Ratio tables

When you reach for it

Beatrix is hanging wallpaper. The packet says to mix 3 scoops of paste powder with 8 liters of water. One wall needs a small batch and the whole room a large one, and every batch must be the same mix. Rather than work each batch out on its own, she lists them:

Every row equals 3 : 8. A table like this is a ratio table: one column for each quantity, and every row the same ratio scaled by some number. Reach for one when a comparison stays fixed and you want many amounts at once, when an amount is awkward to reach in one step, or to test whether a list of pairs keeps one ratio.

Why scaling both parts keeps a ratio the same is scaling a ratio up or down; one missing number in a pair of equal ratios is finding a missing value. A ratio table lines many such pairs up, so the rows can help each other. Part 1 (day one) builds tables and reads their patterns; Part 2 (day two) fills gaps and tests tables.

The procedure

Part 1: building rows by scaling

Every row of a ratio table is the first row multiplied by one number, the same number in both columns. For 9 scoops, from 3 to 9 is \times 3, so the water is 8 \times 3 = 24 liters. The number may be less than 1: half a batch is \displaystyle \times \frac{1}{2}, which gives 1.5 scoops and 4 liters.

A new row may come from any row already in the table, not only the first, because every row is a scaled copy of every other. From the 12 row, a row for 36 scoops is \times 3, so the water is 32 \times 3 = 96 liters.

The headers give each column’s quantity and unit, in the order of the ratio (ratio language): scoops to liters. The rows may come in any order.

Part 1: adding two rows

Two batches poured into one bucket make a bigger batch of the same mix. So adding two rows, column by column, gives another row of the table. The 6 row and the 9 row give

This is allowed because the 6 row is 3 : 8 times 2 and the 9 row is 3 : 8 times 3. By the distributive property,

so the new row is 3 : 8 times 2 + 3 = 5, a row of the same table. Taking a smaller row away from a larger one works the same way: 15 : 40 take away 6 : 16 leaves 9 : 24.

Adding rows is often the quickest road to an awkward amount: 21 = 9 + 12, so 21 scoops take 24 + 32 = 56 liters. It also explains why, in Beatrix’s table, the scoops go up by 3 and the water by 8 each time: each row is the one above it with the first row added on.

Part 1: the patterns down and across

Down. Moving from one row to another multiplies both columns by the same number. From the 3 row to the 12 row the scoops go \times 4, and so does the water: 8 \times 4 = 32. This scaling number changes from one pair of rows to another, but for any two rows it is the same in both columns.

Across. In every row, the water is the same multiple of the powder. In the first row it is \displaystyle 8 \div 3 = \frac{8}{3} = 2\frac{2}{3}. Scaling a row cannot change that multiple: since \displaystyle 3 \times \frac{8}{3} = 8, multiplying 3 and 8 by the same number keeps the second \displaystyle \frac{8}{3} times the first. So \displaystyle 6 \times \frac{8}{3} = 16, \displaystyle 15 \times \frac{8}{3} = 40, and so on down the table (Figure 1).

The across number turns any amount in the first column into its partner in one step. When tables like this are studied as rules, it is called the constant of proportionality.

Part 2: going through a row of one

Lior hires an electric scooter at 13 dollars for every 4 hours. What do 3 hours cost? From 4 to 3 is not a whole scaling number, so first divide both columns by 4 to reach the row with 1 in it, then multiply by 3:

One hour costs 13 \div 4 = 3.25 dollars, and 3 hours cost 3 \times 3.25 = 9.75 dollars. From the row of 1, every row is one multiplication away. Its partner, 3.25, is the across number: every cost is 3.25 times its hours. A cost for one hour is a unit rate, which has a lesson of its own.

The 1 can sit in either column. One dollar buys \displaystyle 4 \div 13 = \frac{4}{13} of an hour, the row \displaystyle \frac{4}{13} : 1.

Part 2: filling a gap

A gap sits in a row whose other number is known. Matching that number with the same column of a full row gives the scaling number, and the scaling number gives the gap. When it is awkward, add or take away rows, go through the row of 1, or use the across number. Every road builds a row of the same table, so all land on the same value; take the friendliest.

Part 2: is it a ratio table at all?

A table of pairs is a ratio table only when every row is the same ratio. The quickest test is the across number in every row. If it is the same in all of them, the table is a ratio table; if one row differs, that row is not a scaled copy of the others.

Part 2: the rows as points on a grid

Kwame plants a hedge at 5 saplings for every 2 meters. Four rows of his ratio table are 5 : 2, 10 : 4, 15 : 6 and 20 : 8. Plot each row as a point on a coordinate grid, the first column across and the second up (Figure 2).

The points lie on one straight line that runs back to the corner (0, 0), itself a row: the first row times 0. Every ratio table with numbers above zero plots this way; Example 6 has a table whose points miss the corner.

Worked examples

Boundaries

Every row must be the same ratio. A row is made by multiplying both numbers by one number, or by adding or taking away whole rows. Adding the same amount to both columns does not make a row: from 3 : 8, adding 1 to each part gives 4 : 9, a different mix. Nor does multiplying two rows together.

The columns keep their order and their units. The first number of every row counts the same thing in the same unit; rows may be listed in any order, but columns cannot swap in some rows.

The first row needs numbers above zero. The row 0 : 0 belongs to every ratio table, but no scaling number leads away from it, so it cannot be the row a table is built from.

The setting may not allow every row. The hedge table has the row 2.5 : 1, two and a half saplings for one meter. It is a true row, but nobody plants half a sapling, so it is a rate to work with rather than a planting plan.

Not every rule makes a ratio table. A price with a fixed charge on top, like Emeka’s shop in Example 6, does not.

Where this goes next

The row of 1 starts unit rates, and the across number becomes the constant of proportionality. A double number line shows the same pairs as matching marks on two lines, and a tape diagram handles a ratio when the total is known. Two rows written as one statement are a proportion, and solving a proportion finds a gap when no friendly row is in sight.

Exercises

Part 1: building rows by scaling

1. Complete the ratio table of saplings to meters that starts 5 : 2, with the rows 25 : \,?, 55 : \,? and 100 : \,?.
2. Complete the ratio table of hours to cost in dollars that starts 4 : 13, with the rows 12 : \,?, 40 : \,? and 2 : \,?.
3. Complete the ratio table that starts 7 : 2, with the rows 14 : \,?, ? : 10 and 70 : \,?.
4. Complete the ratio table that starts 14 : 35, with the rows 2 : \,?, 6 : \,? and 20 : \,?.

Part 1: adding two rows

5. A table of saplings to meters has the rows 5 : 2 and 15 : 6. Using only adding or taking away rows, find the meters for 20, 10 and 40 saplings.
6. A table of hours to cost in dollars has the rows 4 : 13 and 6 : 19.5. By adding rows, find the cost of 10 hours and of 16 hours.
7. A table of scoops to liters has the rows 3 : 8 and 24 : 64. Using only these rows, find the liters for 39 scoops and for 45 scoops.

Part 1: patterns down and across

8. A ratio table has the rows 6 : 21, 10 : 35, 14 : 49 and 22 : 77. (a) Find the across number, second divided by first. (b) Find the scaling number from the 6 row to the 22 row, and show it is the same in both columns.
9. In Beatrix’s table of scoops to liters, first row 3 : 8, what is the across number? Use it to find the liters for 18 scoops and for 7.5 scoops.
10. In Kwame’s hedge, 5 saplings for every 2 meters, how many meters does one sapling fill? How many meters do 60 saplings fill?

Part 2: through a row of one

11. At 13 dollars for every 4 hours, what do 5 hours and 9 hours of scooter hire cost?
12. At 5 saplings for every 2 meters, find the saplings for 1 meter, 16 meters and 26 meters.
13. A ratio table starts 6 : 10. Find the row with 1 in the first column, then the row 9 : \,?. Then find the row with 1 in the second column.

Part 2: filling gaps

14. Fill the gaps in the ratio table 8 : 6, 4 : \,?, ? : 15, 28 : \,?.
15. Fill the gaps in the ratio table of scoops to liters 3 : 8, ? : 28, 0.75 : \,?, ? : 60.
16. Fill the gaps in the ratio table of hours to cost in dollars 4 : 13, ? : 52, 13 : \,?, ? : 55.25.

Part 2: is it a ratio table?

17. Hours to cost in dollars: 2 : 7, 3 : 10, 5 : 16. Is this a ratio table?
18. Is 4 : 6, 6 : 9, 10 : 15, 16 : 24 a ratio table? If so, give its across number.
19. Saplings to meters: 5 : 2, 10 : 4, 15 : 7. Is this a ratio table? If not, which row breaks it, and what should that row be?

Part 2: in context

20. Rosa mixes paste at 4 scoops to 11 liters. Is her paste thicker or thinner than the packet’s 3 scoops to 8 liters?
21. One shop hires scooters at 13 dollars for every 4 hours, another at 20 dollars for every 6 hours. Which is cheaper for 12 hours, and by how much?
22. The rows 3 : 8 and 6 : 16 are plotted as the points (3, 8) and (6, 16). Does the point (4.5, 12) lie on the same line through (0, 0)? Does (9, 25)?

Reasoning

23. Explain why adding two rows of a ratio table always gives another row of the same table. Use the rows 6 : 16 and 9 : 24 of the table that starts 3 : 8.
24. From the row 3 : 8 to the row 6 : 16, the scoops go up by 3 and the liters by 8, yet both columns are multiplied by 2. Explain why the scaling number is the same in both columns while the amount added is not.
25. Can one ratio table contain both the rows 3 : 8 and 8 : 3? Explain.