Proportional Relationships and Solving Proportions Math Worksheet for 12-Year-Olds / Learning Corner

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Instructions

Read each situation carefully and identify the two quantities being compared.
Show your work for every proportion and equation.
Use multiplication or division to find the constant of proportionality.
Check your answer by substituting it back into the original situation.
Complete the Core Practice sections first. Try the Challenge Quest when you are ready.

Helpful reminder: Two quantities are proportional when they have a constant ratio. A proportional relationship can be written as:

A proportion: a/b = c/d
An equation: y = kx

Here, k is the constant of proportionality.

Example: If 3 notebooks cost $6, then each notebook costs $2. The equation is c = 2n, where n is the number of notebooks and c is the cost.

Learning Goals

By the end of this worksheet, you should be able to:

Decide whether two quantities are proportional.
Write a proportion from a real-world situation.
Solve proportions using multiplication and division.
Write and use a proportional equation.
Explain what the constant of proportionality means.

Section 1: Is the Relationship Proportional?

Circle Yes or No. Then write one sentence explaining your choice.

Three notebooks cost $6, and five notebooks cost $10.
Yes / No Explanation: __
A taxi charges $4 to start the ride and $2 for every mile.
Yes / No Explanation: __
A recipe uses 2 cups of flour for 3 batches and 4 cups of flour for 6 batches.
Yes / No Explanation: __
A movie theater charges $9 for one ticket, $18 for two tickets, and $27 for three tickets.
Yes / No Explanation: __
A gym charges a $20 membership fee plus $5 per visit.
Yes / No Explanation: __
A cyclist travels 12 miles in 1 hour, 24 miles in 2 hours, and 36 miles in 3 hours.
Yes / No Explanation: __

Section 2: Find the Constant of Proportionality

For each situation, find the constant of proportionality, k. Then write an equation.

Example: Four pens cost $8.
k = 8 ÷ 4 = 2 dollars per pen.
Equation: c = 2p

Six movie tickets cost $42.

k = ____

Let t = number of tickets and c = cost. Equation: ____
A runner travels 15 miles in 3 hours.

k = ____ miles per hour

Let h = hours and d = distance. Equation: ____
Five batches of cookies require 20 cups of flour.

k = ____ cups per batch

Let b = batches and f = cups of flour. Equation: ____
A printer makes 180 pages in 3 minutes.

k = ____ pages per minute

Let m = minutes and p = pages. Equation: ____

Section 3: Writing Proportions

Write a proportion that represents each situation. You do not need to solve it yet.

Four apples cost $3. How much would 10 apples cost? Let x be the cost of 10 apples.

Proportion: ____
A car travels 5 miles in 2 minutes. How far will it travel in 8 minutes? Let x be the distance.

Proportion: ____
On a map, 1 inch represents 12 miles. How many miles does 3.5 inches represent? Let x be the distance.

Proportion: ____
A smoothie recipe uses 3 cups of fruit for 2 servings. How many cups are needed for 7 servings? Let x be the number of cups.

Proportion: ____

Section 4: Solve the Proportion

Use cross multiplication or an equivalent method. Show your work.

x/5 = 12/15

x = ____

Work: _____
7/9 = x/27

x = ____

Work: _____
4/x = 10/25

x = ____

Work: _____
3.5/7 = x/14

x = ____

Work: _____
16/x = 4/3

x = ____

Work: _____
A recipe uses 3/4 cup of oats for 1 batch. How many cups are needed for 8 batches? Write and solve a proportion.

Proportion: ____

Answer: ____ cups

Section 5: Write a Proportional Equation

For each situation, identify the variables and write an equation in the form y = kx.

Six tickets cost $42.

Let x = ____ and y = ____

Equation: __
A car travels 55 miles every hour.

Let t = ____ and d = ____

Equation: __
A bakery uses 4 cups of flour for every 3 loaves of bread. Let l represent the number of loaves and f represent cups of flour.

Equation: __
A machine fills 18 bottles in 2 minutes. Let m represent minutes and b represent bottles.

Equation: __
A school club earns $8 for every car it washes. Let c represent cars washed and e represent earnings.

Equation: __

Section 6: Use the Equation

Write an equation first. Then use it to answer the question.

A music service charges $6 per month. How much will 9 months cost?

Equation: __

Answer: $____
A hiker walks 3 miles every hour. How far will the hiker walk in 4.5 hours?

Equation: __

Answer: ____ miles
A school printer prints 60 pages per minute. How many pages can it print in 7 minutes?

Equation: __

Answer: ____ pages
A craft project needs 2.5 feet of ribbon for each gift box. How much ribbon is needed for 12 boxes?

Equation: __

Answer: ____ feet
A proportional relationship has the equation y = 4x. Find y when x = 6, and explain what the 4 means.

y = ____

The 4 means: ___

Section 7: Sort and Explain

Write each situation under Proportional or Not Proportional. Then explain your choice for one situation in each group.

A babysitter earns $12 for every hour worked.
A phone plan costs $30 each month plus $5 for each extra gigabyte.
Five pounds of oranges cost $10, and 8 pounds cost $16.
A parking garage charges $8 to enter plus $3 for each hour.
A machine makes 24 parts in 3 hours and 40 parts in 5 hours.
A store gives every customer a $5 coupon, then charges $10 per item.

Proportional:

Not Proportional:

Explanation for one proportional situation: __

Explanation for one nonproportional situation: ____

Challenge Quest

A student says, “The equation y = 2.5x means that y is always 2.5 more than x.” Is the student correct? Explain what the equation really means.

Create your own real-world proportional situation. Include:

Two quantities
A constant of proportionality
A proportion
An equation in the form y = kx
One question and its answer

Situation: ____

Proportion: __

Equation: ____

Question: ____

Answer: __

Quick Reflection

Which method helps you most: finding a unit rate, writing a proportion, or using an equation? Why?

Give one example of a proportional relationship you might use outside of school.

Answer Key

Yes. The cost per notebook is always $2.
No. The taxi has a starting fee, so the cost does not begin at zero.
Yes. Both ratios simplify to 2/3 cup per batch.
Yes. The cost is $9 per ticket.
No. The membership fee is a starting amount added to the visit cost.
Yes. The cyclist travels 12 miles per hour.
k = 42 ÷ 6 = 7. Equation: c = 7t.
k = 15 ÷ 3 = 5. Equation: d = 5h.
k = 20 ÷ 5 = 4. Equation: f = 4b.
k = 180 ÷ 3 = 60. Equation: p = 60m.
3/4 = x/10.
5/2 = x/8.
1/12 = 3.5/x.
3/2 = x/7.
x/5 = 12/15; 15x = 60; x = 4.
7/9 = x/27; 9x = 189; x = 21.
4/x = 10/25; 10x = 100; x = 10.
3.5/7 = x/14; 7x = 49; x = 7.
16/x = 4/3; 4x = 48; x = 12.
3/4 = x/8; 4x = 24; x = 6 cups.
x = number of tickets, y = cost. Equation: y = 7x.
t = hours, d = distance. Equation: d = 55t.
Equation: f = 4/3l.
Equation: b = 9m.
Equation: e = 8c.
Equation: c = 6m. For 9 months, c = 6(9) = $54.
Equation: d = 3h. For 4.5 hours, d = 3(4.5) = 13.5 miles.
Equation: p = 60m. For 7 minutes, p = 60(7) = 420 pages.
Equation: r = 2.5b. For 12 boxes, r = 2.5(12) = 30 feet.
y = 4(6) = 24. The 4 means that y is 4 times x, or that the relationship has a constant of proportionality of 4.
The student is incorrect. y = 2.5x means that y is 2.5 times x, not 2.5 more than x.
Answers will vary. A complete answer must include a realistic proportional situation, a correct constant of proportionality, a matching proportion, an equation such as y = 3x, and a correctly solved question.
Answers will vary but should include a reasonable explanation.
Answers will vary. Possible examples include hourly pay, distance traveled at a constant speed, recipe ingredients, or price per item.

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Proportional Relationships and Solving Proportions Math Worksheet for 12-Year-Olds

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