Proportional Data Examples: 45 Problems with Answers

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

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

Concept Recap

Proportional data expresses quantities as fractions or percentages of a whole, enabling fair comparison across groups of different sizes.

Raw counts can mislead when groups differ in size — saying "100 people in City A vs. 100 in City B have a disease" ignores that City A may be ten times larger.

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: Proportional data expresses each quantity as a fraction or percent of its own total so different-sized groups compare fairly.

Common stuck point: The procedure for proportional data is the easy part; the trap is comparing raw counts across groups of different sizes. Asking "Am I expressing a count as a fraction of its own total so different-sized groups compare fairly?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I expressing a count as a fraction of its own total so different-sized groups compare fairly?

Worked Examples

Example 1

easy

In a survey, 45 out of 180 students prefer online learning. Calculate the sample proportion

\hat{p}

and interpret it.

Answer

\hat{p} = \frac{45}{180} = 0.25 = 25\%

First step

Sample proportion formula:

\hat{p} = \frac{x}{n}

Full solution

2
Substitute: $\hat{p} = \frac{45}{180} = 0.25$
3
Convert to percentage: $0.25 \times 100 = 25\%$
4
Interpret: 25% of sampled students prefer online learning; this estimates the true population proportion

Sample proportion

\hat{p} = x/n

is the fundamental estimate of a population probability from data. It is always between 0 and 1. The sample proportion is an unbiased estimator of the true population proportion

p

Example 2

medium

A poll finds

\hat{p} = 0.52

supporting a candidate from

n=400

voters. Calculate the standard error of

\hat{p}

and construct an approximate 95% confidence interval.

Example 3

medium

A school's lunch survey:

80

400

students chose pasta. Another school reports

25

100

students chose pasta. Which proportion is higher, and by how many percentage points?

Example 4

medium

A survey reports '

60\%

of voters agree' but only

5

voters were polled. Explain whether this is a meaningful summary.

Example 5

hard

In a population of

10{,}000

2\%

have a disease. A test has

90\%

sensitivity and

90\%

specificity. Estimate the number of true positives and false positives.

Example 6

hard

A retailer claims '

50\%

off' on a coat originally priced at $240, then adds a

10\%

tax on the discounted price. Find the final price and the effective discount as a proportion of the original.

Example 7

medium

In year 1, defect rate is

4\%

; in year 2 it falls to

3\%

. Compute (a) the absolute change and (b) the relative change.

Practice Problems

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

Example 1

easy

In a class of 25 students, 15 passed the test. Calculate

\hat{p}

and find the number expected to pass in a class of 100 using this estimate.

Example 2

hard

Compare two proportions: Group A: 30/120 support policy. Group B: 45/150 support policy. Test if the proportions differ using a z-test for difference in proportions.

Example 3

easy

Out of

25

students,

20

passed. What proportion passed?

Example 4

easy

A pie chart slice is

25\%

. What is its angle in degrees?

Example 5

easy

City A:

50

sick of

100

. City B:

30

sick of

200

. Compare the proportions sick.

Example 6

easy

A breakdown shows

40\%, 35\%, 30\%

for three categories. What is wrong?

Example 7

easy

Is '

100\%

of patients recovered' impressive if only

1

patient was treated? Why?

Example 8

easy

Convert the proportion

3/8

to a percentage.

Example 9

easy

200

voters,

30\%

chose candidate X. How many voters is that?

Example 10

easy

Which is a fairer comparison of two differently sized classes' performance: number who passed, or proportion who passed?

Example 11

medium

Class A:

18

20

passed. Class B:

80

100

passed. Compute both proportions and state which is higher and by how many percentage points.

Example 12

medium

A survey breakdown:

45\%

25\%

20\%

C, and 'other.' What percent is 'other,' and how many of

400

respondents chose 'other'?

Example 13

medium

A drug 'reduces risk by

50\%

.' Baseline risk is

2\%

. What is the new absolute risk, and the absolute risk reduction?

Example 14

medium

Two pollsters report candidate support: Poll 1 says

52\%

from

1000

people; Poll 2 says

60\%

from

25

people. Which is more trustworthy and why?

Example 15

medium

A pictograph uses one icon

= 10

cars. Row A shows

4.5

icons, row B shows

3

icons. How many cars each, and what proportion of the total is A?

Example 16

medium

Sales mix: product P is

30\%

of $2M total revenue. Next year P is

40\%

of $1.5M total. Did P's revenue rise or fall?

Example 17

medium

A test is

95\%

accurate. In a population where

1\%

have a disease, of

10{,}000

people, estimate how many true positives vs false positives (assume

95\%

sensitivity and

95\%

specificity).

Example 18

medium

A recipe for

4

servings needs

300

g flour. Scale it proportionally for

6

servings.

Example 19

medium

In a class, the ratio of boys to girls is

3:5

. If there are

32

students, how many are girls?

Example 20

challenge

Prove that if two groups have proportions

p_1

and

p_2

of a trait, the combined proportion always lies between

p_1

and

p_2

, and equals their size-weighted average.

Example 21

challenge

A '

200\%

increase' is reported. Starting value is

50

. Find the new value, and explain why '

200\%

increase' differs from '

200\%

of.'

Example 22

challenge

Show that comparing two proportions by their RATIO (relative) vs their DIFFERENCE (absolute) can rank interventions differently, using

p

0.1\to0.05

q

0.5\to0.4

Example 23

easy

In a survey of

250

commuters,

60

said they bike to work. What is the sample proportion

\hat{p}

that bike?

Example 24

easy

A pie chart slice represents

\hat{p} = 0.40

. What is the central angle of that slice in degrees?

Example 25

easy

Town X reports

30

rainy days out of

100

. Town Y reports

20

rainy days out of

50

. Which town had the higher proportion of rainy days?

Example 26

easy

A pie chart breakdown shows

35\%

25\%

15\%

C. What percent must be 'other'?

Example 27

medium

1{,}200

ticket holders,

54\%

are season members. How many ticket holders are NOT season members?

Example 28

medium

A poll of

625

adults reports

\hat{p} = 0.48

favoring a tax measure. Estimate the standard error of

\hat{p}

Example 29

medium

In a town,

20\%

2{,}000

households own pets. Of those,

30\%

own cats. How many households own cats?

Example 30

medium

A vaccine is reported to reduce risk by

40\%

. Baseline risk is

5\%

. Find the new absolute risk and the absolute risk reduction in percentage points.

Example 31

medium

In a pictograph, one icon represents

20

books. Library A shows

3.5

icons and Library B shows

5

icons. What proportion of the combined collection is in Library A?

Example 32

medium

A company's revenue mix: Product A was

25\%

of $8M last year and is

30\%

of $10M this year. Did Product A's revenue rise or fall, and by how much?

Example 33

hard

In group A,

20

50

approve a policy (

40\%

); in group B,

35

100

approve (

35\%

). Compute the pooled approval proportion when the groups are combined.

Example 34

hard

A z-test compares

\hat{p}_1 = 0.50

(n=200) vs

\hat{p}_2 = 0.40

(n=200). Compute the pooled

\hat{p}

and the test statistic

z

Example 35

hard

A poll reports

\hat{p} = 0.55

with a margin of error of

\pm 0.03

95\%

confidence. State the confidence interval and whether it supports a majority.

Example 36

hard

In a class,

60\%

are girls. Of the girls,

70\%

play sports; of the boys,

80\%

play sports. What proportion of the entire class plays sports?

Example 37

challenge

Two cities each report a

50\%

approval rate. City A polled

40

residents, City B polled

4{,}000

. Compare their

95\%

margins of error (approximate

z=2

Example 38

challenge

Simpson's paradox setup: hospital A treats

90

100

healthy patients and

30

100

sick patients successfully; hospital B treats

80

100

healthy and

20

100

sick. Show that A's overall rate exceeds B's, and confirm A wins within each subgroup as well.

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Related Concepts

Percent as Ratio Normalization (Statistics)

Background Knowledge

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

percent as ratio