Benchmark percents and mental percent
Half of a crowd, or a tenth of it, takes no working at all. How far can a handful of easy percents like these take you without pencil and paper?
Where you are 8 in Percent
Why this exists
Ilya’s magic show plays to an audience of 360. How many people are half the audience? Almost anyone says 180 at once. A tenth of the audience? 36, just as fast. Neither needs a pencil, because halving and taking a tenth are things we already do in our heads.
Most percents look less friendly. 37\% of 600, or 15\% of 460, seem to call for a written method. But 37\% is three lots of 10\% and seven lots of 1\%, and 10\% and 1\% of a number are as easy to find as half of it. A few easy percents, the benchmark percents, are enough to build most others in the head, and to say roughly how big any percent of anything is.
Definition
The benchmark percents are 1\%, 10\%, 25\%, 50\%, 75\% and 100\%. Four of them are the Benchmark fractions a quarter, a half, three quarters and one whole, the familiar fractions we use as reference points for judging others. The other two are a tenth and a hundredth. Each has a quick way to find it of a quantity.
| Percent | Fraction | Of a quantity, found by |
|---|---|---|
| 100\% | 1 | taking the whole |
| 50\% | \displaystyle \frac{1}{2} | halving |
| 25\% | \displaystyle \frac{1}{4} | halving, then halving again |
| 75\% | \displaystyle \frac{3}{4} | adding 50\% and 25\% |
| 10\% | \displaystyle \frac{1}{10} | dividing by 10: each digit moves one place right |
| 1\% | \displaystyle \frac{1}{100} | dividing by 100: each digit moves two places right |
The digit moves are Multiplying and dividing by ten, one hundred and one thousand: dividing by 10 makes every digit worth a tenth as much, so each slides one place to the right. So 10\% of 4200 is 420, 1\% of it is 42, and 10\% of 63 is 6.3.
Two ways of working grow from these.
Building a percent up means writing it as a sum of benchmark pieces, finding each piece and adding them. 10\% and 1\% build any whole-number percent, the way tens and ones build any whole number: 37\% is three 10\% pieces and seven 1\% pieces. Sometimes taking a piece away is quicker: 99\% is 100\% with 1\% taken away.
Estimating with percents means swapping an awkward percent or quantity for a nearby benchmark or round number, to get an answer that is close rather than exact. The sign \approx means “is about”:
Intuition
Figure 1 is a percent bar for Ilya’s audience of 360, with the benchmarks marked. Folding the bar in half gives the 50\% mark, folding again gives 25\% and 75\%, and cutting off a tenth gives 10\%.
1\% is too thin to see at that size, which is what Figure 2 is for. Its top bar is cut into ten 10\% pieces. Take one piece, stretch it out and cut it into ten again: each new piece is 1\%, a tenth of a tenth. Percents have place value just as whole numbers do. 37\% is three tens and seven ones of percent, so 37\% of a quantity is three tenths of it and seven hundredths of it, together.
Estimating is reading the bar by eye. A group of 85 people sits just left of the 25\% mark in Figure 1, where 90 is, so they are a little under a quarter of the audience, with no working at all.
Results
Ten percent and one percent. 10\% of a quantity is the quantity divided by 10, and 1\% of it is the quantity divided by 100. So 1\% of a quantity is a tenth of 10\% of it.
Pieces of the same whole add and take away. 30\% of a quantity and 7\% of the same quantity together make 37\% of it. 100\% of a quantity with 1\% of it taken away leaves 99\% of it.
Halving and doubling. Half the percent gives half the amount, and double the percent gives double the amount: 5\% is half of 10\%, and 20\% is double 10\%.
Which way an estimate leans. A larger percent of the same quantity is larger, and the same percent of a larger quantity is larger. So an estimate made by rounding both the percent and the quantity up is too big, and one made by rounding both down is too small.
Why it is true
\displaystyle 10\% = \frac{10}{100} = \frac{1}{10}, so 10\% of a quantity is a tenth of it: the quantity divided by 10. In the same way \displaystyle 1\% = \frac{1}{100}, and since 100 = 10 \times 10, dividing by 100 is dividing by 10 twice. That is why 1\% is a tenth of 10\%.
A percent of a quantity is that many hundredths of it. One hundredth of 600 is 6, so 37\% of 600 is 37 sixes, and by the The distributive property
which is 30\% of 600 and 7\% of 600 added. The same split works for any percent and any quantity. On the bar it is two pieces laid end to end: 0\% to 30\% and 30\% to 37\% make 0\% to 37\%. Taking a piece away is the same thing run backwards.
Halving works for the same reason: 5 hundredths of a quantity is half of 10 hundredths of it. And for estimates, more hundredths means more, and a hundredth of a larger quantity is larger, so rounding the percent up and the quantity up both push the answer up.
Worked examples
The benchmarks of an audience
Ilya’s magic show has an audience of 360. How many people are 50\%, 25\%, 75\% and 10\% of the audience?
Solution50\% is half: 360 \div 2 = 180 people. 25\% is half of a half: 180 \div 2 = 90 people. 75\% is 50\% and 25\% together: 180 + 90 = 270 people. 10\% is the audience divided by 10, which moves each digit of 360 one place right: 36 people. These are the marks of Figure 1.
Check. 75\% and 25\% make the whole audience: 270 + 90 = 360.
Thirty-seven percent of the summer’s hires
Gael hires out boats on a lake. Over the summer his boats were hired 600 times, and 37\% of the hires were rowing boats. How many rowing-boat hires were there?
SolutionSplit 37\% into three 10\% pieces and seven 1\% pieces, as in Figure 2. 10\% of 600 is 60 (each digit one place right), and 1\% of 600 is 6 (two places right). Then
and the two pieces together are 180 + 42 = 222. There were 222 rowing-boat hires.
Check. 37 hundredths of 600, with each hundredth worth 6, is 37 \times 6 = 222.
Halving and doubling ten percent
Jadzia runs the ticket booth at a carnival. On Friday she sold 460 ride tickets: 15\% were for the ghost train and 20\% for the dodgems. How many tickets were sold for each ride?
SolutionEverything starts from 10\% of 460, which is 46. Half of that is 5\%, which is 23, so
and 69 tickets were for the ghost train. Double 10\% is 20\%, so 2 \times 46 = 92 tickets were for the dodgems.
Check. 20\% is \displaystyle \frac{20}{100} = \frac{1}{5}, and a fifth of 460 is 460 \div 5 = 92.
Working down from a benchmark
Ilya’s Saturday show sold 99\% of the 700 seats in a theater. His Sunday show, in a hall of 250 seats, filled 48\% of them. How many seats were filled at each show?
Solution99\% is just under 100\%, so take 1\% away from the whole. 1\% of 700 is 7, and 700 - 7 = 693 seats were sold on Saturday.
48\% is just under 50\%, so start from half and take 2\% away. Half of 250 is 125. 1\% of 250 is 2.5 (each digit two places right), so 2\% is 5, and 125 - 5 = 120 seats were filled on Sunday. Building 48\% up instead would need four tens and eight ones, 4 \times 25 + 8 \times 2.5, which reaches the same 120 by a longer road. Starting from the nearer benchmark is shorter.
Check. 48 hundredths of 250, with each hundredth worth 2.5, is 48 \times 2.5 = 120.
An estimate, and which way it leans
Jadzia’s carnival had 3150 visitors on Saturday, and 74\% of them rode the big wheel. Estimate how many rode it, and say whether the exact number is more or less than the estimate.
Solution74\% is close to the benchmark 75\%, and 3150 is close to 3200, which halves twice without fuss. Half of 3200 is 1600, a quarter is 800, and three quarters is 1600 + 800 = 2400:
About 2400 visitors rode the big wheel. Both roundings went up, 74\% to 75\% and 3150 to 3200, and each makes the answer larger, so 2400 is too big: the exact number is less.
Check. Work it exactly as 75\% with 1\% taken away. Half of 3150 is 1575, a quarter is 787.5, so 75\% is 1575 + 787.5 = 2362.5. 1\% of 3150 is 31.5, so 74\% is 2362.5 - 31.5 = 2331. That is 2331 riders, less than 2400 as predicted.
About what percent?
At a show in a bigger hall, 118 of Ilya’s audience of 480 were children. About what percent of the audience were children?
SolutionHold 118 up against the benchmarks of 480. Half of 480 is 240 and a quarter is 120, so 25\% of the audience is 120 people. 118 is just under 120: the children were a little under 25\% of the audience.
To pin it down further, 1\% of 480 is 4.8, so 24\% is 120 - 4.8 = 115.2. Since 118 lies between 115.2 and 120, the children were between 24\% and 25\% of the audience, and nearer 25\%, because 118 is 2 short of 120 but 2.8 past 115.2. So about 25\%.
Boundaries
Pieces need not be whole. Taking a tenth or a hundredth always works, but the result is a whole number only when the quantity allows it: 1\% of 250 seats is 2.5, as in Example 4. A piece with a decimal is fine on the way to an answer. A final count must still make sense: 1\% of 360 people is 3.6, which is not a whole number of people, so it can only be a step on the way.
Building up is quick only near a benchmark. 37\% of 600 splits neatly; 67\% of 83 would need six 10\% pieces of 8.3 and seven 1\% pieces of 0.83. When the percent and the quantity are both awkward, the written method of Finding a percent of a quantity is quicker, and a benchmark estimate is the way to check its answer.
An estimate is not an answer. It says roughly how big. Where the exact number matters, work it out. And an estimate leans a known way only when both roundings went the same way: rounding 26\% down to 25\% but 1190 up to 1200 pushes in opposite directions, and the estimate alone cannot say whether it is high or low.
Above one hundred and below one. Building up goes on past 100\%: 110\% of a quantity is the whole and 10\% more, as Percents above 100 and below 1 explains. Below 1\%, halving still works: 0.5\% is half of 1\%.
Where this goes next
Finding a percent of a quantity writes the same work as one multiplication: 37\% of 600 is 0.37 \times 600, and building up from 10\% and 1\% is that multiplication done in pieces. Converting among fractions, decimals and percents links many more fractions to percents than the benchmarks do, such as \displaystyle \frac{1}{5} = 20\% and \displaystyle \frac{1}{8} = 12.5\%, which gives more pieces to build with. Discounts, sales tax, tips and commission leans on 10\%, 25\% and 50\% all the time, for a price with a quarter taken off or a quick estimate of what is left to pay. Finding the whole from a part and a percent runs the benchmarks backwards: if 10\% of a crowd is 18 people, the whole crowd is 10 \times 18 = 180. And Percent change measures growth against a starting amount, where a quick 10\% is often the first estimate.
Exercises
Benchmark percents of a quantity
- Find 50\%, 25\% and 75\% of 520.
- Find 10\% and 1\% of 3700.
- Koa’s carnival stall took 85 dollars in its first hour. Find 10\% and 1\% of 85 dollars.
- Gael has 64 boats on the lake. How many boats are 25\% of them, and how many are 75\%?
- Name the benchmark percent equal to each fraction: \displaystyle \frac{1}{2}, \displaystyle \frac{1}{10}, \displaystyle \frac{3}{4} and \displaystyle \frac{1}{100}.
Building up from ten percent and one percent
- Find 30\% of 260.
- Find 6\% of 350.
- Find 43\% of 900.
- Find 15\% of 180.
- Find 35\% of 140, using 25\% and 10\%.
- Find 90\% of 70 by taking a piece away from 100\%.
- Find 98\% of 450.
- On Sunday 1600 people came to the carnival, and 12\% of them bought candyfloss. How many people bought candyfloss?
- Hiring a rowing boat on the lake for a day costs 45 dollars, and a motor boat costs 120\% of that. What does the motor boat cost for a day?
Estimating
- Estimate 51\% of 798.
- Estimate 9\% of 590. Without working out the exact answer, say whether it is more or less than your estimate.
- Koa’s magic show took 410 dollars in ticket money, and 76\% of it paid for the hall. Estimate how many dollars paid for the hall, and say whether the exact amount is more or less than your estimate.
- Femke’s boat hire has 120 boats, and 31 of them are out on the lake. About what percent of the boats are out? Between which two whole-number percents does it lie?
- Of the 95 raffle tickets Jadzia sold at the carnival, 9 won a prize. About what percent of the tickets won? Between which two whole-number percents does it lie?
Reasoning
- Explain why moving every digit of a quantity one place to the right gives 10\% of it.
- Explain why half of 10\% of a quantity is always 5\% of it.
- Find 75\% of 84 in two ways: as 50\% and 25\% added, and as 100\% with 25\% taken away. Explain why the two ways must agree.
- Without working out 11\% of 300, decide whether it is more or less than 30, and explain. Then find it.
Answers
- 260, 130 and 390. Half of 520 is 260, half again is 130, and 260 + 130 = 390.
- 370 and 37. Each digit moves one place right for 10\% and two places right for 1\%.
- 8.50 dollars and 0.85 dollars, which is 85 cents. 85 \div 10 = 8.5 and 85 \div 100 = 0.85.
- 16 boats and 48 boats. Half of 64 is 32, half again is 16, and 32 + 16 = 48.
- 50\%, 10\%, 75\% and 1\%.
- 78. 10\% of 260 is 26, and 3 \times 26 = 78.
- 21. 1\% of 350 is 3.5, and 6 \times 3.5 = 21. (Or 5\% is half of 35, which is 17.5, and 17.5 + 3.5 = 21.)
- 387. 10\% of 900 is 90 and 1\% is 9: 4 \times 90 + 3 \times 9 = 360 + 27 = 387.
- 27. 10\% of 180 is 18, half of that is 9, and 18 + 9 = 27.
- 49. 25\% of 140 is 35 (half is 70, half again 35), 10\% is 14, and 35 + 14 = 49.
- 63. 10\% of 70 is 7, and 70 - 7 = 63.
- 441. 1\% of 450 is 4.5, so 2\% is 9, and 450 - 9 = 441.
- 192 people. 10\% of 1600 is 160, 1\% is 16, so 2\% is 32, and 160 + 32 = 192.
- 54 dollars. 120\% is 100\% and 20\%. 10\% of 45 is 4.5, so 20\% is 9, and 45 + 9 = 54.
- About 400. 51\% \approx 50\% and 798 \approx 800, and half of 800 is 400.
- About 60, and the exact answer is less. 9\% \approx 10\% and 590 \approx 600, and 10\% of 600 is 60. Both roundings went up, so the estimate is too big.
- About 300 dollars, and the exact amount is more. 76\% \approx 75\% and 410 \approx 400, and three quarters of 400 is 300. Both roundings went down, so the estimate is too small.
- About 26\%, between 25\% and 26\%. 25\% of 120 is 30, and 1\% is 1.2, so 26\% is 31.2; 31 lies between 30 and 31.2, and nearer 31.2, since it is 0.2 short of 31.2 but 1 past 30.
- A little under 10\%, between 9\% and 10\%. 10\% of 95 is 9.5, and 1\% is 0.95, so 9\% is 8.55; 9 lies between 8.55 and 9.5.
- \displaystyle 10\% = \frac{10}{100} = \frac{1}{10}, so 10\% of a quantity is the quantity divided by 10. Dividing by 10 makes every digit worth a tenth of what it was, and a tenth of a place’s value is the value of the place to its right, so each digit slides one place right.
- 10\% of a quantity is 10 hundredths of it, and 5\% is 5 hundredths of it. 5 hundredths is half of 10 hundredths, whatever the quantity is.
- 63 both ways. 50\% of 84 is 42 and 25\% is 21, so 42 + 21 = 63; and 84 - 21 = 63. They must agree because 50\% + 25\% and 100\% - 25\% are the same 75\%: the same three quarters of the bar, reached from its two ends.
- More than 30. 10\% of 300 is 30, and 11\% is one more hundredth of the same quantity, so it is larger. Exactly, 1\% of 300 is 3, so 11\% is 30 + 3 = 33.