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Tape diagrams and sharing in a ratio

Two gardeners agree to split every harvest 3 to 5. When 40 kg of potatoes come out of the ground, how much does each of them take home?

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A fraction of a whole numberGrade 4Part-to-part and part-to-whole ratiosMath 6
Where you are 5 in Equivalent ratios and ratio tables
Fractions, ratio and percent3
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Equivalent ratios and ratio tables5
Tape diagrams and sharing in a ratio
The double number lineScaling a ratio up or downFinding a missing value in a pair of equal ratiosRatio tables

Why this exists

Fenna and Maeve rent an allotment together. Maeve does more of the digging, so they agree to split every harvest in the ratio 3 : 5, Fenna to Maeve. One Saturday they lift 40 kg of potatoes.

The ratio says how the two shares compare, not how many kilograms each person takes, and 40 is neither 3 nor 5. Splitting in half would be easy: two equal pieces. Splitting 3 to 5 becomes just as easy once the 3 and the 5 are turned into pieces of one size, because equal pieces can be counted out. A tape diagram is the picture that does this.

Definition

Day 1: the tape and its blocks

A tape diagram for a part-to-part ratio draws one strip, or tape, for each quantity in the ratio, one under the other with their left ends lined up. Each strip is cut into blocks, as many as that quantity’s number in the ratio, and every block in every strip is the same size. For Fenna and Maeve’s 3 : 5, Fenna’s strip has 3 blocks and Maeve’s has 5 (Figure 1). The order of the words in the ratio (What a ratio is: language and notation) says which strip gets which number.

\text{Fenna}
\text{Maeve}
\text{whole harvest}
Figure 1The ratio 3 : 5 as a tape. Every block, in either strip, is the same size.

Because the blocks are all the same size, each one stands for the same amount, whatever that amount turns out to be. Two more counts come straight from the picture:

  • the whole tape is all the blocks together, 3 + 5 = 8 blocks, and stands for the total;
  • the difference is the blocks one strip has beyond the other, 5 - 3 = 2 blocks.

Sharing an amount in a given ratio means splitting a total into shares whose ratio is the given one. On Day 1 the amount you know is the total or one of the shares.

Day 2: differences and three shares

The same blocks work when the amount you know is the difference between two shares (“Maeve got 14 kg more”). A ratio with three numbers, such as 3 : 5 : 7, gets three strips, and the whole tape is 3 + 5 + 7 = 15 blocks. In every case the thinking is the same: see which blocks the known amount covers, share it equally among those blocks, then count blocks.

Intuition

A tape diagram turns a ratio into a count of equal pieces. Sharing 40 kg in the ratio 3 : 5 sounds awkward; sharing 40 kg into 8 equal blocks is plain division, and then Fenna takes 3 of the blocks and Maeve takes 5. However big a block turns out to be, Fenna always has 3 of them to Maeve’s 5, so the shares are always in the ratio 3 : 5.

The tape is also a picture of fractions. Fenna holds 3 of the 8 equal blocks, so her share is \displaystyle \frac{3}{8} of the harvest and Maeve’s is \displaystyle \frac{5}{8}. Finding a share is then finding a fraction of a whole number.

Tape, table or double number line? A ratio table and a The double number line hold the same pairs of shares as a tape. Their columns are the two shares, though, so a total or a difference has to be worked out from each pair. On a tape the total and the difference are blocks you can see and count, which makes it the quickest picture when a problem gives a total or a “how many more”. When the problem gives one share and asks for the other, any of the three works. When the two quantities are different kinds, such as lanterns and meters, a total means nothing, and the double number line is the natural picture.

Results

Every share is a whole number of equal blocks, so whichever amount is known gives the value of one block. For Fenna and Maeve’s 3 : 5:

Amount you knowBlocks it coversOne block is
the totalall 8the total divided by 8
Fenna’s share3her share divided by 3
Maeve’s share5her share divided by 5
the difference2the difference divided by 2

Each share is then its number of blocks times one block, and each share is the fraction “its blocks over all the blocks” of the total: \displaystyle \frac{3}{8} for Fenna, \displaystyle \frac{5}{8} for Maeve.

Why it is true

Shares in the ratio 3 : 5 form a ratio equivalent to 3 : 5, which means they are 3 and 5 both multiplied by the same number (Scaling a ratio up or down). Call that number the value of one block. Fenna’s share is 3 times it, which is 3 blocks, and Maeve’s is 5 times it, which is 5 blocks. So the tape draws exactly what the ratio says.

The total is 3 blocks plus 5 blocks. Adding 3 copies of an amount to 5 copies of the same amount gives 3 + 5 = 8 copies; this is the distributive property. In the same way 5 blocks take away 3 blocks leaves 2 blocks. So the total is 8 times one block and the difference is 2 times one block, and dividing a known amount by its number of blocks undoes that multiplication and gives back the block.

There is only one answer. A different block value would change the total, so once the total (or a share, or the difference) is fixed, the block is fixed, and with it every share.

Worked examples

Examples 1 to 3 go with the first day, 4 to 7 with the second.

Example 1

Sharing the potatoes

Fenna and Maeve split their harvest in the ratio 3 : 5, Fenna to Maeve. They lift 40 kg of potatoes. How much does each take?

Solution

Draw Fenna’s strip with 3 blocks and Maeve’s with 5, all the same size (Figure 1). The 40 kg is the whole harvest, so it fills the whole tape, all 8 blocks. Equal blocks share it equally:

40 \div 8 = 5 \text{ kg in each block}
5
5
5
\text{Fenna}
5
5
5
5
5
\text{Maeve}
\text{40 kg}
Figure 240 kg fills all 8 blocks, so each block is 5 kg.

Fenna’s 3 blocks hold 3 \times 5 = 15 kg and Maeve’s 5 blocks hold 5 \times 5 = 25 kg.

Check. 15 + 25 = 40, and 15 : 25 is 3 : 5 with both numbers multiplied by 5.

Example 2

Prize money from one share

Quinn and Thea write a story together and win a prize. Quinn wrote more of it, so they split the prize in the ratio 7 : 4, Quinn to Thea. Quinn’s share is 56 dollars. How much does Thea get, and how big was the prize?

Solution

Quinn’s strip has 7 blocks and Thea’s has 4. This time the known amount is Quinn’s share, so it fills only his 7 blocks:

56 \div 7 = 8 \text{ dollars in each block}

Thea’s 4 blocks hold 4 \times 8 = 32 dollars. The prize is the whole tape, 7 + 4 = 11 blocks, so it was 11 \times 8 = 88 dollars, which agrees with 56 + 32 (Figure 3).

8
8
8
8
8
8
8
\text{Quinn}
8
8
8
8
\text{Thea}
\text{56 dollars}
\text{88 dollars}
Figure 3Quinn’s 56 dollars fills his 7 blocks, so each block is 8 dollars.

Check. 56 : 32 is 7 : 4 with both numbers multiplied by 8.

Example 3

Honey as a fraction of the jars

Zeno and Annika keep bees together. Annika owns more of the hives, so they share the honey jars in the ratio 2 : 9, Zeno to Annika. The season fills 77 jars. What fraction of the jars is Zeno’s, and how many jars does each get?

Solution

The tape has 2 + 9 = 11 equal blocks, and Zeno holds 2 of them, so his share is \displaystyle \frac{2}{11} of the jars and Annika’s is \displaystyle \frac{9}{11}. One block is \displaystyle \frac{1}{11} of the jars:

77 \div 11 = 7 \text{ jars}

So \displaystyle \frac{2}{11} of 77 is 2 \times 7 = 14 jars for Zeno, and \displaystyle \frac{9}{11} of 77 is 9 \times 7 = 63 jars for Annika. The tape and the fraction are two ways of saying the same thing: one block is one eleventh of the total.

Check. 14 + 63 = 77, and 14 : 63 is 2 : 9 with both numbers multiplied by 7.

Example 4

How many more

Fenna and Maeve still share in the ratio 3 : 5. This week Maeve’s share of the courgettes is 14 kg more than Fenna’s. How much does each get, and how much did they pick?

Solution

With the strips lined up on the left, the first 3 blocks of each match, and Maeve’s last 2 blocks are what she has beyond Fenna (Figure 4). Those 2 blocks are the 14 kg:

\text{Fenna}
\text{Maeve}
\text{14 kg more}
Figure 4The first 3 blocks of each strip match. The shaded 2 blocks are what Maeve has beyond Fenna.
14 \div 2 = 7 \text{ kg in each block}

Fenna gets 3 \times 7 = 21 kg and Maeve gets 5 \times 7 = 35 kg. They picked 8 \times 7 = 56 kg in all. The total was never given: the difference alone fixed the size of a block, and the block fixes everything else.

Check. 35 - 21 = 14, and 21 : 35 is 3 : 5 with both numbers multiplied by 7.

Example 5

Three shares

Quinn, Thea and Zeno win 120 dollars in a quiz and split it in the ratio 3 : 5 : 7, the number of rounds each of them answered. How much does each get?

Solution

Three people get three strips, of 3, 5 and 7 blocks (Figure 5). The prize fills the whole tape, 3 + 5 + 7 = 15 blocks:

120 \div 15 = 8 \text{ dollars in each block}
8
8
8
\text{Quinn}
8
8
8
8
8
\text{Thea}
8
8
8
8
8
8
8
\text{Zeno}
\text{120 dollars}
Figure 5A three-part ratio gets three strips: 3 + 5 + 7 = 15 blocks of 8 dollars.

Quinn gets 3 \times 8 = 24 dollars, Thea 5 \times 8 = 40 dollars and Zeno 7 \times 8 = 56 dollars.

Check. 24 + 40 + 56 = 120. Every pair keeps its part of the ratio: Quinn to Thea is 24 : 40, which is 3 : 5 multiplied by 8.

Example 6

A block that is not a whole number

Annika and Fenna enter a jar of honey and a basket of vegetables together at the village show and win 50 dollars, which they split in the ratio 3 : 5, Annika to Fenna. How much does each get?

Solution

The tape has 3 + 5 = 8 blocks, and 50 does not share into 8 whole dollars. It does not have to: money can be split into cents, and dividing gives a decimal block.

50 \div 8 = 6.25 \text{ dollars in each block}

Annika gets 3 \times 6.25 = 18.75 dollars and Fenna gets 5 \times 6.25 = 31.25 dollars.

Check. 18.75 + 31.25 = 50. Jars of honey cannot be split this way; see Boundaries.

Example 7

Evening up the honey

Zeno and Annika share 66 jars in the ratio 2 : 9. Afterwards Annika decides they should have the same number of jars. How many jars must she hand to Zeno?

Solution

The tape has 11 blocks, so one block is 66 \div 11 = 6 jars. Zeno has 2 \times 6 = 12 jars and Annika 9 \times 6 = 54.

Handing jars over moves them from one strip to the other but does not change the whole tape: still 11 blocks. Equal shares mean half of that each, 5.5 blocks, marked by the dashed line in Figure 6. Zeno has 2 blocks, so he needs 5.5 - 2 = 3.5 more, and they come off Annika’s 9, leaving her 5.5 too.

6
6
\text{Zeno}
6
6
6
6
6
6
6
6
\text{Annika}
\text{33 each}
\text{21 jars}
Figure 6Equal shares stop at the dashed line, 5.5 blocks each. Annika’s shaded 3.5 blocks move up to fill Zeno’s dashed ones.
3.5 \times 6 = 21 \text{ jars}

Annika hands over 21 jars. Notice that the answer was 3.5 blocks before the total was used; the total only says how much a block is worth.

Check. Zeno ends with 12 + 21 = 33 jars and Annika with 54 - 21 = 33, and 33 is half of 66.

Boundaries

The ratio must compare the shares with each other. “Fenna gets 3 kg of every 5 kg” is a part-to-whole ratio: the whole tape is 5 blocks, Fenna has 3 and Maeve 2. Only a part-to-part ratio puts its numbers straight onto the strips.

Totals and differences need one kind of quantity. Kilograms add to kilograms, so a tape can show the total harvest. With 3 lanterns for every 10 meters of path, the blocks in the two strips would stand for different things, and “lanterns plus meters” means nothing. The double number line fits that kind of ratio.

Whole items need a block that gives whole shares. Money can be shared to the cent and potatoes to any weight, but jars come whole. Sharing 2 : 9 works out in whole jars for 77 or 66 jars, and fails for some totals, such as the 100 jars of Exercise 22.

Large numbers in the ratio. For 23 : 41, drawing 64 blocks is slow. Draw two long strips labeled “23 blocks” and “41 blocks”; the reasoning about one block is unchanged.

Where this goes next

Comparing two ratios asks which of two splits gives a bigger share, and the fractions a tape shows, such as Fenna’s \displaystyle \frac{3}{8}, are one way to decide. The same splitting done on a line segment, so that a point cuts it into two pieces in a given ratio, is Partitioning a segment in a given ratio, where the equal blocks become equal steps along the segment. A tape cut into 100 equal blocks is a picture of a percent (What percent means: a rate per hundred). And the move at the heart of every example here, finding what one block is worth, grows into the unit rate.

Exercises

Day 1: sharing a total, or working from one share

  1. Draw a tape diagram for Fenna and Maeve’s 3 : 5. How many blocks are there in all, and what fraction of every harvest is Fenna’s?
  2. Fenna and Maeve (3 : 5) pick 64 kg of apples. How much does each take?
  3. Quinn and Thea (7 : 4, Quinn to Thea) win a prize of 132 dollars. How much does each get?
  4. Zeno and Annika (2 : 9, Zeno to Annika) share their jars, and Zeno receives 16 jars. How many jars does Annika get, and how many were there in all?
  5. In another 7 : 4 split, Thea receives 28 dollars. How much does Quinn get, and what was the prize?
  6. Maeve and Annika share the potatoes from a second plot in the ratio 1 : 5, Maeve to Annika. The plot gives 60 kg. How much does each take?
  7. A prize of 121 dollars is split 7 : 4, Quinn to Thea. What is one block worth, what fraction of the prize is Thea’s, and how much does she get?
  8. Zeno and Annika (2 : 9) fill 132 jars. What fraction of the jars is Zeno’s, and how many jars does each get?
  9. Fenna and Maeve (3 : 5) share some beans, and Maeve’s share is 30 kg. How much does Fenna get, and how much was picked?
  10. Quinn says: “Thea and I split 55 dollars 7 : 4, and I got 30 dollars.” Can that be right?

Day 2: differences, three shares and decimal blocks

  1. Fenna and Maeve (3 : 5) share the onions, and Maeve gets 18 kg more than Fenna. How much does each get, and what is the total?
  2. In a 7 : 4 split, Quinn gets 27 dollars more than Thea. How much does each get, and what was the prize?
  3. Zeno and Annika (2 : 9) share their jars, and Annika gets 35 more jars than Zeno. How many does each get, and how many were there?
  4. Fenna, Maeve and Thea share 90 kg of onions in the ratio 3 : 5 : 2. How much does each take?
  5. Quinn, Thea and Zeno split a 200 dollar prize in the ratio 7 : 4 : 5. How much does each get?
  6. Zeno, Annika and Maeve share honey jars in the ratio 2 : 9 : 4, and Maeve receives 24 jars. How many jars does each of the others get, and how many were there?
  7. Fenna, Maeve and Quinn share a harvest in the ratio 1 : 3 : 4, and Quinn gets 15 kg more than Fenna. How much does each get, and what is the total?
  8. Annika and Fenna split a 30 dollar prize in the ratio 3 : 5. How much does each get?
  9. Zeno and Annika share 44 jars in the ratio 2 : 9. How many jars must Annika then hand to Zeno for them to have the same number?
  10. Fenna and Maeve share 48 kg of potatoes in the ratio 3 : 5. How much must Maeve then hand to Fenna for them to have the same amount?
  11. Quinn and Thea split a prize 7 : 4. After Quinn spends 15 dollars of his share, the two of them have the same amount. How big was the prize, and how much was Thea’s share?

Reasoning

  1. Zeno says that he and Annika shared 100 jars exactly in the ratio 2 : 9. Explain why that cannot be true.
  2. Explain why every block in both strips of a tape diagram must be the same size. What would go wrong if Fenna’s blocks were drawn twice as long as Maeve’s?
  3. Explain why, in every 3 : 5 share, the difference between the two shares is one quarter of the total.
  4. Quinn, Thea and Zeno split a prize 3 : 5 : 7. Zeno says he gets “almost half”. What fraction of the prize does he get exactly, and is it more or less than half?
Answers
  1. 8 blocks; \displaystyle \frac{3}{8}. Fenna’s strip has 3 blocks and Maeve’s 5, and Fenna holds 3 of the 3 + 5 = 8 equal blocks.
  2. Fenna 24 kg, Maeve 40 kg. One block is 64 \div 8 = 8 kg; 3 \times 8 and 5 \times 8.
  3. Quinn 84 dollars, Thea 48 dollars. One block is 132 \div 11 = 12 dollars; 7 \times 12 and 4 \times 12.
  4. Annika 72 jars; 88 jars in all. Zeno’s 16 jars fill 2 blocks, so a block is 8 jars; 9 \times 8 = 72 and 11 \times 8 = 88.
  5. Quinn 49 dollars; the prize was 77 dollars. Thea’s 28 dollars fill 4 blocks, so a block is 7 dollars; 7 \times 7 = 49 and 11 \times 7 = 77.
  6. Maeve 10 kg, Annika 50 kg. The tape has 1 + 5 = 6 blocks of 60 \div 6 = 10 kg.
  7. 11 dollars; \displaystyle \frac{4}{11}; 44 dollars. One block is 121 \div 11 = 11, and Thea holds 4 of the 11 blocks: 4 \times 11 = 44.
  8. \displaystyle \frac{2}{11}; Zeno 24 jars, Annika 108 jars. One eleventh of 132 is 12; 2 \times 12 and 9 \times 12.
  9. Fenna 18 kg; 48 kg picked. Maeve’s 30 kg fill 5 blocks, so a block is 6 kg; 3 \times 6 and 8 \times 6.
  10. No. One block is 55 \div 11 = 5 dollars, so Quinn’s 7 blocks are 35 dollars and Thea’s 4 are 20 dollars, not 30 and 25.
  11. Fenna 27 kg, Maeve 45 kg; total 72 kg. The 2 extra blocks are 18 kg, so a block is 9 kg; 3 \times 9, 5 \times 9, 8 \times 9.
  12. Quinn 63 dollars, Thea 36 dollars; prize 99 dollars. Quinn has 7 - 4 = 3 extra blocks, worth 27, so a block is 9 dollars.
  13. Zeno 10 jars, Annika 45 jars; 55 jars in all. Annika has 9 - 2 = 7 extra blocks, worth 35, so a block is 5 jars.
  14. Fenna 27 kg, Maeve 45 kg, Thea 18 kg. The tape has 3 + 5 + 2 = 10 blocks of 90 \div 10 = 9 kg.
  15. Quinn 87.50 dollars, Thea 50 dollars, Zeno 62.50 dollars. The tape has 7 + 4 + 5 = 16 blocks of 200 \div 16 = 12.5 dollars.
  16. Zeno 12 jars, Annika 54 jars; 90 jars in all. Maeve’s 24 jars fill 4 blocks, so a block is 6 jars; 2 \times 6, 9 \times 6, 15 \times 6.
  17. Fenna 5 kg, Maeve 15 kg, Quinn 20 kg; total 40 kg. Quinn has 4 - 1 = 3 blocks more than Fenna, worth 15 kg, so a block is 5 kg.
  18. Annika 11.25 dollars, Fenna 18.75 dollars. One block is 30 \div 8 = 3.75 dollars; 3 \times 3.75 and 5 \times 3.75.
  19. 14 jars. A block is 44 \div 11 = 4 jars, so Zeno has 8 and Annika 36. Equal shares are 5.5 blocks each, so Annika hands over 9 - 5.5 = 3.5 blocks: 3.5 \times 4 = 14. Both end with 22.
  20. 6 kg. A block is 48 \div 8 = 6 kg, so Fenna has 18 kg and Maeve 30 kg. Equal shares are 4 blocks each, so Maeve hands over 1 block, 6 kg. Both end with 24 kg.
  21. The prize was 55 dollars; Thea’s share was 20 dollars. Quinn’s spending removes exactly his 7 - 4 = 3 extra blocks, so 3 blocks are 15 dollars and a block is 5; 11 \times 5 = 55 and 4 \times 5 = 20.
  22. The tape has 11 blocks, so each would be \displaystyle 100 \div 11 = 9\frac{1}{11} jars, and Zeno’s 2 blocks would be \displaystyle 18\frac{2}{11} jars. Jars come whole, so no exact 2 : 9 share of 100 jars exists. Whole shares need a total that 11 divides, such as 99 or 110.
  23. The tape works by giving every block one value, so that counting blocks counts amounts. If Fenna’s blocks were twice as long, her 3 blocks would really hold as much as 6 of Maeve’s, and the picture would show 6 : 5, not 3 : 5. Sharing 40 kg into 8 blocks would then give Fenna and Maeve amounts that are not in the ratio 3 : 5.
  24. The total is 8 blocks and the difference is 5 - 3 = 2 blocks, whatever a block is worth. Two blocks out of eight is \displaystyle \frac{2}{8} = \frac{1}{4} of the total.
  25. \displaystyle \frac{7}{15}, a little less than half. Zeno holds 7 of the 3 + 5 + 7 = 15 blocks, and half of the tape would be 7.5 blocks.
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