Math 6 · 6.RP.A.3 · 6.RP.A.3A

Finding a missing value in a pair of equal ratios

Two ratios are meant to be equal, and you know three of their four numbers. What must the fourth one be?

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Finding a missing value in a pair of equal ratios

Ratio tables

When you reach for it

A pond keeper’s rule says 5 goldfish for every 800 liters of water. Clio’s pond holds 3200 liters. How many goldfish can it take? The rule is a ratio, 5 : 800, and Clio’s pond has to keep the same comparison, so the two ratios are equal:

5 : 800 = \,? : 3200

Three of the four numbers are known. This lesson finds the fourth.

Reach for this move whenever a comparison is fixed (a recipe, a rule, a mixture, a ration of food) and the amounts change: a bigger picnic, a fuller hutch, a different pond. The fixed comparison is one ratio, the new situation is the other, and the question mark sits in the new one. How to read a ratio and keep its parts in order is in ratio language and notation. Why multiplying both parts by one number keeps a ratio the same is the work of scaling a ratio up or down. This lesson uses that fact the other way round: the second ratio is known to be a scaled copy of the first, and the job is to find out by how much.

The procedure

Across: from one ratio to the other

Equal ratios are scaled copies of each other. One number, the scaling number, multiplies both parts of the first ratio to give the second. So two numbers in the same position (both goldfish, or both liters) are linked by the scaling number, and the position where both numbers are known tells you what it is.

In Clio’s pond the liters are known in both ratios: 800 goes to 3200. Ask “800 times what is 3200?” Division answers that question:

3200 \div 800 = 4

Now apply the same scaling number to the other part, the goldfish. It must be the same number, because a scaled copy multiplies both parts alike:

5 \times 4 = 20

Clio’s pond takes 20 goldfish, and 5 : 800 = 20 : 3200. This is reasoning across, from one ratio to the other: the arrows going down in Figure 1.

Figure 1Across, the scaling number 4 takes each part of the rule to its partner in Clio’s pond. Within, the liters are 160 times the goldfish in both ratios.

Within: inside one ratio

Figure 1 has a second number in it. Inside the rule 5 : 800 the liters are 160 times the goldfish, because 800 \div 5 = 160. Scaling cannot change that. Since 5 \times 160 = 800, multiplying both sides by 4 gives 20 \times 160 = 3200: in every ratio equal to 5 : 800, the second part is 160 times the first. This is reasoning within one ratio, the arrows going sideways. Here 160 is the liters for one goldfish, a unit rate.

Within gives a second road to a missing amount. For 9 goldfish the liters are 9 \times 160 = 1440. Across would need the scaling number 9 \div 5 = 1.8, which works but is clumsier. The two roads always reach the same answer, so take whichever has the friendlier number.

When the scaling number is not whole

Rabbits in a hutch eat hay at 4 rabbits to 9 kilograms a week. How much hay do 6 rabbits need? Across, 6 \div 4 = 1.5. Within, 9 \div 4 = 2.25. Neither is whole.

Either one still works, but there is a gentler road: scale down to a friendlier ratio first, then scale up. The numbers 4 and 6 share the factor 2, so divide both parts of 4 : 9 by 2:

4 : 9 = 2 : 4.5

From 2 rabbits to 6 is now a whole scaling number, \times 3:

2 : 4.5 = 6 : 13.5

So 6 rabbits need 13.5 kilograms of hay a week. Dividing by 2 and then multiplying by 3 is the same as multiplying by \displaystyle \frac{3}{2}, which is 1.5, the across number from the start. The road through 2 : 4.5 only splits it into two easy steps (Figure 2).

Figure 2From 4 rabbits to 6 is not a whole scaling number, so go down to 2 : 4.5 first and then up by 3.

The friendlier ratio to aim for is one whose part divides exactly into the amount you were given. Dividing by a factor the two numbers share is the usual way there. Going all the way down to one part, here 1 : 2.25, always works.

Checking

The answer completes a ratio that must equal the first, so check by scaling: multiply each part of the original ratio by the scaling number and see that both parts land. For the rabbits, 4 \times 1.5 = 6 and 9 \times 1.5 = 13.5. Then check that the answer fits the setting: goldfish and sandwiches come in whole numbers, hay does not.

Worked examples

Example 1

Across with a whole scaling number

Hana packs a picnic with 7 cheese sandwiches for every 3 tomato sandwiches. (a) She makes 21 cheese sandwiches. How many tomato sandwiches go with them? (b) Noemi uses the same ratio and makes 15 tomato sandwiches. How many cheese sandwiches does she make?

Solution

(a) The cheese is known in both ratios: 7 goes to 21, and 21 \div 7 = 3, so the scaling number is 3. Apply it to the tomato part: 3 \times 3 = 9. Hana makes 9 tomato sandwiches, and 7 : 3 = 21 : 9.

(b) Now the tomato is known in both: 3 goes to 15, and 15 \div 3 = 5. The missing amount is in the first place, which changes nothing; apply \times 5 to the cheese, 7 \times 5 = 35. Noemi makes 35 cheese sandwiches. Check: 7 : 3 = 35 : 15, both parts times 5.

Example 2

Choosing across or within

The pond rule is 5 goldfish for every 800 liters. (a) Dmitri wants to keep 9 goldfish. How many liters must his pond hold? (b) Idris’s pond holds 4000 liters. How many goldfish does the rule allow?

Solution

(a) The question is 5 : 800 = 9 : \,?. Across, the scaling number is 9 \div 5 = 1.8. Within, the liters are 800 \div 5 = 160 times the goldfish, a whole number, so use that: 9 \times 160 = 1440. Dmitri’s pond must hold 1440 liters. Across gives the same, 800 \times 1.8 = 1440.

(b) The question is 5 : 800 = \,? : 4000. Across, 4000 \div 800 = 5, so the goldfish are 5 \times 5 = 25. Within works too, run backwards: the liters are 160 times the goldfish, so the goldfish are 4000 \div 160 = 25. Idris’s pond takes 25 goldfish.

Example 3

A scaling number that is not whole

Teodor’s rabbits eat hay at 4 rabbits to 9 kilograms a week. His hutches will soon hold 14 rabbits. How much hay will they eat in a week?

Solution

The question is 4 : 9 = 14 : \,?. Across, 14 \div 4 = 3.5; within, 9 \div 4 = 2.25. Neither is whole, so go through a friendlier ratio. The numbers 4 and 14 share the factor 2, and dividing both parts by 2 gives 4 : 9 = 2 : 4.5. From 2 to 14 is \times 7, a whole number:

2 : 4.5 = 14 : 31.5

Teodor needs 31.5 kilograms of hay a week. Check with the single across number: 4 \times 3.5 = 14 and 9 \times 3.5 = 31.5.

Example 4

Down to one rabbit, with fractions

Rabbits eat hay at 4 rabbits to 9 kilograms a week. (a) How much hay does one rabbit eat in a week? (b) Clio keeps 3 rabbits. How much hay do they eat in a week?

Solution

(a) The question is 4 : 9 = 1 : \,?. From 4 to 1 is a scaling down: divide both parts by 4, which is multiplying by \displaystyle \frac{1}{4}. The hay is \displaystyle 9 \div 4 = \frac{9}{4} = 2\frac{1}{4} kilograms. This is also the within number: in every ratio equal to 4 : 9, the hay is \displaystyle 2\frac{1}{4} times the rabbits.

(b) The numbers 3 and 4 share no factor except 1, so the friendliest ratio is the one just found, \displaystyle 1 : 2\frac{1}{4}. Scale it up by 3:

3 \times 2\frac{1}{4} = 6 + \frac{3}{4} = 6\frac{3}{4}

Clio’s rabbits eat \displaystyle 6\frac{3}{4} kilograms a week. Check with the across number \displaystyle \frac{3}{4}: \displaystyle 4 \times \frac{3}{4} = 3 and \displaystyle 9 \times \frac{3}{4} = \frac{27}{4} = 6\frac{3}{4}.

Example 5

An answer that does not fit the setting

Noemi’s pond holds 2000 liters. Using the rule of 5 goldfish for every 800 liters, how many goldfish can she keep?

Solution

The question is 5 : 800 = \,? : 2000. Across, 2000 \div 800 = 2.5, not whole. The numbers 800 and 2000 share the factor 400, so scale down by 2 to reach a 400: 5 : 800 = 2.5 : 400. From 400 to 2000 is \times 5:

2.5 : 400 = 12.5 : 2000

The equal ratio says 12.5 goldfish, and half a goldfish is not a thing a pond can hold. The rule sets the least water each goldfish needs, so test the whole numbers on either side with the within number 160: 12 goldfish need 12 \times 160 = 1920 liters, which fits in 2000, and 13 need 13 \times 160 = 2080, which does not. Noemi can keep at most 12 goldfish. The arithmetic gave 12.5; the setting decided which whole number answers the question.

Example 6

Are two ratios equal?

Hana’s picnic plan says 7 cheese sandwiches for every 3 tomato. Teodor packed 28 cheese and 12 tomato. Idris packed 35 cheese and 16 tomato. Which basket keeps Hana’s ratio?

Solution

Treat one number in each basket as missing and work out what the ratio says it should be. Teodor: 7 : 3 = 28 : \,?, and 28 \div 7 = 4, so the tomato should be 3 \times 4 = 12. He packed 12, so 28 : 12 equals 7 : 3. Idris: 7 : 3 = 35 : \,?, and 35 \div 7 = 5, so the tomato should be 3 \times 5 = 15. He packed 16, so 35 : 16 is not equal to 7 : 3; Idris has one tomato sandwich more than the ratio calls for. Only Teodor’s basket keeps the ratio.

Boundaries

The comparison has to be fixed. The move finds a value that keeps two ratios equal, so it only answers questions where the ratio really stays the same. Ages do not: when Noemi is 4 her cousin is 9, and when Noemi is 8 her cousin is 13, not 18, because both ages grow by the same amount rather than by the same factor.

The parts must line up. The first part of one ratio is matched with the first part of the other. In 7 cheese to 3 tomato, 21 cheese gives 9 tomato; reading the 21 as tomato gives 49 cheese, an answer to a different question.

The units must match across. If the rule is in kilograms and the amount you have is in grams, convert first; a scaling number between 9 kilograms and 9000 grams would be meaningless.

Every part must be more than zero. A scaling number turns the known amount in the first ratio into the known amount in the second. If the first is 0, no number does that.

The answer may not fit the things being counted. Equal ratios can call for 12.5 goldfish or \displaystyle 18\frac{2}{3} sandwiches. The ratio is still right; the setting decides whether to round down, round up, or say the ratio cannot be kept exactly.

Where this goes next

A ratio table lists many equal ratios at once, and every new row in it is a missing value found by scaling. A double number line draws the same pairs as matching marks on two lines. Going down to one part, as with one rabbit’s hay, is the start of unit rates. Two equal ratios written as one statement are called a proportion, and solving a proportion finds the missing value when the scaling number is not friendly at all.

Exercises

Across with a whole scaling number

7 : 3 = 56 : \,?
4 : 9 = \,? : 45
5 : 8 = 40 : \,?
42 : 18 = 7 : \,?
At another picnic the ratio is 7 cheese sandwiches to 3 tomato sandwiches, and there are 18 tomato sandwiches. How many cheese sandwiches are there?

Within one ratio

3 : 12 = 7 : \,?
With the pond rule of 5 goldfish for every 800 liters, how many liters do 11 goldfish need?
4 : 20 = \,? : 35

Through a friendlier ratio

4 : 9 = 10 : \,?
6 : 9 = 8 : \,?
5 : 8 = \,? : 12
10 : 4 = 25 : \,?
7 : 3 = \,? : 4.5

Fractions

4 : 9 = 5 : \,?
5 : 8 = \,? : 2

In context

Rabbits eat hay at 4 rabbits to 9 kilograms a week. A rescue center puts out 54 kilograms of hay a week. How many rabbits does that feed?
At the same rate, how much hay do 18 rabbits eat in a week?
With the pond rule of 5 goldfish for every 800 liters, how many goldfish can live in a pond of 1000 liters?
Idris wants exactly 8 tomato sandwiches and the ratio of 7 cheese to 3 tomato. Can he keep the ratio with whole sandwiches?
Dmitri keeps 3 goldfish in a 400 liter pond. Does that meet the rule of 5 goldfish for every 800 liters? If not, how many goldfish would?

Reasoning

Explain, using a scaling number, why 4 : 9 and 6 : 11 are not equal ratios.
Explain why reasoning across and reasoning within always give the same missing amount. Use 5 : 800 = 20 : \,? to show the idea.
To solve 4 : 9 = 6 : \,?, Hana divides by 2 and then multiplies by 3, while Clio multiplies by 1.5 once. Explain why they must get the same answer.

Answers

24. The scaling number is 56 \div 7 = 8, and 3 \times 8 = 24.
20. The scaling number is 45 \div 9 = 5, and 4 \times 5 = 20.
64. The scaling number is 40 \div 5 = 8, and 8 \times 8 = 64.
3. Here the ratio scales down: 42 \div 7 = 6, so divide 18 by 6 too, 18 \div 6 = 3.
42 cheese sandwiches. The scaling number is 18 \div 3 = 6, and 7 \times 6 = 42.
28. Within, 12 is 4 times 3, so the missing part is 7 \times 4 = 28. (Across would need \displaystyle 7 \div 3 = 2\frac{1}{3}.)
1760 liters. Within, each goldfish needs 800 \div 5 = 160 liters, and 11 \times 160 = 1760.
7. Within, 20 is 5 times 4, so the first part is the second divided by 5: 35 \div 5 = 7.
22.5. Scale down by 2 to 2 : 4.5, then up by 5: 4.5 \times 5 = 22.5.
12. Scale down by 3 to 2 : 3, then up by 4: 3 \times 4 = 12.
7.5. Scale down by 2 to 2.5 : 4, then up by 3: 2.5 \times 3 = 7.5.
10. Scale down by 2 to 5 : 2, then up by 5: 2 \times 5 = 10.
10.5. Scale down by 2 to 3.5 : 1.5, then up by 3: 3.5 \times 3 = 10.5.
\displaystyle 11\frac{1}{4}. Scale down by 4 to \displaystyle 1 : 2\frac{1}{4}, then up by 5: \displaystyle 5 \times 2\frac{1}{4} = 10 + \frac{5}{4} = 11\frac{1}{4}.
\displaystyle 1\frac{1}{4}. From 8 to 2 is dividing by 4, so \displaystyle 5 \div 4 = \frac{5}{4} = 1\frac{1}{4}.
24 rabbits. The scaling number is 54 \div 9 = 6, and 4 \times 6 = 24.
40.5 kilograms. Scale down by 2 to 2 : 4.5, then up by 9: 4.5 \times 9 = 40.5.
At most 6 goldfish. The equal ratio gives 1000 \div 160 = 6.25; 6 goldfish need 6 \times 160 = 960 liters, which fits, and 7 need 1120, which does not.
No. From 3 to 8 is \displaystyle \times \frac{8}{3}, so the cheese would be \displaystyle 7 \times \frac{8}{3} = \frac{56}{3} = 18\frac{2}{3}, not a whole number of sandwiches.
No. Three goldfish need 3 \times 160 = 480 liters, more than 400. The equal ratio for 400 liters is 2.5 : 400, so the pond suits at most 2 goldfish.
The scaling number from 4 to 6 is 6 \div 4 = 1.5, and 9 \times 1.5 = 13.5, so 4 : 9 = 6 : 13.5. The second part is 11, not 13.5, so no single number scales 4 : 9 to 6 : 11. (Adding 2 to each part is not scaling.)
Across gives 800 \times 4, and 800 = 5 \times 160, so that is (5 \times 160) \times 4. Within gives 20 \times 160, and 20 = 5 \times 4, so that is (5 \times 4) \times 160. Both multiply the same three numbers, only in a different order and grouping, which the commutative and associative properties allow; both are 3200. Any pair of equal ratios works the same way: each road multiplies the first number by the across number and by the within number.
Dividing by 2 is multiplying by \displaystyle \frac{1}{2}, and multiplying by \displaystyle \frac{1}{2} and then by 3 is multiplying by \displaystyle \frac{1}{2} \times 3 = \frac{3}{2} = 1.5 (associative property). So Hana’s two steps are Clio’s one step split in two: both get 9 \times 1.5 = 13.5.