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Math 7 · 7.G.A.1

Using a scale on maps, plans and models

A map is a hand’s width across, and the land it shows is many kilometers wide. How does a length on the paper tell you a length on the ground?

You need
Scale factorMath 7
Where you are 2 in Scale drawings
Fractions, ratio and percent3
Ratio6
Scale drawings2
Scale factor
Using a scale on maps, plans and models

When you reach for it

Whenever a drawing stands for something too large to draw at full size (a road map, a city map, the plan of a house), or a model is built smaller than the real thing (a model train, a model ship). Each comes with a scale, a statement of how lengths on the drawing match lengths in reality. You reach for this skill when you know a length on one side and want the matching length on the other: “the towns are 3.5 inches apart on the map, so how far apart are they?”, or “the room is 15 feet wide, so how long do I draw that wall?”

One scale serves every length because the drawing is a scale copy: every length has been multiplied by the same number. Scale factor multiplies every length shows that this holds for every length you can measure, straight or curved.

This lesson takes two days. Part 1 reads a stated scale (in words, as a ratio, or as a scale bar) as a rate and uses it in both directions, including when the two sides of the scale are in different units. Part 2 uses the same rate on floor plans and scale models, then redraws a plan at a new scale.

The procedure

Part 1: a scale is a rate

A scale pairs a length on the drawing with the real length it stands for. “One inch represents 25 miles” is the pair 1 inch and 25 miles, and like any ratio it gives a unit rate: 25 miles of ground per inch of map.

Scales are written in three ways.

  • In words, with a unit on each side: “1 inch represents 25 miles”, “1 cm to 3 km”, “\tfrac{1}{4} \text{ inch} = 1 \text{ foot}”. The = sign in the last is a habit of plans: the quarter inch stands for the foot, it does not equal it.
  • As a ratio with no units, such as 1 : 24{,}000: one unit on the drawing for 24{,}000 of the same unit in reality, whatever that unit is.
  • As a scale bar: a line drawn on the map, labeled with the real distance its length stands for. Measure the bar with a ruler and you have a scale in words.

From the drawing to the real thing, multiply the drawing length by the rate. Each inch carries 25 miles, so 2.6 inches carry

2.6 \times 25 = 65 \text{ miles}

From the real thing to the drawing, divide by the rate: how many inches, each worth 25 miles, make up 80 miles?

80 \div 25 = 3.2 \text{ inches}

Both moves are one proportion, solved two ways round. A drawing length and its real length are always in the ratio the scale names,

\frac{1 \text{ in}}{25 \text{ mi}} = \frac{2.6 \text{ in}}{x \text{ mi}}

and the cross products give x = 65. The rate is that proportion solved once, ready for every length on the map. Writing the proportion from the words of a problem is the skill of Setting up proportions from word problems. The unit of the answer comes from the scale: inches of map times miles per inch gives miles.

When the two sides use different units. A word scale such as “1 cm represents 240 m” can be used as it stands, as long as each answer keeps the unit of its own side. To write it as a ratio 1 : n, or to compare it with a map whose scale is a ratio, first put both sides in one unit:

\begin{aligned} 1 \text{ cm} : 240 \text{ m} &= 1 \text{ cm} : 24{,}000 \text{ cm} \\ &= 1 : 24{,}000 \end{aligned}

The conversions used most are 1 \text{ m} = 100 \text{ cm} and 1 \text{ km} = 100{,}000 \text{ cm} in metric units, and 1 \text{ ft} = 12 \text{ in} and 1 \text{ mile} = 5{,}280 \text{ ft} = 63{,}360 \text{ in} in customary units.

In 1 : n, a larger n means each unit of drawing carries more ground, so everything is drawn smaller. A map at 1 : 10{,}000 draws a street 4 times as long as a map at 1 : 40{,}000, because 40{,}000 \div 10{,}000 = 4.

A scale bar survives a copier. When a map is enlarged or shrunk, the bar changes with everything else and stays right, while a scale in words or as a ratio still describes the map as first printed.

Part 2: plans, models and a new scale

A floor plan is a map of a building seen from above. Its scale is usually a fraction of an inch for each foot, and \tfrac{1}{4} \text{ inch} = 1 \text{ foot} is common for a house. As a rate, dividing by a fraction as in Unit rates when the amounts are fractions,

1 \div \tfrac{1}{4} = 4 \text{ feet per inch}

As a ratio it is \tfrac{1}{4} \text{ in} : 12 \text{ in}, and multiplying both sides by 4 gives 1 : 48.

A scale model is a scale copy in three dimensions: length, width and height all stand in the same ratio to the real thing. Model railways are often built at 1 : 87, so a real length is 87 times the model length, in any unit, as long as both are in the same one.

Redrawing at a new scale. A plan may have to be drawn again at a different scale, to fit a smaller sheet or to show more detail. If the old plan is at 1 : a and the new one at 1 : b, a length d on the old plan stands for a real length a \times d, which the new plan draws as (a \times d) \div b. So

\text{new length} = \frac{a}{b} \times \text{old length}

The same number \displaystyle \frac{a}{b} multiplies every length, so the new plan is a scale copy of the old one with scale factor \displaystyle \frac{a}{b}, smaller when b is larger than a. Going through the real lengths is safer when the scales use different units; going straight across with the factor is quicker when there are many lengths.

This lesson keeps to lengths. What a scale does to the area of a room is a different number, worked out in Scale factor, area and volume.

Worked examples

Examples 1 to 4 belong to Part 1, and Examples 5 to 7 to Part 2.

Example 1

A road map with a scale in words

Albrecht’s road map has the scale “1 inch represents 40 miles”.

(a) Two towns are 3.5 inches apart on the map. How far apart are they?

(b) A third town is 90 miles from the first in a straight line. How far apart should the two be on the map?

Solution

(a) Each inch of map stands for 40 miles, so 3.5 inches stand for

3.5 \times 40 = 140 \text{ miles}

(b) Now the real distance is known. How many inches, each worth 40 miles, make up 90 miles?

90 \div 40 = 2.25 \text{ inches}

Check: 2.25 \times 40 = 90. Both answers are straight-line distances; a winding road between the towns is longer, and would be measured along the road on the map.

Example 2

A ratio scale on a hiking map

A hiking map has the scale 1 : 24{,}000.

(a) A straight stretch of trail measures 7.5 cm on the map. How long is it, in kilometers?

(b) A path along a ridge is 3 km long. How long is it on the map?

Solution

(a) One centimeter of map stands for 24{,}000 centimeters of ground, so

7.5 \times 24{,}000 = 180{,}000 \text{ cm}

That is 1{,}800 m, since there are 100 cm in a meter, and so 1.8 km.

(b) Put the real length into centimeters first, so both sides of the ratio use one unit: 3 \text{ km} = 3{,}000 \text{ m} = 300{,}000 \text{ cm}. Then

300{,}000 \div 24{,}000 = 12.5 \text{ cm}

The path is surely curved, but every length on a scale copy is multiplied by the same number, curved ones included. The 12.5 cm is the drawn path’s length, measured by laying a string along it.

Example 3

Reading a scale bar

On a city map, the scale bar is 5 cm long and is labeled 0 to 2 km. Ghislaine measures the straight line from the museum to the station as 6.3 cm.

\text{Museum}
\text{Station}
6.3\text{ cm}
0
1\text{ km}
2\text{ km}
Figure 1The city map in Example 3, drawn in proportion. On the printed map the bar is 5 cm long and stands for 2 km.

(a) How far is the museum from the station?

(b) Write the map’s scale as a ratio 1 : n.

Solution

(a) The bar says 5 cm of map stands for 2 km, so one centimeter stands for

2 \div 5 = 0.4 \text{ km} = 400 \text{ m}

and 6.3 cm stands for 6.3 \times 0.4 = 2.52 km.

(b) In centimeters, 2 \text{ km} = 200{,}000 \text{ cm}, so the bar says 5 \text{ cm} : 200{,}000 \text{ cm}. Dividing both sides by 5,

5 : 200{,}000 = 1 : 40{,}000

Check (a) with this form: 6.3 \times 40{,}000 = 252{,}000 cm, which is 2.52 km.

Example 4

Two maps with different kinds of scale

Brunhilde has two maps of the same area. Map A says “1 inch represents 2 miles”. Map B says 1 : 100{,}000.

(a) Write map A’s scale as a ratio 1 : n.

(b) A straight road is 10 miles long. How long is it on each map, and which map draws it longer?

Solution

(a) A mile is 63{,}360 inches, so 2 miles is 126{,}720 inches, and map A’s scale is

1 : 126{,}720

(b) On map A each inch carries 2 miles, so the road takes 10 \div 2 = 5 inches.

On map B, put the real length into inches first: 10 \text{ miles} = 633{,}600 \text{ in}. Then

633{,}600 \div 100{,}000 = 6.336 \text{ inches}

Map B draws the road longer, as (a) predicts: 100{,}000 is less than 126{,}720, so each inch of map B carries less ground.

Example 5

A floor plan at a quarter inch to the foot

Cornelius has the plan of a house drawn at \tfrac{1}{4} \text{ inch} = 1 \text{ foot}. The living room measures 4\tfrac{1}{2} inches by 3\tfrac{3}{4} inches on the plan.

4\tfrac{1}{2}\text{ in}
3\tfrac{3}{4}\text{ in}
\text{Living room}
\tfrac{3}{4}\text{ in}
\text{Scale: }\tfrac{1}{4}\text{ in} = 1\text{ ft}
Figure 2The living room in Example 5, drawn in proportion, with its plan lengths marked.

(a) How large is the real room?

(b) The doorway is 3 feet wide. How wide is it on the plan?

(c) Write the plan’s scale as a ratio.

Solution

(a) An inch holds four quarter inches, so it stands for four feet:

1 \div \tfrac{1}{4} = 4 \text{ feet per inch}

Multiply each plan length by 4:

\begin{aligned} 4\tfrac{1}{2} \times 4 &= 18 \text{ feet} \\ 3\tfrac{3}{4} \times 4 &= 15 \text{ feet} \end{aligned}

The room is 18 feet by 15 feet.

(b) From the real thing to the plan, divide by the rate: 3 \div 4 = \tfrac{3}{4} inch, or three quarter inches, one for each foot.

(c) A foot is 12 inches, so the scale is \tfrac{1}{4} \text{ in} : 12 \text{ in}, and multiplying both sides by 4 gives 1 : 48. Check: 4\tfrac{1}{2} \times 48 = 216 inches, and 216 \div 12 = 18 feet, as in (a).

Example 6

A model locomotive and a boxcar

Desdemona builds a model railway at 1 : 87.

(a) Her model locomotive is 24 cm long. How long is the real one?

(b) A real boxcar is 12.18 m long and 4.35 m tall. How long and how tall should its model be?

Solution

(a) Every real length is 87 times the model length: 24 \times 87 = 2{,}088 cm, which is 20.88 m.

(b) In centimeters, 12.18 \text{ m} = 1{,}218 \text{ cm} and 4.35 \text{ m} = 435 \text{ cm}. Divide each by 87:

\begin{aligned} 1{,}218 \div 87 &= 14 \text{ cm} \\ 435 \div 87 &= 5 \text{ cm} \end{aligned}

The model boxcar is 14 cm long and 5 cm tall. The height uses the same ratio as the length; a model shrunk by one number lengthwise and another heightwise would look squashed.

Example 7

Redrawing a garden plan at a smaller scale

Evander’s garden plan is drawn at 1 \text{ cm} : 2 \text{ m}. On it the garden is 9 cm by 6.5 cm, and a vegetable bed in one corner is 3 cm by 2 cm. He wants to redraw the plan at 1 \text{ cm} : 5 \text{ m}. What are the four lengths on the new plan?

9\text{ cm}
6.5\text{ cm}
1\text{ cm} : 2\text{ m}
3.6\text{ cm}
2.6\text{ cm}
1\text{ cm} : 5\text{ m}
Figure 3Example 7. Every length on the new plan is \tfrac{2}{5} of the matching length on the old one.
Solution

Through the real lengths. At 2 m per centimeter the garden is 18 m by 13 m and the bed is 6 m by 4 m. On the new plan each centimeter stands for 5 m, so divide each by 5:

\begin{aligned} 18 \div 5 &= 3.6 & 13 \div 5 &= 2.6 \\ 6 \div 5 &= 1.2 & 4 \div 5 &= 0.8 \end{aligned}

The new plan shows the garden as 3.6 cm by 2.6 cm and the bed as 1.2 cm by 0.8 cm.

Straight across. Each length was multiplied by 2 and then divided by 5, which is multiplying by \displaystyle \frac{2}{5} = 0.4: 9 \times 0.4 = 3.6, 6.5 \times 0.4 = 2.6, 3 \times 0.4 = 1.2 and 2 \times 0.4 = 0.8. As ratios the plans are 1 : 200 and 1 : 500, and \displaystyle \frac{200}{500} = \frac{2}{5} is the factor \displaystyle \frac{a}{b} from the procedure, less than 1 because each new centimeter carries more ground.

Boundaries

  • The drawing must be a true scale copy. A diagram marked “not to scale”, or a railway map drawn for clarity, has no single scale. A relief model that exaggerates heights has one scale for heights and another for distances.
  • A flat map of a large part of the Earth has no single scale. A curved surface cannot be flattened without stretching some lengths more than others, so the stated scale is exact only along certain lines. On a city or regional map the error is too small to matter; on a map of a continent it is large.
  • The scale belongs to the drawing as printed. After resizing, only a scale bar stays correct.
  • The ratio form needs one unit on both sides. A word scale with two units must be converted before it is written as 1 : n or compared with one.
  • Lengths only. Areas on a plan do not follow the same number; see Scale factor, area and volume.
  • An answer is only as exact as its measurement. On a map at 1 : 24{,}000, an error of 1 mm on the map is 24{,}000 mm, or 24 m, on the ground.

Where this goes next

“Real length equals rate times drawing length” is a proportional relationship, and the rate is its constant of proportionality: real length graphed against drawing length is a straight line through the origin. Scale factor, area and volume asks what a scale does to areas and volumes. Similar figures puts scale copies on a firm footing, and dilations build a scale copy outward from a single point, the way a projector enlarges a plan. Proportions in similar triangles then measure lengths that cannot be reached, such as a tree’s height from its shadow.

Exercises

Part 1: from the drawing to the real thing

  1. A road map has the scale “1 inch represents 30 miles”. Two towns are 4.25 inches apart on the map. How far apart are they?
  2. A hiking map has the scale 1 : 50{,}000. A straight trail measures 9.4 cm on the map. How long is the trail, in kilometers?
  3. A map’s scale reads “2 cm represents 15 km”. Two villages are 11 cm apart on the map. How far apart are they?

Part 1: from the real thing to the drawing

  1. On a road map at “1 inch represents 60 miles”, how far apart are two cities that are 210 miles apart?
  2. A hiking map has the scale 1 : 20{,}000. How long is a 3.6 km path on the map?
  3. A map’s scale reads “\tfrac{1}{2} inch represents 15 miles”. (a) How many miles does 1 inch represent? (b) How long on the map is a straight road 75 miles long?

Part 1: scale bars and units

  1. A city map has a scale bar 4 cm long, labeled 0 to 600 m. (a) How many meters does 1 cm of the map represent? (b) Write the scale as a ratio 1 : n.
  2. On the map in Exercise 7, the park gate is 7.4 cm from the library. How far apart are they, in kilometers?
  3. Write the scale “1 inch represents 1 mile” as a ratio 1 : n.
  4. Write the scale “1 cm represents 3 km” as a ratio 1 : n.
  5. Map P has the scale “1 cm represents 2 km”. Map Q has the scale 1 : 150{,}000. A lake is 6 km across. How wide is it on each map, and which map shows it larger?

Part 2: floor plans

  1. A house plan is drawn at \tfrac{1}{4} \text{ inch} = 1 \text{ foot}. The kitchen measures 3\tfrac{1}{4} inches by 2\tfrac{1}{2} inches on the plan. How large is the real kitchen?
  2. An architect’s drawing is at \tfrac{1}{8} \text{ inch} = 1 \text{ foot}. How long is a 46 foot wall on the drawing?
  3. A classroom is 9 m long and 7.5 m wide. How large is it on a plan at 1 : 75, in centimeters?

Part 2: scale models

  1. A model railway is at 1 : 87. A model passenger car is 26 cm long. How long is the real car, in meters?
  2. A model ship is built at 1 : 300. (a) The real ship is 270 m long. How long is the model, in centimeters? (b) A mast on the model is 15 cm tall. How tall is the real mast, in meters?
  3. A model car is 18 cm long, and the real car is 4.32 m long. (a) What is the model’s scale, as a ratio 1 : n? (b) The model is 6 cm tall. How tall is the real car?
  4. A real locomotive is 20.8 m long. How long is its model at 1 : 160, and at 1 : 80? Why is one exactly twice the other?

Part 2: redrawing at a new scale

  1. A garden plan at 1 \text{ cm} : 4 \text{ m} is redrawn at 1 \text{ cm} : 2 \text{ m}. Two lengths on the old plan are 3.5 cm and 12.2 cm. What are they on the new plan?
  2. A plan at 1 : 200 is redrawn at 1 : 300. Three lengths on the old plan are 9 cm, 7.5 cm and 12 cm. What are they on the new plan?
  3. An architect’s drawing at \tfrac{1}{8} \text{ inch} = 1 \text{ foot} is redrawn at \tfrac{1}{4} \text{ inch} = 1 \text{ foot}. A wall is 2\tfrac{3}{8} inches long on the old drawing. How long is the real wall, and how long is it on the new drawing?
  4. A plan of a sports hall at 1 : 120 is 42 cm by 30 cm. It must be redrawn to fit on a sheet 28 cm by 21 cm. Of the scales 1 : n that fit, which one draws the hall largest, and how large is the hall on the sheet?

Reasoning

  1. Explain why the scale 1 : 24{,}000 needs no units, while “1 cm represents 240 m” must name a unit on each side. Then show that the two describe the same map.
  2. A map carries a scale bar and also the words “1 cm represents 500 m”. It is copied at twice its size. Which of the two is still correct on the copy, and what should the words say now?
  3. A plan at 1 : a is redrawn at 1 : b. Explain why every length on the new plan is \displaystyle \frac{a}{b} times the matching length on the old one, and why the new plan is smaller when b is larger than a.
Answers
  1. 127.5 miles. Each inch carries 30 miles: 4.25 \times 30 = 127.5.
  2. 4.7 km. 9.4 \times 50{,}000 = 470{,}000 cm, which is 4{,}700 m.
  3. 82.5 km. One centimeter stands for 15 \div 2 = 7.5 km, and 11 \times 7.5 = 82.5.
  4. 3.5 inches. 210 \div 60 = 3.5.
  5. 18 cm. 3.6 \text{ km} = 360{,}000 \text{ cm}, and 360{,}000 \div 20{,}000 = 18.
  6. (a) 30 miles: 15 \div \tfrac{1}{2} = 30. (b) 2\tfrac{1}{2} inches: 75 \div 30 = 2.5.
  7. (a) 150 m: 600 \div 4 = 150. (b) 1 : 15{,}000. 150 \text{ m} = 15{,}000 \text{ cm}, so the scale is 1 \text{ cm} : 15{,}000 \text{ cm}.
  8. 1.11 km. 7.4 \times 150 = 1{,}110 m.
  9. 1 : 63{,}360. A mile is 5{,}280 \times 12 = 63{,}360 inches.
  10. 1 : 300{,}000. 3 \text{ km} = 3{,}000 \text{ m} = 300{,}000 \text{ cm}.
  11. 3 cm on map P and 4 cm on map Q, so map Q shows it larger. On P, 6 \div 2 = 3. On Q, 6 \text{ km} = 600{,}000 \text{ cm} and 600{,}000 \div 150{,}000 = 4. As a ratio map P is 1 : 200{,}000, and the smaller n belongs to Q.
  12. 13 feet by 10 feet. The rate is 4 feet per inch: 3\tfrac{1}{4} \times 4 = 13 and 2\tfrac{1}{2} \times 4 = 10.
  13. 5\tfrac{3}{4} inches. Each foot is drawn as \tfrac{1}{8} inch, so the wall is 46 \times \tfrac{1}{8} = \tfrac{46}{8} = 5\tfrac{3}{4} inches.
  14. 12 cm by 10 cm. 9 \text{ m} = 900 \text{ cm} and 900 \div 75 = 12; 7.5 \text{ m} = 750 \text{ cm} and 750 \div 75 = 10.
  15. 22.62 m. 26 \times 87 = 2{,}262 cm.
  16. (a) 90 cm: 270 \div 300 = 0.9 m. (b) 45 m: 15 \times 300 = 4{,}500 cm.
  17. (a) 1 : 24. 4.32 \text{ m} = 432 \text{ cm}, and dividing both sides of 18 : 432 by 18 gives 1 : 24. (b) 1.44 m: 6 \times 24 = 144 cm.
  18. 13 cm at 1 : 160 and 26 cm at 1 : 80. 20.8 \text{ m} = 2{,}080 \text{ cm}, 2{,}080 \div 160 = 13 and 2{,}080 \div 80 = 26. Dividing by half as large a number gives twice as much, and 80 is half of 160.
  19. 7 cm and 24.4 cm. The factor is \displaystyle \frac{4}{2} = 2. Through the real length: 3.5 cm on the old plan stands for 3.5 \times 4 = 14 m, which is 14 \div 2 = 7 cm on the new one.
  20. 6 cm, 5 cm and 8 cm. The factor is \displaystyle \frac{200}{300} = \frac{2}{3}: \displaystyle 9 \times \frac{2}{3} = 6, \displaystyle 7.5 \times \frac{2}{3} = 5 and \displaystyle 12 \times \frac{2}{3} = 8.
  21. The wall is 19 feet, and 4\tfrac{3}{4} inches on the new drawing. The old rate is 8 feet per inch, so 2\tfrac{3}{8} \times 8 = 19 feet; the new rate is 4 feet per inch, so 19 \div 4 = 4\tfrac{3}{4} inches. The factor is 2, since a quarter inch is twice an eighth.
  22. 1 : 180, which draws the hall 28 cm by 20 cm. The real hall is 5{,}040 cm by 3{,}600 cm. To fit, 5{,}040 \div n \le 28, so n \ge 180, and 3{,}600 \div n \le 21, so \displaystyle n \ge \frac{3{,}600}{21} \approx 171.4. So any 1 : n with n at least 180 fits, and the smallest such n draws largest: 5{,}040 \div 180 = 28 and 3{,}600 \div 180 = 20. The hall then fills the sheet’s length exactly.
  23. A ratio of two lengths in the same unit has no unit: 1 : 24{,}000 says every drawing length is \displaystyle \frac{1}{24{,}000} of the real one, whether both are measured in centimeters, inches or anything else. “1 cm represents 240 m” pairs two different units, and without them the numbers 1 and 240 would give the wrong ratio. They describe the same map because 240 \text{ m} = 24{,}000 \text{ cm}, so 1 \text{ cm} : 240 \text{ m} is 1 \text{ cm} : 24{,}000 \text{ cm}, which is 1 : 24{,}000.
  24. The scale bar. Copying at twice the size doubles every length on the map, the bar included, so the bar still matches the distances drawn beside it. The words do not: on the copy, 2 cm represents 500 m, so they should now say “1 cm represents 250 m”.
  25. A length d on the old plan stands for a real length a \times d, and the new plan draws that real length as \displaystyle (a \times d) \div b = \frac{a}{b} \times d. The number \displaystyle \frac{a}{b} does not depend on d, so every length is multiplied by the same factor. When b > a, the fraction \displaystyle \frac{a}{b} is less than 1, so every length shrinks.
kg1128 · updated 2026-10-07 · CC-BY-SA-4.0