Multiplication as Scaling Examples in Math

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

Understanding multiplication as resizing or scaling a quantity by a factor. Multiplying by 2 doubles, by 0.5 halves, and by 1 leaves unchanged — it stretches or shrinks the original number.

Multiplying by 2 doubles something; by 0.5 cuts it in half; by 3 triples it.

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 sees multiplication as stretching or shrinking an amount: above 1 enlarges, below 1 shrinks, exactly 1 leaves it alone.

Common stuck point: The procedure for multiplication as scaling is the easy part; the trap is assuming multiplying always enlarges. Asking "Is one amount being resized by a factor instead of counted in equal groups?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is one amount being resized by a factor instead of counted in equal groups?

Worked Examples

Example 1

easy

A recipe uses 3 cups of flour. If you make the recipe 4 times bigger, how many cups of flour do you need?

Answer

12 cups

First step

Original amount: 3 cups.

Full solution

2
Scale factor: 4 (making it 4 times bigger).
3
Multiply: $3 \times 4 = 12$ cups.
4
You need 12 cups of flour.

Scaling by a factor means multiplying. Making something 4 times bigger means multiplying by 4.

Example 2

medium

A drawing of a cat is 5 cm tall. You scale it up to be 3 times taller. How tall is the scaled drawing?

Example 3

easy

A recipe calls for

2

cups of milk. Double the recipe. How much milk?

Example 4

medium

A spring is

20

cm long. It is stretched to

1.4

times its length. What is the new length, and how much did it stretch?

Example 5

hard

A bank account grows by

10\%

each year. After

2

years, what is the scaling factor? Starting with $100, what is the balance?

Practice Problems

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

Example 1

easy

A plant is 8 inches tall. After one month it is 2 times taller. How tall is it now?

Example 2

medium

A photo is 4 inches wide. A poster version is 6 times wider. How wide is the poster?

Example 3

easy

What is

2 \times 5

, thought of as doubling

5

Example 4

easy

Multiplying by

1

leaves a number unchanged. What is

7 \times 1

Example 5

easy

What is half of

8

, i.e.

8 \times \frac{1}{2}

Example 6

easy

Tripling means multiplying by

3

. What is

3 \times 4

Example 7

easy

Does multiplying

6

0.5

make it bigger or smaller?

Example 8

easy

What is

10 \times 2

as a doubling?

Example 9

easy

Multiplying by

0

scales any number to what?

Example 10

easy

What is

\frac{1}{2}

10

Example 11

medium

A recipe makes

4

cookies. To triple it, how many cookies result?

Example 12

medium

A photo is

6

cm wide. Scaled by

\frac{1}{2}

, how wide is it?

Example 13

medium

A drink costs

\$3

. Buying

5

of them costs how much, by scaling the price?

Example 14

medium

A number is multiplied by

4

to become

20

. Was the number stretched or shrunk, and to what?

Example 15

medium

Compute

3 \times 0.5

and explain why it is not

3.5

Example 16

medium

A spring is

8

cm long. It is stretched to

1.5

times its length. How long is it now?

Example 17

medium

Doubling a number twice is the same as multiplying by what single factor?

Example 18

challenge

A price is first doubled, then halved. What is the net scaling factor, and what happens to a

\$20

price?

Example 19

challenge

Scaling a number by

\frac{3}{4}

gives

9

. What was the original number?

Example 20

challenge

A square's side is scaled by

3

. By what factor does its area change?

Example 21

medium

A rope is

9

m. A second rope is

\frac{1}{3}

as long. How long is the second rope?

Example 22

medium

A savings of

\$8

grows to

4

times its size. How much is it now?

Example 23

easy

A picture is

12

cm wide. Scale it to

\tfrac{1}{4}

of its width. How wide is it now?

Example 24

easy

A tree is

5

ft tall. After a year it is

4

times taller. How tall now?

Example 25

easy

Does

5\times0.8

make

5

bigger or smaller? What is the result?

Example 26

medium

A photo is

4

in by

6

in. Scaled to

1.5\times

each side, what are the new dimensions?

Example 27

medium

A model car is

\tfrac{1}{20}

the size of the real car. The real car is

5

m long. How long is the model?

Example 28

medium

A book costs $24. The store offers it at

25\%

off. What is the new price?

Example 29

medium

Doubling a number, then doubling again, is the same as multiplying by what?

Example 30

medium

A jar holds

8

cups of flour. A larger jar holds

\tfrac{5}{4}

times that. How much does the larger jar hold?

Example 31

medium

A square's side is doubled. By what factor does its area scale?

Example 32

medium

A drawing

6

in tall is scaled to

9

in tall. What is the scale factor?

Example 33

hard

A price of $50 is first increased by

20\%

, then the new price is decreased by

20\%

. What is the final price?

Example 34

hard

A scientist halves a sample three times in a row. What fraction remains? If the original was

80

g, how much is left?

Example 35

hard

A blueprint uses scale

1:50

(drawing

:

real). A drawn wall is

8

cm. How many meters is the real wall?

Example 36

hard

A cube has side

4

cm. The side is scaled by

\tfrac{1}{2}

. What is the new volume?

Example 37

hard

A photo enlarged by factor

k

has area

9

times the original. What is

k

Example 38

hard

A salary is reduced by

20\%

. By what percent must the reduced salary now grow to return to the original?

Example 39

medium

A rope is

24

ft. A second rope is

\tfrac{2}{3}

as long. How long is the second rope?

Example 40

challenge

A quantity is multiplied by

1.5

, then by

0.5

, then by

2

. What is the net scaling factor, and what is the result if the start is

40

Example 41

challenge

A statue

6

ft tall casts a shadow

9

ft long at the same time a building casts a shadow

60

ft long. How tall is the building (using shadow-length scaling)?

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Related Concepts

Multiplication Scaling Proportionality

Background Knowledge

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

multiplication