Math 7 · 6.RP.A.3C

Solving percent problems with an equation or a proportion

A label says a serving gives 15% of a day’s iron, a candidate won 45% of the votes, a player made 34 of her 80 shots. Each sentence ties three numbers together. Is there one equation that finds whichever of the three is missing?

You need

Solving a proportionMath 7 What percent means: a rate per hundredMath 6

Where you are 3 in The three percent questions

Fractions, ratio and percent3

Percent8

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

Every percent statement ties together three numbers. “Halima received 45\% of the 460 votes cast, which was 207 votes” has a whole, 460 votes, the amount the percent is taken of; a percent, 45\%; and a part, 207 votes, the piece that the percent picks out of the whole. In a real question one of the three is missing, and which one it is changes from problem to problem:

a label gives a nutrient’s daily value and the percent in one serving: how much is in the serving? (the part is missing)
a candidate’s votes and her share of the vote are known: how many votes were cast? (the whole)
a player’s shots made and shots taken are known: what is her shooting percentage? (the percent)

Each can be worked on its own terms: dividing the part by the whole finds the percent, and working backwards from a part finds the whole. This lesson replaces the three separate questions with one relationship,

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

written either as an equation or as a proportion. You reach for it when you would rather translate the sentence than first decide which kind of question it is: put the two numbers you know in their places, put a letter where the missing one goes, and solve.

This lesson is taught over two days. Part 1 writes and solves the percent equation. Part 2 writes the same relationship as a proportion and solves it with the moves from Solving a proportion.

The procedure

Part 1: the percent equation

Why the part is the percent times the whole. A percent is a rate per hundred: p\% means p hundredths, the number \displaystyle \frac{p}{100}. Taking a fraction of an amount means multiplying by it, just as \displaystyle \frac{3}{4} of 20 is \displaystyle \frac{3}{4} \times 20 = 15 (see Multiplying fractions). So p\% of a whole is \displaystyle \frac{p}{100} times the whole, and that amount is the part:

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

For example, 45\% of 460 is \displaystyle \frac{45}{100} \times 460 = 0.45 \times 460 = 207.

Translating a sentence. A percent question is a sentence of the form “this is that percent of that”, with one of the three numbers replaced by a question. Read it phrase by phrase:

Words in the sentence	Become
“is”, “was”, “makes up”	=
“of”	\times
p\%	\displaystyle \frac{p}{100}, or the same number as a decimal
“what number”, “how many”, “what percent”	a letter for the unknown

So “27 is 36\% of what number?” becomes 27 = 0.36 \times w. The sentence may come in any order (“36\% of what number is 27?” says the same thing), so let the words, not their order, decide the places. The number right after “of” is always the whole.

Changing the percent before computing. The percent sign is part of the number’s name, so 36\% is the number 0.36, never 36. Writing 27 = 36w would say the part is thirty-six times the whole. To change a percent, divide by 100, which moves the decimal point two places to the left, or write it as a fraction over 100; Converting among fractions, decimals and percents covers both. A fraction is the better choice when the decimal never ends: \displaystyle 33\frac{1}{3}\% is exactly \displaystyle \frac{1}{3}, while 0.333 is only close to it.

Solving. Where the letter sits decides what has to be done to free it.

The part is missing. In n = 0.35 \times 140 the unknown is already alone, so multiply: n = 49. This is finding a percent of a quantity.
The whole is missing. In 27 = 0.36w the whole has been multiplied by 0.36. Dividing both sides of an equation by the same nonzero number keeps the two sides equal, and dividing by 0.36 undoes the multiplying: \displaystyle w = \frac{27}{0.36} = 75.
The percent is missing. Let x stand for the percent written as a decimal. In 34 = x \times 85, divide both sides by 85: \displaystyle x = \frac{34}{85} = 0.4. The answer is a decimal, so rename it as a percent at the end: \displaystyle 0.4 = \frac{40}{100} = 40\%.

Checking. Put the answer back into the sentence: 36\% of 75 is 0.36 \times 75 = 27, as it should be. A size check catches a misplaced number: a percent below 100\% gives a part smaller than the whole, and a percent above 100\% gives a part larger than the whole, as in Percents above 100 and below 1.

Part 2: the percent proportion

Divide both sides of \displaystyle \text{part} = \frac{p}{100} \times \text{whole} by the whole, which must not be zero:

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

This is a proportion: the part is to the whole as p is to 100. It says what a percent is, a part of a whole renamed as so many out of a hundred. A The double number line shows it, with the whole lined up against 100\% and the part against p\%, as in Example 5.

Write the proportion with the known numbers in their places and a letter for the missing one. The percent goes in as the plain number p, over 100, because the 100 is already written into the proportion. Then solve it in either of the ways from Solving a proportion.

A scale factor. When one denominator is a whole-number multiple of the other, the same factor links the numerators. In \displaystyle \frac{7}{w} = \frac{25}{100}, since 25 \times 4 = 100, the top and bottom on the left are also linked by 4, so w = 7 \times 4 = 28.
Cross products. If \displaystyle \frac{a}{b} = \frac{c}{d}, multiplying both sides by bd gives ad = bc: the cross products are equal. In \displaystyle \frac{21}{w} = \frac{60}{100} this gives 60w = 21 \times 100 = 2{,}100, and dividing by 60 gives w = 35.

The equation and the proportion are one relationship written two ways, so they always give the same answer. The equation is often quicker when the percent is an awkward decimal; the proportion keeps the percent as a whole number and suits numbers that scale easily to 100. Either is correct.

Worked examples

Examples 1 to 4 go with Part 1. Examples 5 to 7 go with Part 2.

Example 1

Three sentences, one equation

Translate each sentence into an equation and solve it.

(a) What number is 35\% of 140?

(b) 27 is 36\% of what number?

Solution

(a) “What number” is the unknown; call it n. “Is” becomes =, 35\% becomes 0.35, and “of” becomes \times:

n = 0.35 \times 140 = 49

As a size check, 35\% is a little more than a third, and a third of 140 is about 47.

(b) The whole is unknown; call it w. The sentence says 27 = 0.36w. Dividing both sides by 0.36 keeps them equal and leaves w alone:

w = \frac{27}{0.36} = \frac{2700}{36} = 75

Check: 0.36 \times 75 = 27.

34 = x \times 85

Dividing both sides by 85 gives \displaystyle x = \frac{34}{85} = \frac{2}{5} = 0.4, so the percent is 40\%. Check: 0.4 \times 85 = 34.

The three equations have the same shape, part equals percent times whole. Only the position of the letter changed, and with it the step that frees the letter.

Example 2

Reading a nutrition label

A nutrition label lists each nutrient as a percent of a daily value, a reference amount for a whole day that the label states.

(a) A muesli bar’s label lists its iron as 15\% of the daily value, and the label’s daily value for iron is 18 milligrams. How many milligrams of iron are in one bar?

(b) A large smoothie’s label lists its vitamin C as 240\% of the daily value, and the label’s daily value for vitamin C is 90 milligrams. How many milligrams of vitamin C are in the smoothie?

Solution

(a) The whole is the daily value, 18 milligrams, and the percent is 15\%. The part, the iron in one bar, is unknown; call it m:

m = 0.15 \times 18 = 2.7

One bar has 2.7 milligrams of iron.

(b) Here \displaystyle 240\% = \frac{240}{100} = 2.4. Again the part is unknown:

m = 2.4 \times 90 = 216

The smoothie has 216 milligrams of vitamin C. The part is larger than the whole, which the percent above 100\% said it would be. Nothing is wrong with that: the daily value is a reference amount to compare with, not a container that the serving has to fit inside.

Example 3

Finding the number of votes cast

In a student council election, Halima received 207 votes, which was 45\% of all the votes cast. How many votes were cast, and how many went to the other candidates?

Solution

The whole, all the votes cast, is unknown; call it w. The sentence “207 is 45\% of w” becomes

207 = 0.45w

The figure shows the same thing: a bar for all the votes, with 45\% of it shaded and that shaded piece worth 207 votes.

Figure 1Halima’s 207 votes are the shaded 45\% of a whole that is not yet known. The equation 207 = 0.45w says exactly what the picture shows.

Divide both sides by 0.45:

w = \frac{207}{0.45} = \frac{20700}{45} = 460

So 460 votes were cast, and the other candidates received 460 - 207 = 253 of them. As a check, the others’ share is 100\% - 45\% = 55\%, and 0.55 \times 460 = 253. The size is sensible too: 45\% is a little under half, so the whole should be a little more than twice 207, which is 414.

Example 4

Shooting percentages

A basketball player’s shooting percentage is the percent of the shots she takes that go in.

(a) Thandiwe took 80 shots over a season and made 34 of them. What was her shooting percentage?

(b) Juno’s shooting percentage over a tournament was \displaystyle 66\frac{2}{3}\%, from 24 shots taken. How many shots did she make?

Solution

(a) The whole is the 80 shots taken and the part is the 34 made. Let x be the percent as a decimal:

34 = x \times 80

Dividing both sides by 80 gives \displaystyle x = \frac{34}{80} = 0.425. That is 42.5 hundredths, so her shooting percentage was 42.5\%. A percent need not be a whole number.

(b) As a decimal, \displaystyle 66\frac{2}{3}\% is 0.6666\ldots, which never ends. As a fraction it is exact: \displaystyle 66\frac{2}{3} = \frac{200}{3}, and dividing by 100 gives \displaystyle \frac{200}{300} = \frac{2}{3}. With m for the shots made,

m = \frac{2}{3} \times 24 = 16

Juno made 16 shots. Rounding the percent to 0.67 first would give 0.67 \times 24 = 16.08, which is not a whole number of shots; the fraction avoids the rounding altogether.

Example 5

A percent of a quantity with a proportion

A recycling drive collected 250 kilograms of material, and 40\% of it was paper. Use a proportion to find how many kilograms of paper the drive collected.

Solution

The whole is 250 kilograms and the percent is 40. The part is unknown; call it n:

\frac{n}{250} = \frac{40}{100}

The right side simplifies to \displaystyle \frac{2}{5}, and 5 \times 50 = 250, so the scale factor from 5 to 250 is 50 and n = 2 \times 50 = 100. Cross products give the same answer: 100n = 40 \times 250 = 10{,}000, so n = 100.

The double number line shows why. The whole 250 kilograms lines up with 100\%. Ten equal steps split both lines, so each step is 25 kilograms on one and 10\% on the other, and 40\%, four steps, lines up with 4 \times 25 = 100 kilograms.

Figure 2The whole 250 kilograms lines up with 100\%. Each tick is 25 kilograms and 10\%, so 40\% lines up with 100 kilograms, which is the proportion \displaystyle \frac{100}{250} = \frac{40}{100}.

The drive collected 100 kilograms of paper.

Example 6

Working back to a daily value

One serving of a yoghurt has 260 milligrams of calcium, which its label lists as 20\% of the daily value. What daily value for calcium is the label using?

Solution

The part is 260 milligrams and the percent is 20. The whole, the daily value, is unknown; call it w:

\frac{260}{w} = \frac{20}{100}

The cross products are equal:

20w = 260 \times 100 = 26{,}000

Dividing both sides by 20 gives w = 1{,}300. A scale factor works too: 20 \times 5 = 100, so w = 260 \times 5 = 1{,}300. Check: 0.2 \times 1{,}300 = 260. The label is using a daily value of 1{,}300 milligrams.

Example 7

Four questions from one recycling drive

At a recycling drive, glass made up 30\% of the material collected, and there were 96 kilograms of glass. There were 80 kilograms of plastic. Paper made up 40\% of the total, and the rest was metal.

(a) How many kilograms were collected in all?

(b) What percent of the total was plastic?

(d) How many kilograms of metal were collected, and what percent of the total was that?

Solution

Each question is the same relationship with the letter in a different place, and the whole found in (a) becomes a known number for the rest.

(a) The whole w is unknown and glass is 30\% of it: 96 = 0.3w. Dividing both sides by 0.3 gives w = 320. The drive collected 320 kilograms.

(b) With the whole known, the proportion for plastic is

\frac{80}{320} = \frac{p}{100}

and the cross products are equal: 320p = 80 \times 100 = 8{,}000. Dividing both sides by 320 gives p = 25. Plastic was 25\% of the total.

(d) Metal is what is left of the whole: 320 - 96 - 80 - 128 = 16 kilograms. Its percent is what is left of 100\%: 100 - 30 - 25 - 40 = 5, so 5\%. The two agree, since 0.05 \times 320 = 16.

Boundaries

The whole cannot be zero. Finding the percent divides by the whole, and the proportion has the whole as a denominator. Nothing is a percent of zero.
A percent of zero says nothing about the whole. If the percent is 0\%, the equation is 0 = 0 \times w, which is true for every w, since 0\% of any whole is 0. And a part other than zero cannot be 0\% of anything: 5 = 0 \times w has no solution. So finding the whole needs a percent other than 0\%.
The percent must be in hundredths. In the equation, p\% goes in as \displaystyle \frac{p}{100}, a decimal or a fraction, never as p. In the proportion the 100 is already written in, so p goes in as the plain number. Using 0.4 over 100 gives an answer a hundred times too small.
The part and the whole are in the same unit. Milligrams against milligrams, votes against votes. If one is in grams and the other in milligrams, convert first.
The whole is the amount the percent is taken of, the number after “of”: the daily value, all the votes cast, all the shots taken. The same part is a different percent of a different whole.
Counts must come out whole. An answer of 47.5 votes, from 38\% of 125 votes, means the 38\% was rounded: no candidate receives half a vote. The true count is 47 or 48, since \displaystyle \frac{47}{125} = 37.6\% and \displaystyle \frac{48}{125} = 38.4\% both round to 38\%. A rounded percent in the equation gives only an estimate.

Where this goes next

The percent equation is the frame for every everyday use of percent. A tip, a tax and a commission are parts, and the bill or the sale is the whole, as in discounts, sales tax, tips and commission and the wider percent in money situations; simple interest multiplies the same percent times whole by a number of years. In percent increase and decrease and Percent change, the part is the amount of change and the whole is the starting value, and in Percent error the part is the size of an error and the whole is the true value. Each time the work is the same: name the part, the percent and the whole, and solve for the one that is missing.

Exercises

Part 1: translate and solve

Write an equation for each sentence and solve it.

What number is 26\% of 450?
42 is 60\% of what number?
What percent of 120 is 54?
7.5\% of what number is 12?
9 is what percent of 750?

Part 1: in context

A bowl of soup has 690 milligrams of sodium, and the label’s daily value for sodium is 2{,}300 milligrams. What percent of the daily value is in the bowl?
In a student council election, 220 votes were cast and the winner received 55\% of them. How many votes did the winner receive?
A netball shooter scored 45 goals in a season, which was 75\% of her attempts. How many attempts did she make?
A recycling drive collected 360 kilograms of material, and aluminum cans made up 15\% of it. How many kilograms of cans did it collect?
A water polo team scored on \displaystyle 12\frac{1}{2}\% of its shots in a season, and it scored 27 goals. How many shots did it take?

Part 2: proportions

Write a proportion for each and solve it.

What number is 64\% of 25?
18 is 24\% of what number?
39 is what percent of 60?
What number is 0.4\% of 2{,}500?

Part 2: in context

A serving of lentils has 8 grams of protein, which the label lists as 16\% of the daily value. What daily value for protein is the label using?
A hockey goalkeeper faced 175 shots in a season and saved 161 of them. What percent of the shots did she save?
A town has 8{,}400 registered voters, and 35\% of them voted in an election.

(a) How many people voted?

(b) The winning candidate received 1{,}617 votes. What percent of the people who voted is that?

(c) What percent of the registered voters is it? 18. A recycling drive collected 172 kilograms of paper, 104 kilograms of plastic, 84 kilograms of glass and 40 kilograms of metal. What percent of the total was each material?

Using both days

A slice of cheese has 3 grams of saturated fat, which its label lists as 15\% of the daily value.

(a) What daily value for saturated fat is the label using?

(b) Another food has 4.5 grams of saturated fat. What percent of the same daily value is that? 20. In a student council election, Desmond received 273 votes, which was 42\% of the votes cast. Orla received 182 votes.

(a) How many votes were cast?

(b) What percent of the votes did Orla receive?

Reasoning

Explain why “p\% of a number” means \displaystyle \frac{p}{100} times the number, and what goes wrong if p itself is put into the equation.
Show that the equation \displaystyle \text{part} = \frac{p}{100} \times \text{whole} and the proportion \displaystyle \frac{\text{part}}{\text{whole}} = \frac{p}{100} say the same thing. Which number must not be zero for the proportion to make sense?
Without solving, decide whether the answer to “35 is 140\% of what number?” is more or less than 35, and explain why.
Tobias says that 20\% of 45 is the same as 45\% of 20. Is he right? Explain using the percent equation, and use the same idea to find 4\% of 75 in your head.
A news report says that a candidate won 52\% of the vote. Explain why this alone does not tell you how many votes she received, and say what one more number would.

Answers

117. n = 0.26 \times 450 = 117.
70. 42 = 0.6w, so \displaystyle w = \frac{42}{0.6} = 70.
45\%. 54 = x \times 120, so \displaystyle x = \frac{54}{120} = 0.45.
160. 0.075w = 12, so \displaystyle w = \frac{12}{0.075} = 160.
1.2\%. 9 = x \times 750, so \displaystyle x = \frac{9}{750} = 0.012, which is 1.2 hundredths.
30\%. 690 = x \times 2{,}300, so \displaystyle x = \frac{690}{2{,}300} = 0.3.
121 votes. n = 0.55 \times 220 = 121.
60 attempts. 45 = 0.75a, so \displaystyle a = \frac{45}{0.75} = 60.
54 kilograms. n = 0.15 \times 360 = 54.
216 shots. \displaystyle 12\frac{1}{2}\% = \frac{12.5}{100} = \frac{1}{8}, so \displaystyle 27 = \frac{1}{8}s and s = 27 \times 8 = 216.
16. \displaystyle \frac{n}{25} = \frac{64}{100}, and 25 \times 4 = 100, so \displaystyle n = \frac{64}{4} = 16.
75. \displaystyle \frac{18}{w} = \frac{24}{100}; cross products give 24w = 1{,}800, so w = 75.
65\%. \displaystyle \frac{39}{60} = \frac{p}{100}; cross products give 60p = 3{,}900, so p = 65.
10. \displaystyle \frac{n}{2{,}500} = \frac{0.4}{100}, and 100 \times 25 = 2{,}500, so n = 0.4 \times 25 = 10.
50 grams. \displaystyle \frac{8}{w} = \frac{16}{100}; cross products give 16w = 800, so w = 50.
92\%. \displaystyle \frac{161}{175} = \frac{p}{100}; cross products give 175p = 16{,}100, so p = 92.
(a) 2{,}940: 0.35 \times 8{,}400 = 2{,}940. (b) 55\%: \displaystyle \frac{1{,}617}{2{,}940} = 0.55. (c) 19.25\%: \displaystyle \frac{1{,}617}{8{,}400} = 0.1925. The same 1{,}617 votes are 55\% of one whole and under 20\% of another.
Paper 43\%, plastic 26\%, glass 21\%, metal 10\%. The total is 172 + 104 + 84 + 40 = 400, and 400 = 4 \times 100, so each percent is the mass divided by 4: \displaystyle \frac{172}{4} = 43, \displaystyle \frac{104}{4} = 26, \displaystyle \frac{84}{4} = 21, \displaystyle \frac{40}{4} = 10. They add to 100.
(a) 20 grams: 3 = 0.15w, so \displaystyle w = \frac{3}{0.15} = 20. (b) 22.5\%: \displaystyle \frac{4.5}{20} = 0.225.
(a) 650: 273 = 0.42v, so \displaystyle v = \frac{273}{0.42} = 650. (b) 28\%: \displaystyle \frac{182}{650} = 0.28. (c) 195 votes, 30\%: 650 - 273 - 182 = 195, and \displaystyle \frac{195}{650} = 0.3; also 100 - 42 - 28 = 30.
“Percent” means “per hundred”, so p\% is p hundredths, the number \displaystyle \frac{p}{100}. Taking a fraction of an amount means multiplying the amount by it, so p\% of a number is \displaystyle \frac{p}{100} times the number. Putting p itself into the equation multiplies by a number a hundred times too big: 36 of a number is thirty-six times it, while 36\% of it is a little over a third of it.
Start from \displaystyle \text{part} = \frac{p}{100} \times \text{whole} and divide both sides by the whole. Dividing both sides by the same nonzero number keeps them equal, and it gives \displaystyle \frac{\text{part}}{\text{whole}} = \frac{p}{100}. Multiplying both sides of the proportion by the whole goes back again, so each statement follows from the other. The whole must not be zero, because it is a denominator and division by zero has no meaning.
Less than 35. The percent is above 100\%, so the part, 35, is larger than the whole, which means the whole is smaller than 35. (In fact \displaystyle \frac{35}{1.4} = 25.)
He is right. 20\% of 45 is \displaystyle \frac{20}{100} \times 45 = \frac{20 \times 45}{100}, and 45\% of 20 is \displaystyle \frac{45}{100} \times 20 = \frac{45 \times 20}{100}. These are equal because 20 \times 45 = 45 \times 20, the commutative property of multiplication; both are \displaystyle \frac{900}{100} = 9. In the same way, 4\% of 75 is 75\% of 4, which is three quarters of 4, so 3.
The equation \text{part} = 0.52 \times \text{whole} has two unknowns, the votes she received and the votes cast, so one equation cannot fix either. Any whole works: 52\% of 500 votes is 260, and 52\% of 5{,}000 votes is 2{,}600. Knowing the number of votes cast (or the number she received) leaves one unknown, and then the equation can be solved.