Finding a percent of a quantity
A box of building bricks says that 35 percent of its bricks are red. How many red bricks is that, and is there one move that answers every question of this kind?
Where you are 8 in Percent
When you reach for it
Reach for this whenever a percent describes a share of a known amount and the question is how much that share comes to. A box holds 720 bricks and 35\% of them are red. A triathlete covers 240 km in a month and 6\% of it is swimming. A puppet theater sold 160 tickets on Saturday and 115\% as many on Sunday. Each time the amount after the word “of” is known and the percent is known; what is missing is the amount the percent picks out.
Benchmark percents and mental percent answered questions like these for friendly percents, by halving or by building up from 10\% and 1\%. That still works, and it stays the best way to estimate. This lesson turns the whole job into one multiplication, which works as well for 35\%, 7.5\% or 115\% as it does for 50\%. The lesson runs over two days: Part 1 multiplies by the percent written as a decimal, and Part 2 multiplies by it written as a fraction, when that is easier.
The procedure
Part 1: a percent is a multiplier
A percent counts hundredths, so \displaystyle 35\% = \frac{35}{100} = 0.35. (Converting among fractions, decimals and percents covers changing between the three forms.) The question “what is 35\% of 720?” asks for thirty-five hundredths of 720. One hundredth of 720 is 720 \div 100 = 7.2, and thirty-five of those hundredths make 35 \times 7.2 = 252. Multiplying 720 by 0.35 counts exactly the same thirty-five hundredths in one go, so
That is the whole move: write the percent as a decimal, then multiply the amount by it. The word “of” turns into “times”, just as it does for a fraction of a whole number, where \displaystyle \frac{3}{4} of 20 is \displaystyle \frac{3}{4} \times 20. The multiplication itself is ordinary decimal multiplication, as in Multiplying decimals and Multiplying multi-digit decimals with the standard algorithm. Writing the decimal is dividing by 100, which moves every digit two places to the right: 6\% = 0.06, 7.5\% = 0.075 and 115\% = 1.15.
Seen this way, a percent is an operator: something done to a number. Taking 35\% of any amount is multiplying that amount by 0.35. The size of the multiplier says what happens to the amount, because multiplying by a number less than 1 takes less than one whole copy of it and multiplying by a number more than 1 takes more than one copy (Multiplying as scaling):
- below 100\% the decimal is less than 1, and the result is smaller than the amount;
- exactly 100\% is 1, and the result equals the amount;
- above 100\% the decimal is more than 1, and the result is larger than the amount, as Percents above 100 and below 1 explains.
A benchmark estimate before multiplying says roughly what answer to expect. 35\% lies between 25\% and 50\%, so 35\% of 720 must lie between a quarter of 720, which is 180, and half of it, which is 360. Closer still, 10\% of 720 is 72, so 30\% is 216 and 40\% is 288, and 252 sits between them. An answer far outside the estimate shows that the multiplication needs another look.
Part 2: choosing a fraction instead
The same percent can also be written as a fraction, and for some percents the fraction is the easier multiplier. \displaystyle 25\% = \frac{25}{100} = \frac{1}{4}, so 25\% of an amount is a quarter of it, and taking it is a single division by 4.
| Percent | Fraction | Taking it means |
|---|---|---|
| 50\% | \displaystyle \frac{1}{2} | dividing by 2 |
| 25\% | \displaystyle \frac{1}{4} | dividing by 4 |
| 75\% | \displaystyle \frac{3}{4} | dividing by 4, then multiplying by 3 |
| 20\% | \displaystyle \frac{1}{5} | dividing by 5 |
| 12.5\% | \displaystyle \frac{1}{8} | dividing by 8 |
| \displaystyle 33\frac{1}{3}\% | \displaystyle \frac{1}{3} | dividing by 3 |
| \displaystyle 66\frac{2}{3}\% | \displaystyle \frac{2}{3} | dividing by 3, then multiplying by 2 |
The fraction and the decimal are the same number, so they always give the same answer; the choice is only about which arithmetic is shorter. The fraction wins when the percent is in this table and the amount divides evenly. The decimal wins when the percent has no small fraction, such as \displaystyle 37\% = \frac{37}{100} or \displaystyle 6\% = \frac{6}{100}. For \displaystyle 33\frac{1}{3}\% and \displaystyle 66\frac{2}{3}\% the decimal never ends (\displaystyle \frac{1}{3} = 0.333\ldots, as Part 2 of Converting among fractions, decimals and percents shows), so the fraction is the only exact way to take them.
Worked examples
Examples 1 to 4 go with Part 1, and Examples 5 to 7 with Part 2.
Red bricks in a box
Rafe’s box of building bricks holds 720 bricks, and 35\% of them are red. How many red bricks are in the box?
SolutionEstimate first. 10\% of 720 is 72, so 30\% is 216 and 40\% is 288. The answer must lie between those two.
Now exactly. 35\% = 0.35, and “of” means multiply:
As whole numbers, 35 \times 720 = 25{,}200. The two decimal places of 0.35 put the point two places in from the right, giving 252.00, which is 252. There are 252 red bricks, and 252 lies between 216 and 288, as the estimate said. Figure 1 shows the same answer on a bar.
Check. The bricks that are not red make up the other 65\%, and 0.65 \times 720 = 468. Together 252 + 468 = 720, the whole box.
A month of triathlon training
Lumi is training for a triathlon. In May she covered 240 km in all: 6\% of it swimming, 72.5\% cycling and the rest running.
(a) How far did she swim?
(b) How far did she cycle?
(c) What percent of her distance was running, and how far did she run?
Solution(a) 6\% is six hundredths, 0.06. As whole numbers 6 \times 240 = 1440, and the two decimal places of 0.06 make that 14.40:
She swam 14.4 km. As an estimate, 10\% of 240 is 24 and 5\% is half of that, 12. Since 6\% is just above 5\%, the answer should be just above 12, and it is.
(b) 72.5\% = 0.725. Splitting 0.725 into 0.7 + 0.025 and multiplying each piece by 240 is the distributive property:
She cycled 174 km. The benchmark 75\% of 240 is 180, and 72.5\% is a little less than 75\%, so a little less than 180 is right.
(c) Swimming, cycling and running together are 100\% of her distance, so running is 100\% - 6\% - 72.5\% = 21.5\%, and
She ran 21.5\% of her distance, 51.6 km.
Check. Subtract the other two distances from the whole: 240 - 14.4 - 174 = 51.6. The two routes agree because percents of the same whole add and subtract just like the amounts they stand for.
Shows that sold more and fewer
Malik’s puppet theater sold 160 tickets for Saturday’s show. Friday’s show sold 85\% of Saturday’s number, and Sunday’s show sold 115\% of it. Before multiplying, say which show sold more than Saturday’s; then find both numbers.
SolutionThe words “of Saturday’s number” make 160 the amount being multiplied. 85\% = 0.85, which is less than 1, so Friday sold fewer than 160. 115\% = 1.15, which is more than 1, so Sunday sold more. Now multiply:
Friday’s show sold 136 tickets and Sunday’s sold 184. Figure 3 shows the two multipliers shrinking and stretching Saturday’s bar.
Check. 15\% of 160 is 0.15 \times 160 = 24. Friday is 15\% below Saturday, 160 - 24 = 136, and Sunday is 15\% above, 160 + 24 = 184.
Bracketing a long multiplication
Orion plans to cover 1{,}850 km in training next season, 77\% of it on his bike. Estimate the cycling distance from below and from above with benchmarks, then find it exactly.
SolutionFor an estimate from below, round both numbers down: 77\% to 75\% and 1{,}850 to 1{,}800. 75\% is \displaystyle \frac{3}{4}, a quarter of 1{,}800 is 450, and three quarters is 1{,}350. Both numbers went down, so the exact answer is more than 1{,}350.
For an estimate from above, round both up: 80\% of 2{,}000. 10\% of 2{,}000 is 200, so 80\% is 1{,}600, and the exact answer is less than that.
Now exactly. As whole numbers, 77 \times 1{,}850 is 70 \times 1{,}850 = 129{,}500 and 7 \times 1{,}850 = 12{,}950 added, which is 142{,}450. The two decimal places of 0.77 make it
Orion plans 1{,}424.5 km on his bike, which sits between 1{,}350 and 1{,}600 as the two estimates promised.
A quarter and an eighth of a brick set
Taro’s brick set has 1{,}096 pieces. Of these, 25\% are flat plates and 12.5\% are wheels. How many plates and how many wheels are there?
Solution\displaystyle 25\% = \frac{25}{100} = \frac{1}{4}, so 25\% of the set is a quarter of it, as Figure 2 shows:
There are 274 plates.
12.5\% is half of 25\%, and \displaystyle 12.5\% = \frac{125}{1000} = \frac{1}{8}. So the wheels are an eighth of the set, 1{,}096 \div 8 = 137, which is also half of 274. There are 137 wheels.
The decimals give the same answers, 0.25 \times 1{,}096 = 274 and 0.125 \times 1{,}096 = 137, but each of those multiplications takes longer than one division. Here the fraction was the easier form.
Balcony seats, where the decimal never ends
Malik’s puppet theater has 291 seats. \displaystyle 33\frac{1}{3}\% of them are in the balcony and \displaystyle 66\frac{2}{3}\% are downstairs. How many seats are in each part?
Solution\displaystyle 33\frac{1}{3}\% is \displaystyle \frac{1}{3}, so the balcony holds a third of the seats:
\displaystyle 66\frac{2}{3}\% is \displaystyle \frac{2}{3}, twice as much, so downstairs holds 2 \times 97 = 194 seats.
Check. 97 + 194 = 291, every seat, since \displaystyle 33\frac{1}{3}\% and \displaystyle 66\frac{2}{3}\% make 100\%.
The decimal for \displaystyle \frac{1}{3} is 0.333\ldots, with threes that never stop, so no decimal written out in full is exactly \displaystyle \frac{1}{3}. Cutting it short to 0.33 gives 0.33 \times 291 = 96.03, which is not even a whole number of seats. The fraction is the exact form here.
A percent of a percent
Lumi ran 55 km in one training week. 60\% of that distance was on trails, and 40\% of her trail distance was uphill. How far did she run uphill on trails, and what percent of the week’s running is that?
SolutionEach “of” is a multiplication, and the two percents are taken of different amounts: the 60\% of the week, the 40\% of the trails.
Trails: \displaystyle 60\% = \frac{3}{5}. A fifth of 55 is 11, so three fifths is 33 km.
Uphill: \displaystyle 40\% = \frac{2}{5} of the 33 km. A fifth of 33 is 6.6, so two fifths is 13.2 km. She ran 13.2 km uphill on trails.
For the second question, write the two steps as one. Regrouping the product is the associative property:
So the uphill trail running is 24\% of the week’s running. Check: 0.24 \times 55 = 13.2, the same distance. Taking 40\% after 60\% leaves 24\% of the start, because the two multipliers multiply.
Boundaries
- The amount after “of” is the one multiplied. In Example 7, 40\% of the trail distance and 40\% of the week’s distance are different amounts. Which amount a percent belongs to has to be read before anything is multiplied.
- Percents of one whole add; a percent of a percent multiplies. Lumi’s May shares, 6\% + 72.5\% + 21.5\% = 100\%, were all of the same 240 km. In Example 7 the 40\% was taken of the part the 60\% picked out, so they combined as 0.4 \times 0.6 = 0.24. Percents of two unrelated wholes do neither: they cannot be added or multiplied into one percent.
- The percent sign is two places of value. 6\% is 0.06 and 7.5\% is 0.075, while the plain decimal 0.6 is 60\%.
- A fraction is quicker only for some percents. 25\%, 50\%, 75\%, 20\%, 12.5\% and \displaystyle 33\frac{1}{3}\% have small fractions; 37\% is \displaystyle \frac{37}{100} and nothing simpler, so the decimal is just as easy. Either form gives the same answer.
- A count must come out whole. Distances and money may have decimal parts, but bricks, seats and tickets may not. A percent of a count that is not a whole number means the percent was rounded, as 0.33 was for \displaystyle \frac{1}{3} in Example 6.
- This finds the part. The move needs the whole and the percent both given; the answer it produces is the part.
Where this goes next
Finding a percent of a quantity is the step inside almost every later use of percent. A circle graph labels each slice with a percent, and Circle graphs multiplies the size of the whole group by it to say how many people the slice stands for. Relative frequency tables go the other way, writing counts as percents of a total, and this multiplication turns them back into counts. Prices and wages come next in Percent in money situations and Discounts, sales tax, tips and commission, where taking 20\% of a price is exactly this move, and Percent change measures a rise or a fall as a percent of where it started.
Example 7 points further ahead too: a percent of a percent is two multipliers multiplied, and that is how repeated changes are combined later on. Running the question backwards, to find the whole or the percent, is Finding the whole from a part and a percent and Finding what percent one number is of another.
Exercises
Part 1: percents as decimal multipliers
- Find 45\% of 280.
- Find 8\% of 375.
- Find 3.5\% of 840.
- Find 140\% of 65.
- Find 90\% of 7.2.
Part 1: in context
- Malik’s puppet theater sold 420 tickets for a week of shows, and 55\% of them were children’s tickets. How many children’s tickets were sold?
- Taro cycled 64 km on Saturday, and 15\% of the route was uphill. How many kilometers were uphill?
- A box of 1{,}500 bricks has 4\% clear bricks. How many clear bricks are in it?
- Rafe’s new brick box holds 160\% of what his old box held, and his old box held 425 bricks. Before multiplying, say whether the new box holds more or fewer. Then find how many it holds.
- Lumi’s Sunday run was 95\% of her 12 km Saturday run. How long was her Sunday run?
Part 1: estimating and checking
- Estimate 48\% of 612 with a benchmark, then find it exactly.
- One of 2.66, 26.6 and 266 is 7\% of 380. Use 10\% and 1\% of 380 to decide which, without multiplying, then check by multiplying.
- Orion swam 790 km over two years, and 23\% of it was in open water. Estimate the open-water distance as 25\% of 800, say whether the exact answer is more or less than the estimate, and then find it.
Part 2: percents as fractions
- Find 25\% of 356 using a fraction.
- Find 20\% of 435 using a fraction.
- Find 12.5\% of 296 using a fraction.
- Find \displaystyle 33\frac{1}{3}\% of 147.
- A puppet theater has 236 seats, and 75\% of them are on the ground floor. How many seats are on the ground floor?
- Taro has 360 bricks, and \displaystyle 66\frac{2}{3}\% of them are red. How many are red? What does 0.67 \times 360 give, and why is it not the answer?
Part 2: choosing a form, and percents of percents
- For each, choose the easier form, a decimal or a fraction, and find the answer: (a) 40\% of 85 (b) 37\% of 200 (c) 87.5\% of 64
- Rafe’s set has 800 bricks. 30\% of them are blue, and 25\% of the blue bricks are flat. How many flat blue bricks are there, and what percent of the whole set is that?
- Malik’s theater sold 275 tickets in its first week. The second week sold 120\% of the first week’s number, and the third week sold 90\% of the second week’s. How many tickets did the third week sell, and what percent of the first week’s number is that?
Reasoning
- Show that 16\% of 25 and 25\% of 16 are equal, and explain why. Use the same idea to find 4\% of 75 in your head.
- Explain why taking 50\% of a quantity and then 50\% of the result leaves 25\% of the quantity, not 0\% of it.
- Explain why finding 1\% of a quantity and multiplying that by 35 gives the same answer as multiplying the quantity by 0.35.
Answers
- 126. 0.45 \times 280 = 126, since 45 \times 280 = 12{,}600.
- 30. 0.08 \times 375 = 30, since 8 \times 375 = 3000.
- 29.4. 3.5\% = 0.035, and 35 \times 840 = 29{,}400; three decimal places give 29.4.
- 91. 1.4 \times 65 = 65 + 26 = 91.
- 6.48. 0.9 \times 7.2 = 6.48, a little less than 7.2 as a multiplier below 1 must give.
- 231 children’s tickets. 0.55 \times 420 = 231.
- 9.6 km. 0.15 \times 64 = 9.6.
- 60 clear bricks. 0.04 \times 1500 = 60.
- More, because 160\% = 1.6 is more than 1. 1.6 \times 425 = 680 bricks.
- 11.4 km. 0.95 \times 12 = 11.4, a little less than 12.
- About 300; exactly 293.76. 48\% \approx 50\% and 612 \approx 600, and half of 600 is 300. Exactly, 0.48 \times 612 = 293.76, since 48 \times 612 = 29{,}376.
- 26.6. 10\% of 380 is 38 and 1\% is 3.8, so 7\% lies between 3.8 and 38, and only 26.6 does. Check: 0.07 \times 380 = 26.6.
- About 200 km, and the exact answer is less, because both 23\% and 790 were rounded up. Exactly, 0.23 \times 790 = 181.7 km.
- 89. \displaystyle 25\% = \frac{1}{4}, and 356 \div 4 = 89.
- 87. \displaystyle 20\% = \frac{1}{5}, and 435 \div 5 = 87.
- 37. \displaystyle 12.5\% = \frac{1}{8}, and 296 \div 8 = 37.
- 49. \displaystyle 33\frac{1}{3}\% = \frac{1}{3}, and 147 \div 3 = 49.
- 177 seats. \displaystyle 75\% = \frac{3}{4}; 236 \div 4 = 59 and 3 \times 59 = 177.
- 240 red bricks. \displaystyle 66\frac{2}{3}\% = \frac{2}{3}; 360 \div 3 = 120 and 2 \times 120 = 240. 0.67 \times 360 = 241.2, which is not exact because 0.67 is \displaystyle \frac{2}{3} cut short, and it is not even a whole number of bricks.
- (a) 34: as a fraction \displaystyle \frac{2}{5}, 85 \div 5 = 17 and 2 \times 17 = 34 (or 0.4 \times 85 = 34). (b) 74: 37\% has no small fraction, so use the decimal, 0.37 \times 200 = 74. (c) 56: \displaystyle 87.5\% = \frac{7}{8}, 64 \div 8 = 8 and 7 \times 8 = 56.
- 60 flat blue bricks, 7.5\% of the set. 0.3 \times 800 = 240 blue bricks, and a quarter of 240 is 60. As one multiplier, 0.25 \times 0.3 = 0.075, which is 7.5\%, and 0.075 \times 800 = 60.
- 297 tickets, 108\% of the first week. Second week 1.2 \times 275 = 330; third week 0.9 \times 330 = 297. As one multiplier, 0.9 \times 1.2 = 1.08, which is 108\%, and 1.08 \times 275 = 297.
- Both are 4: 0.16 \times 25 = 4 and 0.25 \times 16 = 4. Each is 16 \times 25 divided by 100, and the order of a multiplication does not change the product (the commutative property). So 4\% of 75 is 75\% of 4, which is 3.
- Each step multiplies by 0.5, so the two steps together multiply by 0.5 \times 0.5 = 0.25, which is 25\%. The second 50\% is half of what is left, not half of the original, so it cannot take away everything: half of a half is a quarter.
- 1\% of a quantity is the quantity divided by 100, one hundredth of it. Thirty-five of those hundredths is 35 hundredths of the quantity, and 35 hundredths is 0.35. So both ways multiply the quantity by thirty-five hundredths and must agree.