hockey ticket price went from $45 to $54.
(b) A phone’s price dropped from $850 to $680.

Solution

(a) $\dfrac{54 - 45}{45} \times 100\% = \dfrac{9}{45} \times 100\% = 20\%$ , a $20\%$ increase.

(b) $\dfrac{680 - 850}{850} \times 100\% = \dfrac{-170}{850} \times 100\% = -20\%$ , a $20\%$ decrease.

6. (Core) After a $25\%$ discount, a jacket costs $93.75 (before tax). What was the regular price?

Solution

After a $25\%$ discount you pay $75\%$ of the regular price. So $93.75$ is $75\%$ of the regular price:

93.75 \div 0.75 = 125

The regular price was $125. Check: $125 \times 0.75 = 93.75$ . ✓

7. (Core) In a survey, $320$ Grade 9 students each chose their favourite season: $15\%$ chose winter, $20\%$ chose spring, $45\%$ chose summer, and the rest chose fall. How many students chose each season?

Solution

Fall: $100\% - 15\% - 20\% - 45\% = 20\%$ .

Season	Percent	Students
winter	$15\%$	$0.15 \times 320 = 48$
spring	$20\%$	$0.20 \times 320 = 64$
summer	$45\%$	$0.45 \times 320 = 144$
fall	$20\%$	$0.20 \times 320 = 64$

Check: $48 + 64 + 144 + 64 = 320$ . ✓

8. (Challenge) The price of a $200 item goes up $10\%$ one year and another $10\%$ the next year.

(a) Find the price after two years.
(b) Is the total increase $20\%$ ? Explain.

Solution

(a) $200 \times 1.10 = 220$ , then $220 \times 1.10 = 242$ . The price is $242.

(b) No. The total increase is $\dfrac{242 - 200}{200} \times 100\% = 21\%$ . The second $10\%$ is taken of $220, not $200, so it adds $22 instead of $20. With multipliers: $1.10 \times 1.10 = 1.21$ .

9. (Challenge) A pair of headphones costs $80. They’re $25\%$ off, and $13\%$ HST applies. Find the total two ways: (i) discount first, then tax; (ii) tax first, then discount. Explain why the answers match.

Solution

(i) $80 \times 0.75 = 60$ , then $60 \times 1.13 = 67.80$ .

(ii) $80 \times 1.13 = 90.40$ , then $90.40 \times 0.75 = 67.80$ .

Both give $67.80. Each way, the price is just multiplied by $0.75$ and by $1.13$ , and you can multiply numbers in any order: $80 \times 0.75 \times 1.13 = 80 \times 1.13 \times 0.75$ .