Math 6 · 6.RP.A.3A

Deciding which of two ratios is larger

Two mixes are made with different amounts of everything. How can you tell which one is stronger?

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Equivalent ratios and ratio tablesGrade 6 Comparing fractionsGrades 3 to 4

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What a ratio is3

Deciding which of two ratios is larger

Part-to-part and part-to-whole ratios What a ratio is: language and notation

When you reach for it

Bodhi makes tea with 5 spoons of tea leaves to 6 cups of water. Imani uses 4 spoons to 5 cups. Whose tea is stronger?

Bodhi uses more leaves, but he also uses more water, so neither number settles it on its own. The two ratios have to be put on the same footing first: the same amount of water in both, or the same amount of leaves. Then a single comparison tells the whole story.

Reach for this whenever two mixes, rates or shares are described with different amounts: two pots of tea, two compost heaps, two trays of seeds. The ratios may be part to part (leaves to water) or part to whole (seeds that sprouted to seeds planted), and both kinds are compared the same way. Part 1 (day one) matches one term of the two ratios; Part 2 (day two) compares shares of a whole as fractions and handles decimals.

The procedure

Part 1: why scaling is allowed

Multiplying both numbers of a ratio by the same number gives the same mix (scaling a ratio). 10 spoons to 12 cups tastes exactly like 5 spoons to 6 cups: it is two pots of Bodhi’s tea side by side. So each tea may be scaled to any size without changing how strong it is, and the whole skill is choosing sizes at which the two teas share one number.

Part 1: matching the second term

Make the water the same. Bodhi’s water goes 6, 12, 18, 24, 30, \dots and Imani’s goes 5, 10, 15, 20, 25, 30, \dots. The first amount on both lists is 30, a common multiple of 6 and 5. Scale Bodhi’s ratio by 5 and Imani’s by 6:

5 : 6 = 25 : 30 \qquad 4 : 5 = 24 : 30

In the same 30 cups of water, Bodhi uses 25 spoons and Imani 24. More leaves in the same water make stronger tea, so Bodhi’s is stronger (Figure 1).

Figure 1Scaled to the same 30 cups of water, Bodhi’s tea has 25 spoons of leaves and Imani’s has 24.

Part 1: matching the first term

The leaves can be matched instead. A common multiple of 5 and 4 is 20, so scale Bodhi’s ratio by 4 and Imani’s by 5:

5 : 6 = 20 : 24 \qquad 4 : 5 = 20 : 25

Now the leaves are the same and Imani’s go into more water, 25 cups against 24. The same leaves in more water make weaker tea, so again Bodhi’s is stronger. Matching either term always gives the same verdict (Exercise 21 asks why), so match whichever term needs the smaller numbers.

A ratio table for each tea does the same job: list rows until one number in the same column turns up in both tables. Any common multiple works, not only the smallest; 60 cups would settle it as well as 30, with larger numbers.

Part 1: what “larger” and “stronger” mean

Once one term is matched, compare the other one, and read it against the question. With the water matched, the stronger tea has more leaves. With the leaves matched, the stronger tea has less water. A question about the weaker tea turns both of these round.

The order of the ratio matters too (ratio language). Written leaves to water, Bodhi’s 5 : 6 is the larger ratio: with the second terms matched, its first term is larger. Written water to leaves, the same teas are 6 : 5 and 5 : 4. Matched at 20 spoons they are 24 : 20 and 25 : 20, so now Imani’s is the larger ratio. The teas did not change, only the way they were written. “Larger ratio” depends on the order in which the ratio is written; “stronger” is about the tea itself.

Part 2: shares of a whole as fractions

Ezra and Giulia sow seeds in two trays. In Ezra’s tray 7 of every 10 seeds sprouted; in Giulia’s, 9 of every 13. Which tray had the larger share sprout?

Sprouted to planted is a part-to-whole ratio, and a part-to-whole ratio is a fraction of the whole: \displaystyle \frac{7}{10} and \displaystyle \frac{9}{13}. These compare the way fractions do, over a common denominator, here 130:

\frac{7}{10} = \frac{91}{130} \qquad \frac{9}{13} = \frac{90}{130}

Ezra’s tray did better, by one seed in every 130. This is Part 1 in another form: the common denominator is a matched number of seeds planted, 91 : 130 against 90 : 130. Matching the sprouted seeds instead, 63 : 90 against 63 : 91, gives the same verdict: as many seeds sprouted in Ezra’s tray from fewer planted.

Part 2: from part to part to a share of the whole

Faisal’s compost mix is 2 buckets of greens (kitchen scraps and grass) to 3 buckets of browns (dry leaves and card). Jarrah’s is 3 buckets of greens to 4 of browns. Which mix has the larger share of greens?

These are part-to-part ratios. The whole is the two parts added together. Faisal’s batch is 2 + 3 = 5 buckets, so greens are \displaystyle \frac{2}{5} of it; Jarrah’s is 3 + 4 = 7 buckets, so greens are \displaystyle \frac{3}{7} of it. Over the common denominator 35:

\frac{2}{5} = \frac{14}{35} \qquad \frac{3}{7} = \frac{15}{35}

Jarrah’s mix has the larger share of greens (Figure 2). Matching the browns agrees: 2 : 3 = 8 : 12 and 3 : 4 = 9 : 12, so Jarrah has more greens with the same browns. That is no accident: more greens for the same browns means the greens fill more of the whole (Exercise 22).

Figure 2Each bar is one whole batch, with the greens shaded. Jarrah’s greens, \tfrac{3}{7} of his batch, reach a little further than Faisal’s \tfrac{2}{5}.

The question decides which share to look at. The share of browns is \displaystyle \frac{3}{5} = \frac{21}{35} for Faisal and \displaystyle \frac{4}{7} = \frac{20}{35} for Jarrah, so asked which mix is richer in browns, the answer is Faisal’s.

Part 2: decimals, and a road for later

A ratio with decimals, such as 1.5 spoons to 2 cups, scales to whole numbers first: times 2 gives 3 : 4, the same tea. Then match a term as before.

One more way to put ratios on the same footing is to scale each so that one term is 1, for instance the leaves in a single cup of water. Comparing that way, through unit rates and the price of one item, is the subject of unit price.

Worked examples

Example 1

One tea scales to meet the other (Part 1)

Faisal makes tea with 3 spoons of leaves to 5 cups of water, and Giulia with 7 spoons to 10 cups. Whose tea is stronger?

Solution

The water amounts are 5 and 10, and 10 is already a multiple of 5, so only Faisal’s ratio needs scaling. Times 2, 3 : 5 = 6 : 10, the same tea. In the same 10 cups, Faisal uses 6 spoons and Giulia 7. More leaves in the same water, so Giulia’s tea is stronger.

Check by matching the leaves at 21: 3 : 5 = 21 : 35 and 7 : 10 = 21 : 30. Giulia puts the same leaves in less water, so hers is stronger again.

Example 2

Which tea is weaker? (Part 1)

Ezra makes tea with 4 spoons to 7 cups, and Jarrah with 6 spoons to 11 cups. Use ratio tables to decide whose tea is weaker.

Solution

List rows of each table until a number of spoons appears in both.

Ezra’s spoons	Ezra’s cups	Jarrah’s spoons	Jarrah’s cups
4	7	6	11
8	14	12	22
12	21

At 12 spoons, Ezra uses 21 cups and Jarrah 22. The same leaves in more water make weaker tea, so Jarrah’s tea is weaker.

Check by matching the water at 77: 4 : 7 = 44 : 77 (times 11) and 6 : 11 = 42 : 77 (times 7). Jarrah has fewer leaves in the same water, so his is weaker.

Example 3

Written water to leaves (Part 1)

Giulia writes her tea as 13 cups of water to 6 spoons of leaves, and Imani writes hers as 11 cups to 5 spoons. (a) Whose tea is stronger? (b) Whose ratio is larger?

Solution

This time the leaves are the second term. A common multiple of 6 and 5 is 30, so scale Giulia’s ratio by 5 and Imani’s by 6:

13 : 6 = 65 : 30 \qquad 11 : 5 = 66 : 30

(a) With 30 spoons each, Giulia uses 65 cups and Imani 66. The same leaves in less water, so Giulia’s tea is stronger.

(b) With the second terms matched, the first terms are 65 and 66, so Imani’s ratio, 11 : 5, is the larger. A larger ratio of water to leaves means more water for each spoon, which is the weaker tea.

Example 4

Three teas in order (Part 1)

Bodhi’s tea is 5 : 6, Imani’s 4 : 5 and Jarrah’s 7 : 9, spoons to cups. Order them from strongest to weakest.

Solution

Match all three waters at once. The amounts are 6, 5 and 9. Counting up in nines, 9, 18, 27, \dots, the first amount that 5 and 6 both go into is 90 (18 and 36 are not multiples of 5, and 45 is not a multiple of 6), so 90 is a multiple of all three. Scale by 15, 18 and 10:

\begin{aligned} 5 : 6 &= 75 : 90 \\ 4 : 5 &= 72 : 90 \\ 7 : 9 &= 70 : 90 \end{aligned}

In the same 90 cups the leaves are 75, 72 and 70 spoons, so the order is Bodhi, Imani, Jarrah. Check the last two by their leaves: 4 : 5 = 28 : 35 and 7 : 9 = 28 : 36, so Jarrah’s has more water for the same leaves.

Example 5

Count against share (Part 2)

Bodhi and Imani each sow a tray. In Bodhi’s tray, seeds that sprouted to seeds that did not are 5 : 3; in Imani’s, 7 : 4. Bodhi sowed 24 seeds and Imani 22. (a) Which tray had the larger share sprout? (b) How many seeds sprouted in each tray?

Solution

(a) These are part-to-part ratios, so first find each whole. Bodhi’s seeds come in groups of 5 + 3 = 8, so the share that sprouted is \displaystyle \frac{5}{8}; Imani’s come in groups of 7 + 4 = 11, so hers is \displaystyle \frac{7}{11}. Over the common denominator 88:

\frac{5}{8} = \frac{55}{88} \qquad \frac{7}{11} = \frac{56}{88}

Imani’s tray had the larger share sprout.

(b) Bodhi’s 24 seeds are 24 \div 8 = 3 groups, so 5 \times 3 = 15 sprouted. Imani’s 22 seeds are 22 \div 11 = 2 groups, so 7 \times 2 = 14 sprouted. Bodhi had more seeds sprout, 15 against 14, but Imani’s tray had the larger share. “More seeds” and “a larger share” are different questions, and a ratio answers the second.

Example 6

Compost with decimals (Part 2)

Giulia’s compost mix is 1.5 buckets of greens to 2.5 buckets of browns. Ezra’s is 2.5 buckets of greens to 4.5 of browns. Which mix is richer in browns?

Solution

Scale both by 2 to clear the decimals: Giulia’s is 3 : 5 and Ezra’s is 5 : 9, the same mixes in whole numbers. “Richer in browns” asks about the share of browns in the whole. Giulia’s whole is 3 + 5 = 8 buckets and Ezra’s 5 + 9 = 14, so the shares of browns are \displaystyle \frac{5}{8} and \displaystyle \frac{9}{14}. Over the common denominator 56:

\frac{5}{8} = \frac{35}{56} \qquad \frac{9}{14} = \frac{36}{56}

Ezra’s mix is richer in browns. Check by matching the greens at 15: 3 : 5 = 15 : 25 and 5 : 9 = 15 : 27, so Ezra has more browns with the same greens.

Example 7

A tea between two others (Part 2)

Imani finds Bodhi’s tea, 5 : 6, a little too strong and her own, 4 : 5, a little too weak. Find a tea, spoons to cups, whose strength lies between them.

Solution

Matched at 30 cups, the teas have 25 and 24 spoons. No whole number lies between 24 and 25, so scale further. At 60 cups, 5 : 6 = 50 : 60 and 4 : 5 = 48 : 60. Then 49 spoons to 60 cups lies between: in the same water it has more leaves than Imani’s and fewer than Bodhi’s. (It is the same tea as 24.5 : 30.)

Another answer: brew one pot of each, Bodhi’s with 5 spoons in 6 cups and Imani’s with 4 spoons in 5 cups, and pour both into one big pot. The big pot holds the tea from all the leaves in all the water, 5 + 4 = 9 spoons in 6 + 5 = 11 cups, so it is a 9 : 11 tea. Matched at 330 cups, 5 : 6 = 275 : 330, 4 : 5 = 264 : 330 and 9 : 11 = 270 : 330. Since 264 < 270 < 275, the poured pot lies between as well.

Boundaries

Match the same thing in the same place. Water is matched with water and leaves with leaves, and both ratios must be written in the same order. If one tea is written leaves to water and the other water to leaves, turn one of them round first.

The units must agree. Spoons are compared with spoons and cups with cups. A tea measured in a different unit has to be converted before any matching.

The gap between the parts decides nothing. 2 : 3 and 9 : 10 both have one cup more than spoons, yet matched at 30 cups they are 20 : 30 and 27 : 30, far apart in strength. Ratios are put on the same footing by multiplying, never by adding or taking away.

A share needs the whole. Greens to browns 2 : 3 means greens are \displaystyle \frac{2}{5} of the mix, not \displaystyle \frac{2}{3}. A part-to-part ratio becomes a fraction of the whole only after the parts are added.

A ratio compares mixes, not amounts. The tray with the larger share need not have more seeds sprout, as Example 5 shows; the stronger tea need not be the bigger pot.

The terms must be above zero. Matching uses common multiples of the terms, and a term of 0 has none to offer.

Where this goes next

Comparing through one unit, the leaves in a single cup or the price of a single item, is unit price. A double number line lines the matched amounts up as marks on two lines, and a tape diagram shows a part-to-part ratio and its whole at once. Every comparison here rests on equivalent ratios, which later grow into proportions and percentages.

Exercises

Part 1: matching one term

Which tea is stronger: 3 spoons of leaves to 5 cups of water, or 5 spoons to 9 cups?
Which tea is stronger: 7 spoons to 12 cups, or 5 spoons to 8 cups?
Match the leaves to decide which tea is stronger: 6 spoons to 11 cups, or 9 spoons to 16 cups.
Which tea is weaker: 8 spoons to 10 cups, or 11 spoons to 14 cups?
Two compost mixes are 5 buckets of greens to 9 of browns, and 4 buckets of greens to 7 of browns. Which has more greens for the same amount of browns?
Make a ratio table for 2 spoons to 9 cups and one for 3 spoons to 13 cups, until the two share a number of spoons. Which tea is stronger?
Are 6 spoons to 8 cups and 9 spoons to 12 cups the same strength?

Part 1: the order of the terms, and three at once

Two teas are written water to leaves: 7 cups to 2 spoons, and 10 cups to 3 spoons. (a) Which tea is stronger? (b) Which ratio is larger?
Order these teas from strongest to weakest: 2 : 3, 3 : 5 and 5 : 8, spoons to cups.

Part 2: shares of the whole

In one tray 11 of 15 seeds sprouted; in another, 7 of 10. Which tray had the larger share sprout?
Which is the larger share: 13 sprouted of 20 planted, or 16 of 25?
In tray A, seeds that sprouted to seeds that did not are 4 : 1; in tray B they are 7 : 2. Which tray had the larger share sprout?
Two compost mixes are 3 : 5 and 2 : 3, greens to browns. (a) Which has the larger share of greens? (b) What fraction of each mix is browns?
Check your answer to Exercise 13(a) by matching the browns.

Part 2: decimals and counts

Which tea is stronger: 1.5 spoons to 2 cups, or 2.5 spoons to 3.5 cups?
Which compost mix is richer in greens: 0.5 bucket of greens to 1.5 of browns, or 1 bucket of greens to 2.5 of browns?
In Jarrah’s tray 18 of 30 seeds sprouted; in Bodhi’s, 14 of 22. Which tray had more seeds sprout, and which had the larger share sprout?
Show that 5 spoons to 7 cups is stronger than 2 : 3 and weaker than 3 : 4, spoons to cups.
Imani has 5 spoons of leaves and measures water in whole cups. What is the fewest cups of water she can use to make tea weaker than 4 spoons to 5 cups?

Reasoning

The teas 3 : 4 and 7 : 8, spoons to cups, both have one cup more than spoons. Are they the same strength? Explain.
Explain why matching the water and matching the leaves always pick the same tea as stronger.
One compost mix has the larger ratio of greens to browns. Explain why it must also have the larger share of greens in the whole mix.
One tea has the larger ratio of water to leaves. What does that say about its strength? Explain.

Answers

3 : 5 is stronger. At 45 cups, 3 : 5 = 27 : 45 and 5 : 9 = 25 : 45: more leaves in the same water.
5 : 8 is stronger. At 24 cups, 7 : 12 = 14 : 24 and 5 : 8 = 15 : 24.
9 spoons to 16 cups is stronger. At 18 spoons, 6 : 11 = 18 : 33 and 9 : 16 = 18 : 32: the same leaves in less water.
11 spoons to 14 cups is weaker. At 70 cups, 8 : 10 = 56 : 70 and 11 : 14 = 55 : 70: fewer leaves in the same water.
4 : 7. At 63 buckets of browns, 5 : 9 = 35 : 63 and 4 : 7 = 36 : 63.
3 : 13 is stronger. The rows are 2 : 9, 4 : 18, 6 : 27 and 3 : 13, 6 : 26. At 6 spoons, 26 cups is less water than 27.
Yes. At 24 cups both are 18 : 24.
(a) 10 cups to 3 spoons. At 6 spoons, 7 : 2 = 21 : 6 and 10 : 3 = 20 : 6: less water for the same leaves. (b) 7 : 2, since 21 is more than 20 with the second terms matched.
2 : 3, then 5 : 8, then 3 : 5. At 120 cups they are 80 : 120, 75 : 120 and 72 : 120.
The first tray: \displaystyle \frac{11}{15} = \frac{22}{30} and \displaystyle \frac{7}{10} = \frac{21}{30}.
13 of 20: \displaystyle \frac{13}{20} = \frac{65}{100} and \displaystyle \frac{16}{25} = \frac{64}{100}.
Tray A. The wholes are 5 and 9, so the shares are \displaystyle \frac{4}{5} = \frac{36}{45} and \displaystyle \frac{7}{9} = \frac{35}{45}.
(a) 2 : 3. Greens are \displaystyle \frac{3}{8} = \frac{15}{40} of the first mix and \displaystyle \frac{2}{5} = \frac{16}{40} of the second. (b) Browns are \displaystyle \frac{5}{8} of the first mix and \displaystyle \frac{3}{5} of the second.
At 15 buckets of browns, 3 : 5 = 9 : 15 and 2 : 3 = 10 : 15. The 2 : 3 mix has more greens with the same browns, as in 13(a).
1.5 spoons to 2 cups. Times 2 the teas are 3 : 4 and 5 : 7; at 28 cups they are 21 : 28 and 20 : 28.
The second. Times 2 the mixes are 1 : 3 and 2 : 5. At 2 buckets of greens they are 2 : 6 and 2 : 5, so the second has fewer browns with the same greens. The shares of greens are \displaystyle \frac{1}{4} = \frac{7}{28} and \displaystyle \frac{2}{7} = \frac{8}{28}.
Jarrah’s tray had more seeds sprout, 18 against 14. Bodhi’s had the larger share: \displaystyle \frac{18}{30} = \frac{3}{5} = \frac{33}{55} and \displaystyle \frac{14}{22} = \frac{7}{11} = \frac{35}{55}.
At 84 cups, 2 : 3 = 56 : 84, 5 : 7 = 60 : 84 and 3 : 4 = 63 : 84. Since 56 < 60 < 63, it lies between.
7 cups. Match the leaves at 20: 4 : 5 = 20 : 25, so a weaker tea has more than 25 cups with 20 spoons. With 6 cups, 5 : 6 = 20 : 24, less water, so stronger. With 7 cups, 5 : 7 = 20 : 28, more water, so weaker. Fewer than 6 cups is stronger still.
No. At 8 cups, 3 : 4 = 6 : 8, against 7 : 8, so 7 : 8 is stronger. Scaling does not keep the gap between the terms: 3 : 4 times 2 is 6 : 8, the same tea with a gap of 2.
Suppose that with the water matched, tea A has more leaves than tea B. To give A the same leaves as B, its row must be scaled down, by a number less than 1, and that shrinks A’s water below the shared amount too. So with the leaves matched, A has less water than B, which again makes A stronger. For example, 25 : 30 scaled by \displaystyle \frac{24}{25} is 24 : 28.8, against Imani’s 24 : 30.
Match the browns. The mix with the larger ratio then has more greens with the same browns, so its whole is larger. The browns are the same amount out of a larger whole, so they are a smaller share of it (same numerator, larger denominator). The greens make up the rest of the whole, so their share is larger.
It is the weaker tea. Match the leaves: the larger ratio of water to leaves has more water with the same leaves, and the same leaves in more water make weaker tea.