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The double number line

A map says that 2 centimeters stand for 7 kilometers. How can you see at once what 5 centimeters stands for, or which distance goes with any other?

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Why this exists

Farah’s hiking map has a note in the corner: 2 cm on the map stand for 7 km on the ground. A trail on the map measures 4 cm, twice as long, so it is 14 km. That much is scaling a ratio. But the next trail measures 5 cm, and 5 is not a whole number of 2s.

Map distances and ground distances are both lengths, and lengths live on a number line. So draw two number lines, one for each amount, and line them up so that every map distance sits straight above the ground distance it stands for. Then “what goes with 5 cm?” becomes a question about a point you can see, sitting between two marks you already know. That picture is a double number line.

Definition

Day 1: two lines that share zero

A double number line is two number lines drawn one above the other, each with its own label and unit, set up so that:

  • both lines start at 0, and the two zeros sit one above the other;
  • the first pair of marks shows one ratio (see What a ratio is: language and notation), its first amount on the top line straight above its second amount on the bottom line: in Figure 1, 2 cm above 7 km;
  • each line copies its own step again and again at equal spacing, so the next marks are two steps of each amount, then three steps, and so on: 4 above 14, then 6 above 21.

Two numbers that sit one above the other line up, and every pair that lines up is a ratio equivalent to the first pair.

0
0
2
7
4
14
6
21
8
28
10
35
\text{map (cm)}
\text{ground (km)}
Figure 1Farah’s map scale as a double number line. Each top mark sits straight above its partner: 2 cm with 7 km, 4 cm with 14 km, and so on.

In Figure 1 the step on the top line is 2 cm and the step on the bottom line is 7 km. The two steps take the same room on the page even though 7 is bigger than 2: each line has its own scale. What matters is that one step on the top line always matches one step on the bottom line.

Day 2: splitting a gap into equal parts

A double number line can also be cut finer. Splitting the gap from 0 to the first top mark into equal parts splits the gap from 0 to the first bottom mark into the same number of equal parts, because the two gaps are the same stretch of the diagram. This is partitioning, and the smaller steps it makes reach values that no whole number of the first step would hit, such as 5 cm on Farah’s map.

Intuition

Picture walking along the diagram from left to right. On Farah’s map each step carries you 2 cm further along the top line and 7 km further along the bottom line, at the same time. After 3 steps you have gone 6 cm and 21 km together. Stop halfway through a step and you have gone half of each step more: 1 cm and 3.5 km. Every point on the top line has a partner directly below it, which is why the diagram can answer questions that land between marks.

A double number line and a ratio table hold the same pairs: each column of marks is one row of a table. What the line adds is the picture.

  • Size and order show. A value that falls between two marks is seen there, so you can estimate before you work anything out.
  • In-between values are natural. A table lists only the rows you wrote; the line has a point for every amount.
  • The quantities are already lines. Distance, time and length of path really do run along a line, so the diagram looks like the situation.

A table is the better tool when the numbers are far apart (10 m and 1000 m cannot share a sensible line) or when you need many pairs in no particular order.

Results

Pairs that line up are equivalent ratios. Any whole number of steps along, the top mark is the first top amount times that number, and the bottom mark is the first bottom amount times the same number:

2 : 7 = 4 : 14 = 6 : 21 = \cdots

Splitting the first gap into equal parts gives a smaller step that lines up too. Both amounts of the first pair are divided by the number of parts. Cutting Farah’s first gap into 2 parts:

2 : 7 = (2 \div 2) : (7 \div 2) = 1 : 3.5

Halfway lines up with halfway. The point halfway between two marks on one line sits over the point halfway between their partners on the other line.

Why it is true

The mark 3 steps along shows 3 copies of the first pair, and the same holds for any whole number of steps. Taking the same number of copies of both amounts is multiplying both by that number, and multiplying both parts of a ratio by the same number gives an equivalent ratio. That is the rule of Scaling a ratio up or down.

For splitting, cut the gap from 0 to the first top mark into equal parts, say 4 of them, so each part is the top amount divided by 4. Directly below, the same stretch of the diagram runs from 0 to the first bottom mark, cut into the same 4 parts, each the bottom amount divided by 4. Taking 4 copies of the small pair gives back the first pair. So the small pair is the first pair scaled down, and it is equivalent to it. Any number of parts works the same way.

Halfway works the same way. On Farah’s map, the point halfway between the marks 2 and 3 steps along is 2 steps plus half a step. On the top line that is 2 \times 2 + 1 = 5 cm, and below it is 2 \times 7 + 3.5 = 17.5 km: two copies of the pair plus half a copy, on both lines at once.

Worked examples

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

Example 1

Building a map line

Farah’s map uses 2 cm on the map for 7 km on the ground. Build a double number line out to 10 cm. A road on the map measures 8 cm: how long is it on the ground? A town is 35 km away: how far is that on the map?

Solution

Draw the two lines with their zeros together, since no distance on the map stands for no distance on the ground. Put 2 on the top line above 7 on the bottom line, then copy each step: 4 above 14, 6 above 21, 8 above 28, 10 above 35, as in Figure 1. Each column is a whole number of copies of 2 : 7, so each is equivalent to it.

The road: 8 cm is 4 steps, so read straight down to 28 km. The town: 35 km is 5 steps on the bottom line, and the mark above it reads 10 cm.

Example 2

Reading between the marks

On Farah’s map, how far on the ground are 1 cm, 3 cm and 5 cm?

Solution

None of these sits on a mark of Figure 1, but each sits halfway between two marks. 1 cm is halfway from 0 to 2, so its partner is halfway from 0 to 7:

1 \text{ cm lines up with } 3.5 \text{ km}

3 cm is halfway from 2 to 4, so it lines up with halfway from 7 to 14, which is 10.5 km. 5 cm is halfway from 4 to 6, so it lines up with halfway from 14 to 21, which is 17.5 km. Figure 2 shows all the halfway marks.

0
0
1
3.5
2
7
3
10.5
4
14
5
17.5
6
21
\text{map (cm)}
\text{ground (km)}
Figure 2Halfway marks on the top line sit over halfway marks on the bottom line. The lighter ticks are the new ones.

Check. 1 : 3.5 is 2 : 7 with both parts halved, so it belongs on the line.

Example 3

Which pairs belong

Henrik hangs lanterns along a festival path, 3 lanterns for every 10 meters (Figure 3). Which of these pairs line up: 9 lanterns and 30 m, 12 lanterns and 36 m, 15 lanterns and 50 m? How many lanterns does a 100 m path need?

0
0
3
10
6
20
9
30
12
40
15
50
\text{lanterns}
\text{path (m)}
Figure 33 lanterns for every 10 meters of path.
Solution

9 lanterns is 3 steps along the top line, and 3 steps along the bottom line is 30 m. They line up.

12 lanterns is 4 steps, which lines up with 40 m. The point 36 m sits between the 30 m and 40 m marks, so it is not under the 12 lanterns mark: 12 and 36 do not line up, and 12 : 36 is not equivalent to 3 : 10.

15 lanterns is 5 steps, and 5 steps of 10 m is 50 m. They line up.

For the path, count steps on the bottom line. 100 m is 10 steps of 10 m, so the top line has gone 10 steps of 3 lanterns: the path needs 30 lanterns.

Example 4

Halving the first step

Paloma jogs at a steady pace and covers 3 km in 20 minutes. How long does she take to jog 4.5 km?

Solution

Put minutes on the top line and kilometers below, with 20 above 3. Copies of 3 km give 6, 9 and so on, which jump straight past 4.5. So split the first gap: cutting 0 to 3 km in half cuts the same stretch of 0 to 20 minutes in half, and 1.5 km lines up with 10 minutes. That smaller step fits, because 4.5 is 3 steps of 1.5. Three steps on the top line is

3 \times 10 = 30 \text{ minutes}
0
0
10
1.5
20
3
30
4.5
\text{time (min)}
\text{distance (km)}
Figure 4Halving the gap from 0 to 20 minutes halves the gap from 0 to 3 km too, and the smaller step reaches 4.5 km.

Paloma takes 30 minutes. Check. 30 : 4.5 is 20 : 3 with both parts multiplied by 1.5.

Example 5

Down to one centimeter

Gideon’s map uses 4 cm on the map for 5 km on the ground. How far on the ground is a lake that sits 7 cm from the bridge on the map? How long on the map is a 15 km trail?

Solution

7 is not a whole number of 4s, but every whole number is made of 1s. Cut the first gap into 4 equal parts: 4 cm becomes four steps of 1 cm, and 5 km becomes four steps of 5 \div 4 = 1.25 km (Figure 5).

0
0
1
1.25
2
2.5
3
3.75
4
5
5
6.25
6
7.5
7
8.75
8
10
\text{map (cm)}
\text{ground (km)}
Figure 5Cutting the first 4 cm into four equal parts gives 1 cm for every 1.25 km.

Then 7 cm is 7 small steps, which lines up with 7 \times 1.25 = 8.75 km. A second route agrees: 8 cm lines up with 10 km, and stepping back one centimeter takes away 1.25 km, leaving 8.75 km.

The trail needs no splitting: 15 km is 3 steps of 5 km, so it lines up with 3 steps of 4 cm, which is 12 cm.

Example 6

Choosing the size of the step

Mila jogs 2.5 km in 15 minutes at a steady pace. How far does she get in 24 minutes?

Solution

24 is not a whole number of 15s, so split the first gap. Into how many parts? The small step has to fit into both 15 and 24 a whole number of times, and 3 does: 15 = 5 \times 3 and 24 = 8 \times 3. So cut 0 to 15 minutes into 5 parts of 3 minutes, and 0 to 2.5 km into 5 parts of 2.5 \div 5 = 0.5 km. Then 24 minutes is 8 small steps, which lines up with

8 \times 0.5 = 4 \text{ km}

Cutting all the way down to 1 minute would also work, but each minute would line up with 2.5 \div 15 of a kilometer, an awkward fraction. A step that fits both numbers keeps the arithmetic short.

Check. 24 = 1.6 \times 15 and 4 = 1.6 \times 2.5, so 24 : 4 is the same ratio as 15 : 2.5.

Example 7

Estimate first, then find it exactly

A trail on Farah’s map (2 cm for 7 km) measures 4.8 cm. Estimate the ground distance from the double number line, then find it exactly.

Solution

Estimate. In Figure 2, 4.8 cm sits between the 4 cm and 5 cm marks, very close to 5. Its partner sits between 14 km and 17.5 km, very close to 17.5. So the trail is a little under 17.5 km.

Exact. A step that fits 4.8 a whole number of times is 0.4 cm, and it fits 2 as well, since 2 = 5 \times 0.4. Cut the first gap into 5 parts, so 0.4 cm lines up with 7 \div 5 = 1.4 km. Then 4.8 = 12 \times 0.4, and the trail lines up with

12 \times 1.4 = 16.8 \text{ km}

The exact answer, 16.8 km, is a little under 17.5 km, as the estimate said. A table with rows for 4 cm and 6 cm holds the same pairs, but it does not show that 4.8 sits almost at 5. The line shows it before any arithmetic.

Boundaries

The two amounts must keep the same ratio all the way from zero. A double number line fits a fixed map scale, a steady pace, or lanterns hung evenly. A path that has one extra lantern at the gate before the even spacing begins does not fit: 0 m would go with 1 lantern, so the two zeros cannot line up.

Steady means steady. Paloma’s line assumes she keeps one pace for the whole run. If she slows on a hill, her real times leave the diagram.

A point between marks must make sense in the situation. The lantern line puts 7.5 lanterns over 25 m. The diagram is exact, but lanterns come whole. Distance and time can be split as finely as you like; counts cannot.

Read the numbers, not the ruler. A hand-drawn double number line is a sketch. Its marks are equally spaced in idea, and answers come from the numbers on the marks, not from measuring the page.

Far-apart values do not fit. With 3 lanterns for 10 m, a 5000 m path is 500 steps away. A table, or a single multiplication, is quicker than any line.

Two quantities at a time. A double number line relates exactly two amounts. A third, such as the cost of the lanterns, needs a third line sharing the same zero.

Where this goes next

The most useful mark on any double number line is the one that lines up with 1: Gideon’s 1 cm for 1.25 km, or Paloma’s 1 km for \displaystyle 6\frac{2}{3} minutes in Exercise 12. That pair is a rate for one unit, and Rates and unit rates makes it the main tool. When the bottom line runs from 0 to 100, a double number line becomes a picture of a percent, which is where What percent means: a rate per hundred starts. The same pairs written as rows make a ratio table, and the same scaling done with numbers alone is finding a missing value. When the two amounts are parts of one whole, the equal blocks of a tape diagram do the job instead. Map scales come back in Scale drawings.

Exercises

Day 1: reading and building

Use Figure 1 (Farah’s map, 2 cm for 7 km) for Exercises 1 to 3.

  1. How far on the ground is 6 cm on the map?
  2. How far on the map is a ground distance of 28 km?
  3. How far on the ground is 9 cm on the map?

Use Figure 3 (Henrik’s lanterns, 3 for every 10 m) for Exercises 4 to 6.

  1. Extend the lines to 18 lanterns. How many meters of path line up with 18 lanterns?
  2. How many lanterns line up with 70 m?
  3. What lines up with 25 m? Can a 25 m path be lit at exactly this ratio?
  4. Gideon’s map uses 4 cm for 5 km. Which of these pairs line up on its double number line: (a) 12 cm and 15 km, (b) 10 cm and 12 km, (c) 2 cm and 2.5 km?
  5. Ines jogs 1 km every 6 minutes. Build a double number line with kilometers from 0 to 5 on top and minutes below. How long does she take for 5 km, and for 2.5 km?
  6. A double number line has top marks 0, 4, 8, 12 and bottom marks 0, 5, 10, 15, equally spaced. What pair belongs at the next mark, and what pair lines up halfway between 8 and 12?
  7. Arjun’s map uses 3 cm for 10 km. How far on the ground are 12 cm and 1.5 cm?

Day 2: splitting to find a missing value

  1. Paloma jogs 3 km in 20 minutes (Example 4). How far does she jog in 50 minutes?
  2. How long does Paloma take to jog 1 km?
  3. On Gideon’s map (4 cm for 5 km), how far on the ground is 6 cm?
  4. On Gideon’s map, how far on the map is a ground distance of 20 km?
  5. On Gideon’s map, how far on the map is a ground distance of 3.75 km?
  6. On Farah’s map (2 cm for 7 km), how far on the ground is 0.5 cm?
  7. Henrik hangs 3 lanterns for every 10 m. How many lanterns does a 90 m path need, and how long is a path that uses 24 lanterns?
  8. Mila jogs 2.5 km in 15 minutes (Example 6). How far does she get in 9 minutes?
  9. How long does Mila take to jog 3.5 km?
  10. Tomas jogs 4 km in 26 minutes at a steady pace. How long does he take for 10 km?
  11. On Arjun’s map (3 cm for 10 km), how far on the ground is 4.5 cm?
  12. On Arjun’s map, how far on the map is a ground distance of 25 km?

Reasoning

  1. Explain why the two lines of a double number line must have their zeros one above the other.
  2. A double number line has top marks 0, 2, 4, 6 and bottom marks 0, 7, 14, 20, drawn equally spaced. Explain what is wrong and correct it.
  3. Which would you use, a double number line or a ratio table, to (a) estimate the ground distance of a road measuring 6.3 cm on Arjun’s map, and (b) list the lanterns needed for paths of 10 m, 250 m and 1000 m? Explain.
Answers
  1. 21 km. 6 cm is 3 steps of 2 cm, which lines up with 3 \times 7 = 21 km.
  2. 8 cm. 28 km is 4 steps of 7 km, which lines up with 4 \times 2 = 8 cm.
  3. 31.5 km. 9 cm is halfway from 8 to 10 cm, so it lines up with halfway from 28 to 35 km.
  4. 60 m. 18 lanterns is 6 steps of 3, which lines up with 6 \times 10 = 60 m.
  5. 21 lanterns. 70 m is 7 steps of 10 m, and 7 \times 3 = 21.
  6. 7.5 lanterns. 25 m is halfway from 20 to 30 m, so it lines up with halfway from 6 to 9 lanterns. Lanterns come whole, so a 25 m path cannot use exactly this ratio.
  7. (a) and (c). Pair (a) is 3 steps on both lines, and pair (c) is half a step on both lines. Pair (b) does not line up: 10 cm is 2.5 steps, which lines up with 2.5 \times 5 = 12.5 km, not 12 km.
  8. 30 minutes and 15 minutes. The bottom marks are 0, 6, 12, 18, 24, 30, so 5 km lines up with 30 minutes. 2.5 km is halfway from 2 to 3 km, so it lines up with halfway from 12 to 18 minutes.
  9. 16 and 20; then 10 and 12.5. The next mark adds one more step of 4 and 5. Halfway from 8 to 12 is 10, and halfway from their partners 10 and 15 is 12.5.
  10. 40 km and 5 km. 12 cm is 4 steps of 3 cm, so 4 \times 10 = 40 km. 1.5 cm is half a step, so it lines up with half of 10 km.
  11. 7.5 km. 10 minutes lines up with 1.5 km (Example 4), and 50 minutes is 5 of those steps: 5 \times 1.5 = 7.5.
  12. \displaystyle 6\frac{2}{3} minutes. Cut 0 to 3 km into 3 equal parts; the same stretch cuts 20 minutes into 3 parts of \displaystyle 20 \div 3 = 6\frac{2}{3} minutes.
  13. 7.5 km. Halve the first step, so 2 cm lines up with 2.5 km. 6 cm is 3 of those: 3 \times 2.5 = 7.5.
  14. 16 cm. 20 km is 4 steps of 5 km, which lines up with 4 \times 4 = 16 cm.
  15. 3 cm. 1 cm lines up with 1.25 km (Example 5), and 3.75 = 3 \times 1.25.
  16. 1.75 km. 0.5 cm is a quarter of 2 cm, so cut the first gap into 4 parts: 7 \div 4 = 1.75.
  17. 27 lanterns; 80 m. 90 m is 9 steps of 10 m, so 9 \times 3 = 27. 24 lanterns is 8 steps of 3, so 8 \times 10 = 80.
  18. 1.5 km. 3 minutes lines up with 0.5 km (Example 6), and 9 minutes is 3 of those: 3 \times 0.5 = 1.5.
  19. 21 minutes. 3.5 km is 7 steps of 0.5 km, each 3 minutes: 7 \times 3 = 21.
  20. 65 minutes. Halve the first step: 2 km lines up with 13 minutes. 10 km is 5 of those, and 5 \times 13 = 65.
  21. 15 km. Halve the first step: 1.5 cm lines up with 5 km. 4.5 cm is 3 of those: 3 \times 5 = 15.
  22. 7.5 cm. 5 km lines up with 1.5 cm, and 25 km is 5 of those: 5 \times 1.5 = 7.5.
  23. Zero of one amount goes with zero of the other: no map distance stands for no ground distance, and no minutes of jogging cover no kilometers. Every mark counts copies of the first pair from that shared start. If one line started further along, a column would no longer be a whole number of copies of the same pair, and pairs that line up would not be equivalent.
  24. The last bottom mark should be 21, not 20. 6 is 3 steps of 2, so its partner is 3 steps of 7, which is 21. With 20 the bottom steps would be 7, 7 and 6, which are not equal, yet the marks are drawn equally spaced.
  25. (a) A double number line. 6.3 cm sits just past the 6 cm mark, which lines up with 20 km, so the road is a little over 20 km (exactly 21 km), and the picture shows that before any working. (b) A ratio table. The paths are so different in size that no single line with a sensible step holds them all, while a table lists 3, 75 and 300 lanterns in three rows.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0