Math 7

Solving a proportion

A soup recipe serves 8 and you are cooking for 20. Three of the four amounts are known. How do you find the fourth?

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Finding the whole from a part and a percentMath 7 Setting up proportions from word problemsMath 7 Solving percent problems with an equation or a proportionMath 7

Where you are 3 in Proportion

Fractions, ratio and percent3

Ratio6

Proportion3

What a proportion says

Solving a proportion

Setting up proportions from word problems

When you reach for it

A soup recipe serves 8 people and uses 6 cups of stock. You are cooking the same soup for 20. For it to taste the same, the stock has to compare with the servings exactly as it did before: the ratio of stock to servings must not change. Calling the unknown amount of stock x, that is the proportion

\frac{6}{8} = \frac{x}{20}

with cups of stock on top and servings underneath on both sides.

What a proportion says tested proportions in which all four numbers were known. Here one number is missing, and the question turns round: which number makes the proportion true? Finding it is solving the proportion. In this case there is exactly one answer. The cross products of a true proportion are equal, so 6 \cdot 20 = 8 \cdot x, that is 8x = 120, and only x = 15 does that.

Reach for it whenever two quantities grow together at a fixed rate, and you know one matching pair and one more amount: a recipe scaled up for a crowd, how long a steady typist needs for a longer piece, a pulse counted for part of a minute and turned into beats per minute, a scale drawing turned into real lengths. If you have found a missing value in a pair of equal ratios with a ratio table, this is the same question, written as an equation so that it works for any numbers and not only friendly ones.

The procedure

Take the soup proportion \displaystyle \frac{6}{8} = \frac{x}{20}. There are four ways to finish it. The first two are quick when the numbers are friendly; the last two always work.

Across: the scale factor between the two ratios

Equivalent ratios come from multiplying both parts of one ratio by the same number. Going from 8 servings to 20 multiplies the servings by

\frac{20}{8} = 2.5

so the stock must be multiplied by 2.5 as well, and x = 6 \times 2.5 = 15 cups. This is the reasoning of Equivalent ratios and ratio tables, and it is the fastest move when the scale factor is a whole number or an easy decimal.

Within one ratio: the unit rate

The other multiplier compares the two parts of the same ratio. The recipe has \displaystyle \frac{6}{8} = \frac{3}{4} of a cup of stock for every serving, the unit rate. The pot for 20 has the same amount per serving, so

x = 20 \times \frac{3}{4} = 15

Both multipliers are drawn below. Either gives the answer; choose whichever is easier with the numbers in front of you.

Figure 1Two multipliers solve the same proportion. Across, 8 servings become 20 by multiplying by 2.5. Up, each serving needs \displaystyle \frac{3}{4} of a cup of stock. Either one gives x = 15.

Multiplying both sides when the unknown is on top

When neither multiplier is friendly, treat the proportion as the equation it is. Write it with the unknown on the left, \displaystyle \frac{x}{20} = \frac{6}{8}, and multiply both sides by 20. Multiplying both sides of a true equation by the same number keeps it true (the multiplication property of equality), and on the left \displaystyle 20 \cdot \frac{x}{20} = x:

x = 20 \cdot \frac{6}{8} = \frac{120}{8} = 15

The product \displaystyle 20 \cdot \frac{6}{8} can be grouped as \displaystyle \frac{20}{8} \cdot 6, the scale factor times 6, or as \displaystyle 20 \cdot \frac{3}{4}, the servings times the unit rate. The first two moves are this one with the arithmetic done in a convenient order.

Cross products when the unknown is underneath

Now ask the reverse question: how many servings will 27 cups of stock make? The unknown is a number of servings, so it goes underneath:

\frac{6}{8} = \frac{27}{y}

The cross products of a proportion are equal. For b and d not zero,

\frac{a}{b} = \frac{c}{d} \quad\text{exactly when}\quad ad = bc

What a proportion says proves this and uses it as a test. The reason in one line: multiplying both sides of \displaystyle \frac{a}{b} = \frac{c}{d} by bd, which is not zero, gives ad = cb, and dividing both sides of ad = bc by bd leads back again. So the two equations have exactly the same solutions, and the second has no fractions in it. Here it says

6 \cdot y = 8 \cdot 27, \quad\text{so}\quad 6y = 216

Dividing both sides by 6, which is not zero, gives y = 36: the stock makes 36 servings. Cross products work just as well with the unknown on top, so they are the one move that never needs a choice.

There is a second way when the unknown is underneath. If two nonzero numbers are equal, so are their reciprocals. Turning both ratios over gives \displaystyle \frac{8}{6} = \frac{y}{27}, with the unknown on top again, and \displaystyle y = 27 \cdot \frac{8}{6} = 36.

Checking

A solution can be checked twice. Put it back and test the cross products: 6 \cdot 36 = 216 and 8 \cdot 27 = 216. Then ask whether its size makes sense: 27 cups is between 4 and 5 times 6 cups, so the servings should be between 4 \times 8 = 32 and 5 \times 8 = 40, and 36 is.

Worked examples

Example 1

Unknown on top

Solve each proportion.

(a) \displaystyle \frac{x}{18} = \frac{5}{6} \qquad (b) \displaystyle \frac{21}{35} = \frac{n}{10} \qquad (c) \displaystyle \frac{7}{12} = \frac{m}{30}

Solution

(a) From 6 to 18 is a scale factor of 3, so x = 5 \times 3 = 15. Check: 15 \cdot 6 = 90 and 18 \cdot 5 = 90.

(b) No whole number takes 35 to 10, but the known ratio simplifies: dividing both parts of \displaystyle \frac{21}{35} by 7 gives \displaystyle \frac{3}{5}. A simpler name for the same ratio does not change the proportion, so \displaystyle \frac{3}{5} = \frac{n}{10}, and from 5 to 10 is a factor of 2. So n = 6.

m = 30 \cdot \frac{7}{12} = \frac{210}{12} = 17.5

Check with cross products: 7 \cdot 30 = 210 and 12 \cdot 17.5 = 210. The answer need not be a whole number even when the three given numbers are.

Example 2

Unknown underneath

Solve (a) \displaystyle \frac{45}{y} = \frac{9}{4} \qquad (b) \displaystyle \frac{8}{15} = \frac{20}{k}

Solution

(a) Set the cross products equal:

9y = 45 \cdot 4 = 180

Divide both sides by 9, so y = 20. Check: \displaystyle \frac{45}{20} = 2.25 and \displaystyle \frac{9}{4} = 2.25. Turning both ratios over gives the same answer by the other route: \displaystyle \frac{y}{45} = \frac{4}{9}, so \displaystyle y = 45 \cdot \frac{4}{9} = 20.

(b) Cross products give 8k = 15 \cdot 20 = 300, so \displaystyle k = \frac{300}{8} = 37.5. A scale factor confirms it: 20 is 2.5 times 8, so k must be 2.5 times 15, which is 37.5.

Example 3

Decimals, fractions and negative numbers

Solve (a) \displaystyle \frac{1.2}{p} = \frac{0.4}{0.9} \qquad (b) \displaystyle \frac{w}{-6} = \frac{14}{21} \qquad (c) \displaystyle \frac{\frac{2}{3}}{5} = \frac{q}{12}

Solution

(a) Cross products give 0.4p = 1.2 \times 0.9 = 1.08. Dividing both sides by 0.4 gives p = 2.7. A scale factor shows it too: 1.2 is 3 times 0.4, so p is 3 times 0.9.

(b) As an equation between numbers, a proportion may contain negative numbers, and the moves are unchanged. Simplify \displaystyle \frac{14}{21} = \frac{2}{3}, then multiply both sides by -6:

w = -6 \cdot \frac{2}{3} = -4

Check: \displaystyle \frac{-4}{-6} = \frac{2}{3}, since a negative number divided by a negative number is positive.

(c) The left side is a fraction divided by a whole number, \displaystyle \frac{2}{3} \div 5 = \frac{2}{15}. Then multiply both sides by 12:

q = 12 \cdot \frac{2}{15} = \frac{24}{15} = \frac{8}{5}

The cross products reach the same place without simplifying first: \displaystyle \frac{2}{3} \cdot 12 = 5q, so 8 = 5q and \displaystyle q = \frac{8}{5}.

Example 4

Rice for a community lunch

Chidi’s rice recipe uses 3 cups of uncooked rice for 8 people. He is cooking for a community lunch.

(a) How much rice does he need for 70 people?

(b) The kitchen has 33 cups of rice. How many people would that feed?

Solution

Keep cups of rice on top and people underneath on both sides.

(a) The proportion is \displaystyle \frac{3}{8} = \frac{r}{70}. Multiply both sides by 70:

r = 70 \cdot \frac{3}{8} = \frac{210}{8} = 26.25

He needs 26.25 cups, that is \displaystyle 26\frac{1}{4} cups. As a check on size: 70 people is a little under 9 times 8, so the rice should be a little under 9 \times 3 = 27 cups.

(b) The unknown is now a number of people, so it goes underneath: \displaystyle \frac{3}{8} = \frac{33}{p}. Cross products give 3p = 264, so p = 88. A scale factor agrees: 33 is 11 times 3, so the people are 11 times 8.

(c) Now \displaystyle \frac{3}{8} = \frac{35}{p}, so 3p = 280 and \displaystyle p = 93\frac{1}{3}. The rice is enough for 93 people with a little left over, and not enough for 94. The proportion gives an exact number; the situation decides how to round it, and since every person needs a full portion it has to be rounded down.

Example 5

Typing a long piece

Saoirse types at a steady pace and finishes 340 words in 5 minutes.

(a) How long will a 1428-word essay take her?

(b) How many words does she type in 12.5 minutes?

Solution

(a) With words on top and minutes underneath,

\frac{340}{5} = \frac{1428}{t}

Cross products give 340t = 5 \cdot 1428 = 7140, so \displaystyle t = \frac{7140}{340} = 21 minutes.

The proportion can just as well be written with minutes on top, \displaystyle \frac{5}{340} = \frac{t}{1428}. Then \displaystyle t = 1428 \cdot \frac{5}{340} = \frac{7140}{340} = 21 again. Either arrangement is right, provided both sides follow the same one.

(b) The proportion is \displaystyle \frac{340}{5} = \frac{w}{12.5}. The unit rate is \displaystyle \frac{340}{5} = 68 words a minute, so w = 68 \times 12.5 = 850 words.

Example 6

Counting a pulse for part of a minute

(a) Bram counts his pulse for 15 seconds and gets 19 beats. What is his heart rate in beats per minute?

(b) Someone counts 7 breaths in 20 seconds. How many breaths is that per minute?

Solution

(a) A minute is 60 seconds. With beats on top and seconds underneath,

\frac{19}{15} = \frac{b}{60}

From 15 seconds to 60 is a scale factor of 4, so b = 19 \times 4 = 76 beats per minute. The double number line shows the same thing: the shaded 15 seconds, copied four times, fills the minute.

Figure 2The shaded 15 seconds holds 19 beats. A minute is four copies of it, so it holds four times as many, 76 beats.

This is why a pulse taken for 15 seconds is multiplied by 4. The 4 is the scale factor \displaystyle \frac{60}{15}, and using it assumes the heart keeps the same pace for the rest of the minute.

(b) The proportion is \displaystyle \frac{7}{20} = \frac{n}{60}, and the scale factor is \displaystyle \frac{60}{20} = 3, so n = 21 breaths per minute.

m = 25 \cdot \frac{72}{60} = 25 \cdot 1.2 = 30

There are 30 beats in 25 seconds.

Example 7

Pancakes for a crowd

A pancake recipe makes 12 pancakes from 2 cups of flour, 3 eggs and 1.5 cups of milk. Valentina is making pancakes for a crowd and has 7 eggs.

(a) Using all 7 eggs, how many pancakes can she make, and how much flour and milk does she need?

(b) She then finds she has only 4 cups of flour. How many pancakes can she make now, and how many eggs will she use?

Solution

(a) Eggs and pancakes first: \displaystyle \frac{3}{12} = \frac{7}{p}, so 3p = 84 and p = 28 pancakes. Every amount in the recipe must grow by the same scale factor, or the batter changes, so each of the other ingredients gets its own proportion with 28 pancakes:

\frac{2}{12} = \frac{f}{28}, \quad\text{so}\quad 12f = 56 \quad\text{and}\quad f = 4\tfrac{2}{3}

\frac{1.5}{12} = \frac{m}{28}, \quad\text{so}\quad 12m = 42 \quad\text{and}\quad m = 3.5

She needs \displaystyle 4\frac{2}{3} cups of flour and 3.5 cups of milk. Check with the scale factor from 3 eggs to 7, which is \displaystyle \frac{7}{3}: \displaystyle 2 \cdot \frac{7}{3} = \frac{14}{3} = 4\frac{2}{3} and \displaystyle 1.5 \cdot \frac{7}{3} = 3.5.

(b) Four cups of flour is less than the \displaystyle 4\frac{2}{3} cups that 7 eggs need, so now the flour sets the limit. Flour and pancakes: \displaystyle \frac{2}{12} = \frac{4}{p}, so 2p = 48 and p = 24 pancakes. The eggs for 24 pancakes: \displaystyle \frac{3}{12} = \frac{e}{24} gives e = 6. She makes 24 pancakes and has 1 egg left over. Whichever ingredient runs out first decides the size of the batch, and one proportion per ingredient shows which one that is.

Boundaries

Denominators are never zero, and that includes the unknown. In \displaystyle \frac{a}{b} = \frac{c}{y} the answer cannot be y = 0, and the cross products do not always notice. In \displaystyle \frac{6}{y} = \frac{0}{5} they say 6 \cdot 5 = 0 \cdot y, that is 30 = 0, which is false for every y: there is no solution, because a ratio with 6 on top is never 0. In \displaystyle \frac{3}{4} = \frac{0}{y} they say 3y = 0, so y = 0, the one value a denominator cannot take: again there is no solution. In \displaystyle \frac{0}{y} = \frac{0}{5} they say 0 = 0, which is true for every nonzero y, so there is no single answer to find. With the unknown on top there is always exactly one solution; with the unknown underneath there is exactly one when neither number on top is zero.
The answer is only as good as the proportion. Solving finds the number that makes two ratios equal. Whether the real quantities stay in proportion is an assumption made when the equation is written down. A typist does not keep one pace for hours, a pulse can speed up during the count, and the cooking time of a recipe does not grow with the number of servings. In cases like these the proportion gives an estimate at best.
Both sides must be set up the same way. The two ratios must list the same quantities in the same order and in the same units, as What a proportion says insists: rice over people on both sides, and minutes on both sides rather than minutes on one and seconds on the other. A wrongly arranged proportion still has a solution; it is the solution of a different equation.
Rounding belongs to the situation. The proportion gives an exact number. Whether to round up, round down or keep the fraction depends on what is being counted. Rice can be measured to a quarter of a cup; people come whole, and the food must be enough for every one of them (Example 4).

Where this goes next

Setting up a proportion from words is the other half of every problem in this lesson: deciding what goes on top and what underneath, so that the equation says what the sentence says. Percent problems are proportions with one ratio out of 100. “18 is what percent of 40?” is the proportion \displaystyle \frac{18}{40} = \frac{p}{100}, and solving percent problems and finding the whole from a part and a percent both come down to solving one.

When the unknown sits inside an expression, as in \displaystyle \frac{n}{n + 6} = \frac{2}{5}, the cross products still clear the fractions, but what is left is a longer linear equation; that is equations from proportions. The unit rate from the second move, the same for every matching pair, is the constant of proportionality, and it is how a proportional relationship becomes a single equation. Later, proportions in similar triangles use the same moves to find lengths that cannot be measured directly.

Exercises

Unknown on top

Solve each proportion.

\displaystyle \frac{x}{20} = \frac{3}{4}
\displaystyle \frac{14}{8} = \frac{n}{36}
\displaystyle \frac{a}{45} = \frac{8}{18}
\displaystyle \frac{5}{16} = \frac{m}{40}

Unknown underneath

\displaystyle \frac{63}{b} = \frac{7}{3}
\displaystyle \frac{12}{25} = \frac{30}{c}
\displaystyle \frac{9}{k} = \frac{15}{8}

Decimals, fractions and negative numbers

\displaystyle \frac{2.4}{p} = \frac{0.6}{1.5}
\displaystyle \frac{q}{-10} = \frac{27}{45}
\displaystyle \frac{-35}{50} = \frac{14}{r}
\displaystyle \frac{\frac{3}{4}}{6} = \frac{s}{10}
\displaystyle \frac{1.8}{t} = \frac{4.5}{2.5}

Recipes, typing and pulses

A flatbread recipe that makes 10 pieces uses 750 g of flour. How much flour is needed for 36 pieces?
A stew for 4 people uses 3 onions. Zainab is making the same stew for a party of 30. How many onions does she need?
A cake recipe that serves 12 uses 280 g of sugar. Hiroshi has 700 g of sugar. How many servings of the cake will that sugar make?
Esme types 195 words in 3 minutes. At the same pace, how long does she take to type 1105 words?
Ravi types 288 words in 4 minutes and Leilani types 350 words in 5 minutes. (a) Who types faster? (b) In the time Ravi takes to type 1080 words, how many words does Leilani type?
A pulse is counted for 20 seconds and gives 23 beats. What is the heart rate in beats per minute?
Someone at rest breathes 15 times a minute. How many breaths do they take in 40 seconds?
One person’s pulse is counted twice. Over 10 seconds the count is 13 beats; over 30 seconds it is 37 beats. Find the rate in beats per minute that each count gives, and suggest why they differ.

Reasoning

Explain why the proportion \displaystyle \frac{x}{b} = \frac{c}{d} can be solved by multiplying both sides by b, and show that the answer agrees with the one the cross products give.
Show that \displaystyle \frac{10}{y} = \frac{0}{3} has no solution. What does that say about a ratio with 10 on top?
In \displaystyle \frac{a}{b} = \frac{c}{y}, with a, b and c positive, what happens to y when c is doubled, and what happens when a is doubled? Use a soup recipe to say why that makes sense.
A heart rate is often found by counting beats for 15 seconds and multiplying by 4. (a) Use a proportion to explain where the 4 comes from. (b) What would you multiply by after counting for 20 seconds, or for 6 seconds? (c) Explain why a count that is out by one beat matters more for the 6-second count.
Show that turning both ratios over does not change the solution: with a, b and c not zero, \displaystyle \frac{a}{b} = \frac{c}{y} and \displaystyle \frac{b}{a} = \frac{y}{c} have the same solution.

Answers

x = 15. From 4 to 20 is a factor of 5, and 3 \times 5 = 15.
n = 63. Multiplying both sides by 36 gives \displaystyle n = 36 \cdot \frac{14}{8} = \frac{504}{8} = 63. Or simplify \displaystyle \frac{14}{8} = \frac{7}{4}; from 4 to 36 is a factor of 9, and 7 \times 9 = 63.
a = 20. \displaystyle \frac{8}{18} = \frac{4}{9}, and from 9 to 45 is a factor of 5, so a = 4 \times 5 = 20.
m = 12.5. \displaystyle m = 40 \cdot \frac{5}{16} = \frac{200}{16} = 12.5. Check: 5 \cdot 40 = 200 and 16 \cdot 12.5 = 200.
b = 27. Cross products give 7b = 63 \cdot 3 = 189, so b = 27. Or: 63 is 9 times 7, so b is 9 times 3.
c = 62.5. Cross products give 12c = 25 \cdot 30 = 750, so c = 62.5.
k = 4.8. Cross products give 15k = 9 \cdot 8 = 72, so k = 4.8. Check: \displaystyle \frac{9}{4.8} = 1.875 = \frac{15}{8}.
p = 6. Cross products give 0.6p = 2.4 \times 1.5 = 3.6, so p = 6. Or: 2.4 is 4 times 0.6, so p is 4 times 1.5.
q = -6. \displaystyle \frac{27}{45} = \frac{3}{5}, so \displaystyle q = -10 \cdot \frac{3}{5} = -6. Check: \displaystyle \frac{-6}{-10} = \frac{3}{5}.
r = -20. Cross products give -35r = 50 \cdot 14 = 700, so \displaystyle r = \frac{700}{-35} = -20. Check: \displaystyle \frac{14}{-20} = -0.7 and \displaystyle \frac{-35}{50} = -0.7.
\displaystyle s = \frac{5}{4}. The left side is \displaystyle \frac{3}{4} \div 6 = \frac{1}{8}, so \displaystyle s = 10 \cdot \frac{1}{8} = \frac{5}{4}.
t = 1. Cross products give 4.5t = 1.8 \times 2.5 = 4.5, so t = 1.
2700 g. \displaystyle \frac{750}{10} = \frac{f}{36}, and \displaystyle \frac{750}{10} = 75 g a piece, so f = 36 \times 75 = 2700.
23 onions. \displaystyle \frac{3}{4} = \frac{n}{30} gives \displaystyle n = 30 \cdot \frac{3}{4} = 22.5. Onions are bought whole and she needs at least 22.5, so she needs 23.
30 servings. \displaystyle \frac{280}{12} = \frac{700}{s} gives 280s = 8400, so s = 30. Or: 700 is 2.5 times 280, and 2.5 \times 12 = 30.
17 minutes. \displaystyle \frac{195}{3} = \frac{1105}{t} gives 195t = 3315, so t = 17. Her unit rate is 65 words a minute.
(a) Ravi, at \displaystyle \frac{288}{4} = 72 words a minute against Leilani’s \displaystyle \frac{350}{5} = 70. (b) 1050 words. Ravi’s time: \displaystyle \frac{288}{4} = \frac{1080}{t} gives 288t = 4320, so t = 15 minutes. Leilani in 15 minutes: \displaystyle \frac{350}{5} = \frac{w}{15} gives w = 1050.
69 beats per minute. \displaystyle \frac{23}{20} = \frac{b}{60}, and the scale factor from 20 seconds to 60 is 3, so b = 69.
10 breaths. \displaystyle \frac{15}{60} = \frac{n}{40}, so \displaystyle n = 40 \cdot \frac{15}{60} = 10.
78 and 74 beats per minute. \displaystyle \frac{13}{10} = \frac{b}{60} gives b = 13 \times 6 = 78, and \displaystyle \frac{37}{30} = \frac{b}{60} gives b = 37 \times 2 = 74. A count catches only whole beats, so a beat falling just inside or just outside the window changes the count by one; the short count multiplies that by 6, the long count only by 2. The heart rate may also have changed between the two counts.
Multiplying both sides of a true equation by the same number keeps it true, and \displaystyle b \cdot \frac{x}{b} = x because b \ne 0, so \displaystyle x = b \cdot \frac{c}{d} = \frac{bc}{d}. The cross products of the same proportion say xd = bc, and dividing both sides by d, which is not zero, gives \displaystyle x = \frac{bc}{d} again.
The cross products give 10 \cdot 3 = 0 \cdot y, that is 30 = 0, which is false whatever y is, so there is no solution. A quotient is 0 only when its top is 0, so a ratio with 10 on top is never 0, for any nonzero y.
The cross products give ay = bc, so \displaystyle y = \frac{bc}{a}. Doubling c doubles the top of that fraction, so y doubles. Doubling a doubles the bottom, so y is halved. With a cups of stock for b servings, y is how many servings c cups make: twice the stock makes twice the servings, while a recipe that needs twice the stock for the same b servings makes only half as many servings from the same c cups.
(a) With n beats in 15 seconds, \displaystyle \frac{n}{15} = \frac{b}{60}, and the scale factor from 15 seconds to 60 is \displaystyle \frac{60}{15} = 4, so b = 4n. (b) By \displaystyle \frac{60}{20} = 3 after 20 seconds and by \displaystyle \frac{60}{6} = 10 after 6 seconds. (c) The rate is the count times the scale factor, so a count one beat out gives a rate 10 beats per minute out after a 6-second count, but only 4 out after a 15-second count.
The cross products of \displaystyle \frac{a}{b} = \frac{c}{y} give ay = bc, and those of \displaystyle \frac{b}{a} = \frac{y}{c} give bc = ay. These are the same equation, so the two proportions have the same solution, \displaystyle y = \frac{bc}{a}. That value is not zero, because b and c are not, so it is allowed as a denominator in the first proportion.