Solve problems with rates, percentages, and proportions

Ontario Grade 9 Mathematics

Number

Solving Real-World Problems Using Ratios, Rates, and Percent

Have you ever compared the price of two sizes of orange juice to find the better deal? Or figured out how much a jacket costs after a 30% discount? Both of those situations use rates, percentages, and proportions — three closely connected tools that let you make fair comparisons and solve everyday problems. This lesson builds on your Grade 8 work with fractions, ratios, and basic algebra. By the end, you will be able to set up and solve any problem that involves these three ideas, and you will know how to check that your answer makes sense.

What you will learn

Explain the difference between a ratio, a rate, and a proportion.
Set up and solve a proportion to find an unknown quantity.
Calculate percent of a number, percent change, and find an original value given a percentage.
Choose the right strategy (proportion or percent equation) to solve a word problem involving rates and percentages.
Verify that an answer is reasonable by checking it against the original problem.

What Are Ratios, Rates, and Proportions?

A ratio compares two quantities that share the same unit. For example, if a recipe uses 2 cups of flour and 3 cups of oats, the ratio of flour to oats is CAD 2:3. Because both quantities are measured in cups, the unit cancels out and the ratio is just a number comparison.

A rate compares two quantities that have different units. Speed is a familiar rate: 60 kilometres per hour compares distance (km) to time (h). Unit rate is the special case where the second quantity equals 1 — for example, CAD 3.50 per litre of milk. Unit rates make it easy to compare options fairly.

A proportion is a statement that two ratios or two rates are equal. Writing

\frac{a}{b} = \frac{c}{d}

says that the ratio

a

to

b

equals the ratio

c

to

d

. You use a proportion whenever one ratio is known and you need to find a missing value in an equivalent ratio. The key property of a proportion is that the cross-products are equal:

a \times d = b \times c

. This property comes directly from multiplying both sides of the equation by

b \times d

, so it is just algebra you already know.

\frac{a}{b} = \frac{c}{d} \implies a \times d = b \times c

A ratio compares quantities with the same unit; a rate compares quantities with different units.
A unit rate has 1 as the second quantity and makes comparisons straightforward.
A proportion states that two ratios (or rates) are equal: $\frac{a}{b} = \frac{c}{d}$ .
Cross-products are equal in any true proportion: $a \times d = b \times c$ .

Solving Problems with Proportions

Many real problems give you one complete ratio and one incomplete ratio. Your job is to find the missing value. The cleanest way is to write a proportion and use the cross-product property.

Here is the step-by-step method. Step 1: Identify the two ratios and make sure the same type of quantity is in the same position (top or bottom) in both fractions. Step 2: Write the proportion with the unknown as a variable, for example

n

. Step 3: Cross-multiply to produce a one-step or two-step equation. Step 4: Solve for

n

. Step 5: Check by substituting your answer back into the original proportion to confirm both sides are equal.

Example situation: A car travels 150 km using 12 L of fuel. At the same rate, how far can it travel on 20 L? Set up the proportion so kilometres are always on top and litres are always on the bottom:

\frac{150}{12} = \frac{d}{20}

. Cross-multiplying gives

150 \times 20 = 12 \times d

, so

3000 = 12d

, and

d = 250

km. Check:

\frac{150}{12} = 12.5

and

\frac{250}{20} = 12.5

. ✓

Always label your variable and include units in your answer. Writing 'Let

d

= distance in kilometres' keeps your work clear and avoids mixing units accidentally.

\frac{\text{known quantity 1}}{\text{known quantity 2}} = \frac{\text{same type as 1}}{\text{same type as 2}}

Place the same type of quantity in the same position in both fractions before cross-multiplying.
Define your variable with a label and units before you write the proportion.
Cross-multiply, then solve the resulting equation using inverse operations.
Always substitute your answer back in to verify both sides of the proportion are equal.

Working with Percentages

A percentage is a ratio out of 100. Writing 35% means

\frac{35}{100}

, which equals 0.35 as a decimal. You can always convert between the three forms: percent ↔ fraction ↔ decimal.

There are three core percentage calculations you need to know. First, finding a percent of a number: multiply the decimal form of the percent by the whole. For example, 35% of 80 is

0.35 \times 80 = 28

. Second, finding what percent one number is of another: divide the part by the whole, then multiply by 100. For example, 28 is what percent of 80?

\frac{28}{80} \times 100 = 35\%

. Third, finding the original whole when a percent and the resulting part are known: divide the part by the decimal form of the percent. For example, 28 is 35% of what number?

\frac{28}{0.35} = 80

.

Percent change measures how much a value has increased or decreased relative to the original value. The formula is

\text{percent change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100

. A positive result means an increase; a negative result means a decrease. For example, if a bus pass rises from CAD 100 to CAD 115, the percent increase is

\frac{115 - 100}{100} \times 100 = 15\%

.

You can also use a proportion to solve percentage problems. Because percent means 'out of 100,' you can always write

\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}

and solve for the unknown piece. This connects the percent work directly to the proportion method from the previous section.

\text{Percent change} = \frac{\text{new value} - \text{original value}}{\text{original value}} \times 100

Percent means out of 100: $p\% = \frac{p}{100}$ .
Percent of a number: multiply the whole by the decimal equivalent of the percent.
Percent change formula: $\frac{\text{new} - \text{original}}{\text{original}} \times 100$ .
A proportion approach also works: $\frac{\text{part}}{\text{whole}} = \frac{p}{100}$ .

Comparing Rates to Make Decisions

A common real-life task is deciding which option gives better value. To compare rates fairly, convert each one to a unit rate — the cost, speed, or quantity per one unit.

For example, Store A sells 4 granola bars for CAD 5.20 and Store B sells 6 granola bars for CAD 7.50. To compare, find the cost per 1 bar at each store. Store A:

\frac{5.20}{4} = 1.30

CAD per bar. Store B:

\frac{7.50}{6} = 1.25

CAD per bar. Store B is cheaper per bar, so it is the better deal.

You can also scale both rates up to a common quantity instead of going down to 1. Both methods give the same conclusion, but the unit rate method is usually faster. The important thing is that you compare the same type of quantity at every step.

Unit rate =

\frac{total amount}{number of units}

Convert to unit rates (per 1 unit) to compare rates fairly.
Divide the total amount by the number of units to find the unit rate.
Both rates must describe the same type of quantity before you compare them.

Choosing a Strategy and Checking Your Answer

When you read a problem, ask two questions: Are two ratios (or rates) being compared? If yes, set up a proportion. Does the problem involve a percent of a whole, a percent change, or finding an original amount? If yes, use the percent equation or the proportion form

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

.

Some problems involve both ideas. For example: 'Prices went up 8%. A jacket that originally cost CAD 85 now costs how much?' First find 8% of CAD 85:

0.08 \times 85 = 6.80

. Then add:

85 + 6.80 =

CAD 91.80. You could also think of it as 108% of the original:

1.08 \times 85 =

CAD 91.80. Both paths give the same correct answer.

Always check your answer for reasonableness. Ask yourself: Is the answer larger or smaller than the original? Does that match the situation? For a percent increase, the answer must be larger than the original. For a better-value comparison, the unit rate of the better deal must be smaller. If your answer fails this common-sense check, look for a setup error before assuming the arithmetic is wrong.

\frac{\text{part}}{\text{whole}} = \frac{p}{100}

Use a proportion when two equivalent ratios or rates are involved.
Use the percent equation when the problem centres on a part, whole, or percent change.
Check whether your answer is larger or smaller than the original to confirm it makes sense.
Both the proportion method and the percent equation can solve many percentage problems — choose the one that feels clearest to you.

Summary: Three Types of Percentage Calculations

What You Need to Find	What You Know	Method	Example
Percent of a number (the part)	Whole and percent	Multiply: $\text{whole} \times \frac{p}{100}$	35% of 80 → $80 \times 0.35 = 28$
What percent one number is of another	Part and whole	Divide, then × 100: $\frac{\text{part}}{\text{whole}} \times 100$	28 out of 80 → $\frac{28}{80} \times 100 = 35\%$
The original whole	Part and percent	Divide: $\frac{\text{part}}{p \div 100}$	28 is 35% of ? → $\frac{28}{0.35} = 80$
Percent change	Original and new value	$\frac{\text{new} - \text{original}}{\text{original}} \times 100$	CAD 100 → CAD 115: $\frac{15}{100} \times 100 = 15\%$ increase

Worked example

Multi-Step Problem: Discount, Tax, and Best Value

A sporting goods store is selling two water bottles. Bottle X has a regular price of CAD 24.00 and is on sale for 25% off. Bottle Y has a regular price of CAD 30.00 and is on sale for 35% off. After the discount, which bottle costs less? By how many dollars?

Step 1: Find the discount amount for Bottle X.
25% off means you save 25% of the regular price. Convert 25% to a decimal: $25\% = 0.25$ . Multiply by the regular price.
$0.25 \times 24.00 = 6.00$
Step 2: Find the sale price of Bottle X.
Subtract the discount from the regular price to get what you actually pay.
$24.00 - 6.00 = 18.00$
Step 3: Find the discount amount for Bottle Y.
35% off means you save 35% of CAD 30.00. Convert 35% to a decimal: $35\% = 0.35$ .
$0.35 \times 30.00 = 10.50$
Step 4: Find the sale price of Bottle Y.
Subtract the discount from the regular price.
$30.00 - 10.50 = 19.50$
Step 5: Compare the two sale prices.
Find the difference between the two sale prices to answer 'by how many dollars?'
$19.50 - 18.00 = 1.50$

Answer: Bottle X costs CAD 18.00 after the discount. Bottle Y costs CAD 19.50 after the discount. Bottle X costs less — by CAD 1.50.

Check: Reasonableness check: Bottle X starts cheaper and has a smaller percent off, yet it ends up cheaper. Does that make sense? Yes — 25% of a smaller number can still beat 35% of a much larger number. Double-check Bottle Y using a proportion:

\frac{35}{100} = \frac{d}{30}

gives CAD 100d = 1050, so

d = 10.50

. ✓ Sale price: 30.00 - 10.50 = 19.50. ✓

Common mistakes and how to avoid them

Setting up a proportion with mismatched positions — for example, putting kilometres on top in the first fraction but litres on top in the second fraction.

Correction: Always keep the same type of quantity in the same position (top or bottom) in both fractions. Label each fraction with units before cross-multiplying.

Calculating percent change by dividing by the new value instead of the original value.

Correction: The original value is always the denominator in the percent change formula:

\frac{\text{new} - \text{original}}{\text{original}} \times 100

. The original is the starting point of the comparison.

Forgetting to convert the percent to a decimal before multiplying — for example, writing

35 \times 80 = 2800

instead of

0.35 \times 80 = 28

.

Correction: Divide the percent by 100 before multiplying:

35\% = \frac{35}{100} = 0.35

. Multiplying by 35 instead of 0.35 gives an answer 100 times too large.

Taking a percent off the sale price instead of the original price when finding a discount.

Correction: A percent discount is always calculated on the original (regular) price unless the problem clearly states otherwise. Read the problem carefully to identify the whole.

Confusing a ratio with a rate — for example, treating 'km per hour' as a dimensionless ratio and dropping the units.

Correction: A rate always carries two different units. Keep both units in your working. Including units helps you catch setup errors and correctly interpret the answer.

Lesson summary

A ratio compares quantities with the same unit; a rate compares quantities with different units; a proportion states two ratios or rates are equal.
In any proportion $\frac{a}{b} = \frac{c}{d}$ , the cross-products are equal: $a \times d = b \times c$ . Use this to find an unknown value.
Percent means out of 100. Convert to a decimal by dividing by 100 before using it in a calculation.
Percent change is calculated as $\frac{\text{new value} - \text{original value}}{\text{original value}} \times 100$ ; always divide by the original.
To compare rates fairly, convert each to a unit rate (per 1 unit) and then compare the results.
Always verify your answer: check that the size and direction of the result (larger or smaller) match what the problem describes.

Check your understanding

Question 1

A map uses a scale of 3 cm to represent 45 km. How many kilometres does 7 cm represent?

95 km
105 km
135 km
63 km

Show answer and explanation

105 km

Set up the proportion

\frac{3}{45} = \frac{7}{d}

. Cross-multiplying gives CAD 3d = 45 × 7 = 315, so

d = 105

km. Check:

\frac{3}{45} = \frac{1}{15}

and

\frac{7}{105}

=

\frac{1}{15}

. ✓

Question 2

A pair of shoes originally costs CAD 80. The price drops to CAD 60. What is the percent decrease?

20%
33%
25%
75%

Show answer and explanation

25%

Percent change

= \frac{60 - 80}{80} \times 100 = \frac{-20}{80} \times 100 = -25\%

. The negative sign confirms a decrease of 25%. Dividing by the new price of 60 instead of the original 80 is the error that produces the incorrect answer of 33%.

Question 3

Store P sells 5 pens for CAD 3.75. Store Q sells 8 pens for CAD 5.60. Which store has the lower unit price, and what is it?

Store P at CAD 0.75 per pen
Store Q at CAD 0.70 per pen
Both stores charge the same unit price
Store P at CAD 0.70 per pen

Show answer and explanation

Store Q at CAD 0.70 per pen

Store P unit price:

\frac{3.75}{5} = 0.75

CAD per pen. Store Q unit price:

\frac{5.60}{8} = 0.70

CAD per pen. Store Q is cheaper per pen at CAD 0.70.

Question 4

After a 15% tip is added, a restaurant bill totals CAD 46.00. What was the bill before the tip?

CAD 40.00
CAD 39.10
CAD 38.00
CAD 41.40

Show answer and explanation

CAD 40.00

The total represents 115% of the original bill. Let

b

be the original bill. Then

1.15 \times b = 46.00

, so

b = \frac{46.00}{1.15} = 40.00

. The original bill was CAD 40.00. Check:

40.00 \times 0.15 = 6.00

tip, and

40.00 + 6.00 = 46.00

. ✓

Key terms

Ratio: A comparison of two quantities that share the same unit, written as a:b or $\frac{a}{b}$ .
Rate: A comparison of two quantities that have different units, such as kilometres per hour or dollars per litre.
Unit rate: A rate where the second quantity is 1, making it easy to compare different rates directly.
Proportion: A statement that two ratios or two rates are equal, written as $\frac{a}{b} = \frac{c}{d}$ .
Cross-products: In the proportion $\frac{a}{b} = \frac{c}{d}$ , the cross-products are $a \times d$ and $b \times c$ . They are always equal in a true proportion.
Percent: A ratio out of 100, written with the symbol %. For example, $40\% = \frac{40}{100} = 0.40$ .
Percent change: The increase or decrease in a value expressed as a percentage of the original value: $\frac{\text{new} - \text{original}}{\text{original}} \times 100$ .
Discount: An amount subtracted from the original price, usually expressed as a percent of the original price.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 9 Mathematics (MTH1W), expectation B3.5. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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B3.5 · Solve problems with rates, percentages, and proportions

Ontario Grade 9 Mathematics

Solving Real-World Problems Using Ratios, Rates, and Percent

What you will learn

What Are Ratios, Rates, and Proportions?

Solving Problems with Proportions

Working with Percentages

Comparing Rates to Make Decisions

Choosing a Strategy and Checking Your Answer

Summary: Three Types of Percentage Calculations

Multi-Step Problem: Discount, Tax, and Best Value

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Question 4

Key terms

Continue through MTH1W

About this lesson and its review