Math 7 · S.IC.B.6

Finding what percent one number is of another

Twenty-six of forty seeds came up in one bed, and thirty of fifty in another. How do you say how much of each bed came up so that the two can be set side by side?

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Converting among fractions, decimals and percentsMath 6

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Where you are 3 in The three percent questions

Fractions, ratio and percent3

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The three percent questions3

Finding what percent one number is of another

Finding the whole from a part and a percent Solving percent problems with an equation or a proportion

When you reach for it

A count on its own says how many. It does not say how many out of how many, and that is usually the question that matters. If 26 seeds sprouted in one bed and 30 in another, the second bed had more seedlings, but the first bed may still have done better for its size: it all depends on how many seeds each bed was given. With 40 seeds in the first bed and 50 in the second, the first did better, 65\% against 60\%.

A Percent answers “how many out of every hundred”. Turning a part of a whole into a percent puts every whole on the same footing, a hundred, so parts of different wholes can be compared directly. That is why results of surveys, seed trials and sports seasons are so often reported as percents.

You reach for this move whenever you know a part and the whole it comes from, and you want the percent. The question may say it outright (“what percent of the members chose Saturday?”) or more loosely (“what share of its games did the team win?”). There are three numbers in any such situation, the part, the whole and the percent, and the three percent questions each find a different one. This lesson finds the percent. When the percent is given and the whole is missing, see Finding the whole from a part and a percent; when you want one equation that handles all three, see Solving percent problems with an equation or a proportion.

The procedure

Percent means “per hundred”, so p\% is the fraction \displaystyle \frac{p}{100}. To say that a part is p\% of a whole is to say that the part is the same fraction of the whole as p is of 100:

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

So the whole job is to rename the fraction \displaystyle \frac{\text{part}}{\text{whole}} as some number of hundredths. Multiplying both sides by 100 gives the number of hundredths directly:

p = \frac{\text{part}}{\text{whole}} \times 100

The same thing is often written with the percent sign carried along:

\text{percent} = \frac{\text{part}}{\text{whole}} \times 100\%

Multiplying by 100\% does not change the value of anything, because \displaystyle 100\% = \frac{100}{100} = 1. It only changes the name: \displaystyle \frac{13}{20} and 65\% are the same number, written two ways. This is the move from Converting among fractions, decimals and percents, applied to a fraction that comes from a situation rather than one that is handed to you.

Naming the part and the whole

Everything rests on dividing by the right number. The whole is the total that the part is taken out of. In the question “what percent of B is A?” the whole is B, the number that follows “of”, and A is the part. In a survey, the whole is the number of people who answered the question asked. If the question is about only some of them (“what percent of the trail runners chose mornings?”), the whole is just that group.

Sometimes the whole is not printed at all and has to be found first, usually by adding: a team that won 15 games and lost 9, with no ties, played 15 + 9 = 24 games.

Three ways to work out the same fraction

All three give the same answer, because they are all ways of finding how many hundredths \displaystyle \frac{\text{part}}{\text{whole}} is. Which is quickest depends on the numbers.

Scale to hundredths. Multiplying or dividing the top and bottom of a fraction by the same nonzero number gives an equivalent fraction, with the same value. When the whole goes into 100 evenly, or 100 can be reached by dividing, scale straight to a denominator of 100: \displaystyle \frac{9}{25} = \frac{9 \times 4}{25 \times 4} = \frac{36}{100} = 36\%.

Simplify first, then scale. A whole like 40 does not go into 100, but the fraction may simplify to one whose denominator does: \displaystyle \frac{26}{40} = \frac{13}{20} = \frac{65}{100} = 65\%.

Divide. Dividing the part by the whole always works. The quotient is a decimal, and its first two places after the point are the hundredths: 26 \div 40 = 0.65, which is 65 hundredths, or 65\%. Moving the decimal point two places to the right is multiplying by 100, which is exactly the step \times 100 in the formula.

When the division does not end

Often the fraction has no exact two-place decimal. Then the percent can be given exactly, as a mixed number, or rounded, usually to the nearest tenth of a percent (see Rounding decimals). For \displaystyle \frac{2}{3}:

\frac{2}{3} \times 100\% = \frac{200}{3}\% = 66\tfrac{2}{3}\% \approx 66.7\%

The sign \approx is read “is about”. A rounded percent should be reported as “about” the value, since it is not exact.

Percents in data and surveys

When a survey lets each person choose exactly one option, every option’s percent is its count divided by the number who answered. The counts add up to the whole, so the percents add up to 100\%:

\frac{a}{n} + \frac{b}{n} + \frac{c}{n} = \frac{a + b + c}{n} = \frac{n}{n} = 1 = 100\%

Rounded percents may miss 100\% by a tenth or so, which is the rounding and not an error. Data shown in a bar graph work the same way: read each bar as a count, add the counts for the whole, then divide.

Worked examples

Example 1

Wholes that scale to a hundred

Find each percent.

(a) A chess club has 25 members, and 9 of them chose Saturday as club day.

(b) Of 20 bean seeds planted, 17 sprouted.

Solution

In each case the part is the smaller count and the whole is the total it is taken from.

(a) The fraction is \displaystyle \frac{9}{25}. Since 25 \times 4 = 100, multiply top and bottom by 4, which keeps the value:

\frac{9}{25} = \frac{36}{100} = 36\%

So 36\% of the members chose Saturday.

(b) The fraction is \displaystyle \frac{17}{20}, and 20 \times 5 = 100:

\frac{17}{20} = \frac{85}{100} = 85\%

So 85\% of the seeds sprouted.

(c) The fraction is \displaystyle \frac{138}{200}. Here the whole is more than 100, so divide top and bottom by 2 instead:

\frac{138}{200} = \frac{69}{100} = 69\%

So 69\% of Saturday’s loans were novels.

Example 2

Simplifying first

In a sunflower bed, 26 of the 40 seeds planted sprouted. What percent sprouted?

Solution

The part is 26 and the whole is 40. No whole number times 40 gives 100, so scaling directly does not work. But 26 and 40 share a factor of 2:

\frac{26}{40} = \frac{13}{20} = \frac{65}{100} = 65\%

Dividing gives the same thing: 26 \div 40 = 0.65, which is 65 hundredths.

The figure shows why the answer is reasonable. Lay the 40 seeds along a bar and stretch a percent scale along the same bar, so that the whole bed is 100\%. Each 4 seeds is a tenth of the bar, or 10\%, and 26 seeds is six and a half of those tenths: 65\%.

Figure 1The whole bed of 40 seeds is 100\%. The 26 that sprouted reach 65\% of the way along.

So 65\% of the seeds sprouted.

Example 3

A percent that does not come out exactly

A soccer team won 11 of the 18 matches in its season. What percent of its matches did it win?

Solution

The fraction \displaystyle \frac{11}{18} does not simplify, and 18 does not go into 100, so divide: 11 \div 18 = 0.6111\ldots, with the 1 repeating forever. The exact percent is

\begin{aligned} \frac{11}{18} \times 100\% &= \frac{1100}{18}\% \\ &= \frac{550}{9}\% = 61\tfrac{1}{9}\% \end{aligned}

To the nearest tenth of a percent this is about 61.1\%. The team won a little under two thirds of its matches, which fits: two thirds would be 12 of 18.

Example 4

Percents from a survey

The members of a cycling club were each asked to choose one trip for the summer. The bar graph shows the result. What percent of the members who answered chose each trip?

Figure 2The club’s survey in Example 4. Each bar is a count; the percents come from dividing each count by the 30 members who answered.

Solution

Read each bar as a count: lake 14, ridge 9, caves 7. The whole is everyone who answered, which the graph does not print, so add: 14 + 9 + 7 = 30.

\begin{aligned} \text{lake: } \frac{14}{30} \times 100\% &= \frac{140}{3}\% \\ &= 46\tfrac{2}{3}\% \approx 46.7\% \end{aligned}

\text{ridge: } \frac{9}{30} = \frac{3}{10} = \frac{30}{100} = 30\%

\begin{aligned} \text{caves: } \frac{7}{30} \times 100\% &= \frac{70}{3}\% \\ &= 23\tfrac{1}{3}\% \approx 23.3\% \end{aligned}

Each member chose exactly one trip, so the exact percents add to 100\%: 46\tfrac{2}{3} + 30 + 23\tfrac{1}{3} = 100. The rounded ones happen to add to 100.0 as well.

Example 5

Comparing parts of different wholes

Two teams in different leagues finished their seasons. The Harbor team won 36 of its 48 games. The Valley team won 26 of its 32 games. Which team won the larger share of its games?

Solution

Harbor won more games, 36 against 26, but it also played more, so the counts cannot be compared directly. Turn each into a percent, so that both are out of a hundred.

\text{Harbor: } \frac{36}{48} = \frac{3}{4} = \frac{75}{100} = 75\%

\text{Valley: } \frac{26}{32} = \frac{13}{16} = 0.8125 = 81.25\%

For Valley, 16 does not go into 100, so the division 13 \div 16 = 0.8125 does the work; it ends after four places, so the percent is exact. Valley won the larger share of its games, even though Harbor won more of them.

Example 6

Which number is the whole

A library lent 63 books on Monday and 84 on Tuesday.

(a) Monday’s count is what percent of Tuesday’s?

(b) Tuesday’s count is what percent of Monday’s?

Solution

The two questions use the same two numbers, but the word “of” names a different whole each time.

(a) The whole is Tuesday’s count, 84, and the part is 63. Both share a factor of 21:

\frac{63}{84} = \frac{3}{4} = 75\%

(b) Now the whole is Monday’s count, 63, and 84 is compared with it:

\begin{aligned} \frac{84}{63} \times 100\% &= \frac{4}{3} \times 100\% \\ &= \frac{400}{3}\% = 133\tfrac{1}{3}\% \end{aligned}

The answer is above 100\% because 84 is larger than the whole it is measured against; Percents above 100 and below 1 takes up percents like this. Notice too that the two answers are not mirror images: 75\% is 25 below 100\%, while \displaystyle 133\frac{1}{3}\% is \displaystyle 33\frac{1}{3} above it, because the two are measured against different wholes.

Example 7

The same count, three different wholes

A running club asked each of its 50 members two questions: do you run mostly on trails or on roads, and do you prefer to run in the morning or the evening? The two-way table shows the counts.

	Morning	Evening	Total
Trail	12	8	20
Road	9	21	30
Total	21	29	50

(a) What percent of all the members prefer mornings?

(b) What percent of the trail runners prefer mornings?

Solution

Each question names its own whole, and the whole is the group the question is about.

(a) The group is all 50 members, and 21 of them prefer mornings:

\frac{21}{50} = \frac{42}{100} = 42\%

(b) The group is the trail runners only, the 20 in the trail row, and 12 of them prefer mornings:

\frac{12}{20} = \frac{60}{100} = 60\%

\begin{aligned} \frac{12}{21} \times 100\% &= \frac{4}{7} \times 100\% = \frac{400}{7}\% \\ &= 57\tfrac{1}{7}\% \approx 57.1\% \end{aligned}

Parts (b) and (c) use the very same 12 members, yet give different percents, because 12 is being measured against a different group each time. Reading a percent always means asking “percent of what?”

Boundaries

The whole cannot be zero. The percent is \displaystyle \frac{\text{part}}{\text{whole}} \times 100\%, and division by zero has no meaning. A survey that nobody answered has no percents.
The part and the whole must count the same kind of thing, in the same unit. Books lent against books lent, seeds against seeds. If one count is in minutes and the other in hours, convert first; a percent of mismatched units is not a percent of anything.
A part of a group lies between 0\% and 100\%. 0\% means none of the group, 100\% means all of it. A percent above 100\% appears only when the number being compared is not a part of the whole, as with Tuesday’s books against Monday’s.
Percents of a group add to 100\% only when each member is counted exactly once. If people may choose more than one option, the parts overlap and their percents can add to more than 100\%. If some did not answer, the percents of those who did answer add to 100\%, but the percents of the whole club do not.
A rounded percent is not exact. Rounded percents of a group can add to 99.9\% or 100.1\%, and a rounded value should be called “about”.
A percent hides the size of the whole. 1 of 2 and 500 of 1{,}000 are both 50\%. To report a result fully, give the whole as well as the percent.

Where this goes next

The same three numbers, part, whole and percent, keep coming back. Knowing the part and the percent, you can work backwards to the whole in Finding the whole from a part and a percent, and Solving percent problems with an equation or a proportion writes all three questions as one equation, part equals percent times whole. When the number being compared is larger than the whole, the percent goes above 100\%, as in Percents above 100 and below 1.

Finding what percent a change is of the starting amount is the heart of percent increase and decrease and Percent change, and the same division, the extra amount over what something cost, gives a markup. In statistics, the percents inside a two-way table, taken along a row or down a column, are how you look for a link between two answers in a survey.

Exercises

Wholes that scale to a hundred

Find each percent by renaming the fraction as hundredths, without long division.

11 out of 20
7 out of 25
45 out of 50
12 out of 300

Dividing

Find each percent. Where it does not come out exactly, give the exact percent and the percent rounded to the nearest tenth.

27 out of 60
5 out of 8
14 out of 35
2 out of 3
3 out of 400

In context

Of 56 pepper seeds planted, 49 sprouted. What percent sprouted?
A netball team played 30 games in a season: it won 21, lost 7 and drew 2. What percent of its games did it win, and what percent did it draw?
Of the 240 books a library lent on one day, 90 were picture books. What percent were picture books?
A book club asked each of its members to choose one snack for its next meeting: 16 chose fruit, 10 chose crackers and 6 chose popcorn. What percent chose each snack? Check that your percents add to 100\%.
A small library owns 1{,}250 books, and 85 are lent out today. What percent of its books are lent out?

Which number is the whole

A drama club has 40 members, and 16 of them have parts in the spring play. What percent of the club does not have a part?
(a) What percent of 32 is 20? \qquad (b) What percent of 20 is 32?
A reading group asked its 40 members whether they read mostly fiction or non-fiction, and whether they prefer paper books or audiobooks.

	Paper	Audio	Total
Fiction	18	6	24
Non-fiction	7	9	16
Total	25	15	40

(a) What percent of all the members prefer audiobooks?

(b) What percent of the fiction readers prefer audiobooks?

Comparing

In a seed trial, 45 of the 60 seeds in tray A sprouted and 52 of the 65 seeds in tray B sprouted. Which tray had the larger percent sprout?
One season the Lakeside team won 13 of 20 games and the Hillcrest team won 17 of 25. Which team won the larger share of its games?
In March a library lent 480 books, 120 of them graphic novels. In April it lent 600 books, 135 of them graphic novels. In which month were graphic novels the larger share of the books lent?

Reasoning

Explain why dividing the part by the whole and multiplying by 100 gives the percent. Use the meaning of the word “percent”.
A club of 30 members split evenly among three options, 10 each. A report rounds each share to a whole percent and prints 33\%, 33\% and 33\%. Explain why these add to only 99\%, and give the exact percents.
Explain why a percent of a group, such as the percent of a club that chose an option, can never be more than 100\%, yet “84 is what percent of 63?” has an answer above 100\%.
Tamsin knows that 26 out of 40 is 65\%. Without dividing, she says that 52 out of 80 must also be 65\%. Explain why she is right.
In a survey where members could tick every activity they enjoy, the percents for the activities add to 140\%. Explain how this can happen without any mistake in the arithmetic.

Answers

55\%. \displaystyle \frac{11}{20} = \frac{55}{100}, multiplying top and bottom by 5.
28\%. \displaystyle \frac{7}{25} = \frac{28}{100}, multiplying top and bottom by 4.
90\%. \displaystyle \frac{45}{50} = \frac{90}{100}, multiplying top and bottom by 2.
4\%. \displaystyle \frac{12}{300} = \frac{4}{100}, dividing top and bottom by 3.
45\%. 27 \div 60 = 0.45.
62.5\%. 5 \div 8 = 0.625, which is 62.5 hundredths.
40\%. \displaystyle \frac{14}{35} = \frac{2}{5} = \frac{40}{100}.
\displaystyle 66\frac{2}{3}\% \approx 66.7\%. \displaystyle \frac{2}{3} \times 100\% = \frac{200}{3}\%.
0.75\%. 3 \div 400 = 0.0075, which is 0.75 hundredths: less than one percent.
87.5\%. \displaystyle \frac{49}{56} = \frac{7}{8}, and 7 \div 8 = 0.875.
Won 70\%, drew \displaystyle 6\frac{2}{3}\% \approx 6.7\%. \displaystyle \frac{21}{30} = \frac{7}{10} = 70\%, and \displaystyle \frac{2}{30} \times 100\% = \frac{20}{3}\%.
37.5\%. \displaystyle \frac{90}{240} = \frac{3}{8} = 0.375.
Fruit 50\%, crackers 31.25\%, popcorn 18.75\%. The whole is 16 + 10 + 6 = 32: \displaystyle \frac{16}{32} = 0.5, \displaystyle \frac{10}{32} = 0.3125, \displaystyle \frac{6}{32} = 0.1875, and 50 + 31.25 + 18.75 = 100.
6.8\%. 85 \div 1{,}250 = 0.068.
60\%. The members without a part number 40 - 16 = 24, and \displaystyle \frac{24}{40} = \frac{3}{5} = 60\%.
(a) 62.5\%: \displaystyle \frac{20}{32} = \frac{5}{8} = 0.625. (b) 160\%: \displaystyle \frac{32}{20} = 1.6; the answer is above 100\% because 32 is larger than the whole, 20.
(a) 37.5\%: \displaystyle \frac{15}{40} = 0.375. (b) 25\%: \displaystyle \frac{6}{24} = \frac{1}{4}. (c) 60\%: \displaystyle \frac{9}{15} = \frac{3}{5}. Each part uses a different whole: all 40, the 24 fiction readers, the 15 audiobook readers.
Tray B. Tray A: \displaystyle \frac{45}{60} = \frac{3}{4} = 75\%. Tray B: \displaystyle \frac{52}{65} = \frac{4}{5} = 80\%.
Hillcrest. Lakeside: \displaystyle \frac{13}{20} = 65\%. Hillcrest: \displaystyle \frac{17}{25} = \frac{68}{100} = 68\%.
March. March: \displaystyle \frac{120}{480} = \frac{1}{4} = 25\%. April: \displaystyle \frac{135}{600} = 0.225 = 22.5\%. April lent more graphic novels, but they were a smaller share of a larger whole.
“Percent” means “per hundred”, so a percent p\% is the fraction \displaystyle \frac{p}{100}. Saying a part is p\% of a whole means \displaystyle \frac{\text{part}}{\text{whole}} = \frac{p}{100}. Multiplying both sides by 100 gives \displaystyle p = \frac{\text{part}}{\text{whole}} \times 100: the number of hundredths the fraction is worth.
Each option’s exact share is \displaystyle \frac{10}{30} = \frac{1}{3}, which is \displaystyle 33\frac{1}{3}\%, and \displaystyle 3 \times 33\frac{1}{3} = 100. Rounding each share down to 33\% throws away \displaystyle \frac{1}{3}\% three times, which is the missing 1\%. The exact percents still add to 100\%.
In a group, a part is some of the members, so it is at most the whole: the fraction \displaystyle \frac{\text{part}}{\text{whole}} is at most \displaystyle \frac{\text{whole}}{\text{whole}} = 1 = 100\%. In “84 is what percent of 63?”, 84 is not a part of 63 but a separate number compared with it, and since 84 > 63 the fraction \displaystyle \frac{84}{63} is above 1, giving \displaystyle 133\frac{1}{3}\%.
\displaystyle \frac{52}{80} is \displaystyle \frac{26}{40} with top and bottom both multiplied by 2, and multiplying top and bottom by the same nonzero number gives an equivalent fraction with the same value. The same value is the same percent, 65\%.
Each member who ticks several activities is counted once in each of them, so the counts for the activities add to more than the number of members. Divided by the number of members, they add to more than 1, that is, more than 100\%. Percents add to exactly 100\% only when every member is counted exactly once.