Scaling Examples in Math: 45 Problems with Answers

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Scaling.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

Changing the size of a quantity by multiplying by a factor, making it proportionally larger (factor $> 1$ ) or smaller (factor $< 1$ ).

Zooming in or out—everything gets bigger or smaller by the same factor.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Scaling resizes a quantity by multiplying by a factor — bigger if the factor is over 1, smaller if it is between 0 and 1.

Common stuck point: The procedure for scaling is the easy part; the trap is adding a fixed amount instead of multiplying. Asking "Is every part multiplied by the same factor (not increased by a fixed amount)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is every part multiplied by the same factor (not increased by a fixed amount)?

Worked Examples

Example 1

easy

A map has a scale of

1:25{,}000

. Two cities are

8

cm apart on the map. What is the actual distance in kilometres?

Answer

The actual distance is

2

km.

First step

The scale

1:25{,}000

means

1

cm on the map represents

25{,}000

cm in reality.

Full solution

2
Actual distance $= 8 \times 25{,}000 = 200{,}000$ cm.
3
Convert to kilometres: $200{,}000 \text{ cm} \div 100{,}000 = 2$ km.

A map scale is a ratio expressing how much the real world has been shrunk. Multiplying the map measurement by the scale ratio gives the real-world measurement in the same units, which can then be converted as needed.

Example 2

medium

A recipe for

4

servings uses

2.5

cups of oats,

1.5

cups of milk, and

\frac{1}{4}

cup of honey. Scale the recipe up to

10

servings.

Example 3

medium

A recipe for

6

cookies uses

1.5

cups of flour and

0.5

cup of sugar. Scale it to make

9

cookies.

Example 4

medium

A cube has side length

2

cm. The cube is scaled by a factor of

3

. Find the new volume.

Example 5

medium

A model airplane has wingspan

30

cm. The real airplane has wingspan

36

m. Find the scale of the model.

Example 6

hard

Two similar containers hold

5

L and

40

L. What is the ratio of their corresponding heights?

Example 7

hard

A cylindrical tank of height

2

m holds

500

L. A similar tank, scaled by a linear factor of

1.5

, holds how many litres?

Example 8

hard

A small cake serves

8

people and uses

200

g of butter. A similar cake (geometrically scaled by a factor of

\tfrac{3}{2}

in every linear dimension) is baked. How many grams of butter does it use?

Example 9

challenge

A statue is built at

\tfrac{1}{8}

scale of a real human (every linear dimension is one-eighth). The model weighs

0.4

kg (assume same material density as a real human, whose mass is about

80

kg). Verify whether the

\tfrac{1}{8}

linear scale is consistent with these masses.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

A model car is built at a scale of

1:18

. The model is

24

cm long. How long is the actual car in metres?

Example 2

medium

A square has side length

5

cm. It is scaled by a factor of

3

. By what factor does the area increase?

Example 3

easy

Scale

6

by a factor of

3

Example 4

easy

Scale

20

by a factor of

\frac{1}{2}

Example 5

easy

A photo

4

cm wide is scaled by factor

2.5

. New width?

Example 6

easy

If every ingredient is tripled, by what factor is a recipe scaled?

Example 7

easy

Scale

\$15

up by a factor of

4

Example 8

easy

A model is

\frac{1}{10}

the size of the real car. The real car is

5

m. Model length?

Example 9

easy

Doubling a number gives

14

. What was the number?

Example 10

easy

Scale the ratio

2:3

up by a factor of

5

Example 11

medium

A square has side

3

cm. If each side is scaled by

2

, what happens to the area?

Example 12

medium

A cube of side

2

is scaled by factor

3

. How does its volume change?

Example 13

medium

A recipe for

4

people uses

200

g of flour. Scale it for

6

people.

Example 14

medium

A map scale is

1:50000

. A road is

4

cm on the map. Real length in km?

Example 15

medium

Scaling

x

\frac{3}{4}

gives

9

. Find

x

Example 16

medium

Two photos are similar; the small one is

6

cm tall, the large

15

cm. What is the scale factor from small to large?

Example 17

medium

A balloon's radius is scaled by

\frac{1}{2}

. Its surface area changes by what factor?

Example 18

challenge

Two similar triangles have areas

16

and

25

. What is the ratio of their corresponding sides?

Example 19

challenge

A model car is built at scale

1:20

. It uses

0.5

kg of material. Assuming the same density and shape, how much material would the full-size car need?

Example 20

challenge

If scaling a quantity by

a

then by

b

gives the same result as scaling by

12

, and

a=3

, find

b

Example 21

medium

A drawing is scaled by

\frac{2}{3}

. A line that was

9

cm becomes how long?

Example 22

medium

A recipe is scaled by

\frac{1}{2}

, then by

3

. What single factor results?

Example 23

easy

Scale

8

by a factor of

4

Example 24

easy

Scale

30

by a factor of

\tfrac{1}{3}

Example 25

easy

A drawing

7

cm wide is scaled by a factor of

4

. What is the new width?

Example 26

easy

A rope is

24

m long. Scale its length by

\tfrac{1}{4}

. What is the new length?

Example 27

easy

Scale the ratio

4:5

by a factor of

3

Example 28

medium

A blueprint uses a scale of

1:50

. A wall measures

6

cm on the blueprint. How long is the actual wall in metres?

Example 29

medium

A rectangle with width

3

cm and height

5

cm is scaled by a factor of

4

. Find the new perimeter.

Example 30

medium

A triangle's area is

12

^2

. If every side is scaled by a factor of

5

, what is the new area?

Example 31

medium

A map has a scale of

1:100{,}000

. The distance between two towns on the map is

4.5

cm. What is the actual distance in km?

Example 32

medium

A bag of dog food lasts

12

days for

1

dog. Scaled for

4

dogs eating the same amount each, how many days does it last?

Example 33

medium

A square has side

6

cm. By what scale factor must its side be enlarged so that the new area is

144

^2

Example 34

hard

A photo is enlarged so its width grows from

4

in to

10

in. By what factor does its area increase?

Example 35

hard

Triangle

A

has sides

3

4

5

. Triangle

B

is similar and has the longest side

12

. Find the perimeter of triangle

B

Example 36

hard

A model of a building is built at scale

1:200

. The model has volume

0.5

L. What is the volume of the actual building in m

^3

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Related Concepts

Multiplication Ratios Similarity Proportional Relationships

Background Knowledge

These ideas may be useful before you work through the harder examples.

multiplicationratios