Solve multistep rational-number, ratio, rate, and percent problems

Ontario Grade 8 Mathematics

Number

Ontario Grade 8 Mathematics | B2.1

A multistep problem needs more than one calculation. You might track a temperature change, share a total in a ratio, or find a discount and then the new price. A rational number is a number that can be written as a fraction. Whole numbers, fractions, decimals, and negative numbers are rational numbers. Start by identifying what each number means. Then choose calculations that fit the story.

What you will learn

Choose calculations that match the meaning of a problem.
Solve multistep problems with rational numbers, ratios, rates, and percents.
Show clear steps and check whether an answer makes sense.

1. Identify the quantities

Before calculating, find the information given and what the question asks you to find. Include units, which are labels such as badges, hours, or degrees Celsius. The same number can describe different things, so units help keep them clear.

A ratio compares quantities using parts. For example, a ratio of 2 to 3 describes 2 equal parts of one quantity for every 3 equal parts of another. A rate compares quantities with different units, such as kilometres per hour. A percent describes an amount out of 100, so 25% means 25 out of 100.

Rational-number problems can include positive and negative values. On a number line, moving right increases the value and moving left decreases it. Adding a positive number moves right; adding a negative number moves left. Subtracting a positive number moves left.

Name the quantities and their units.
A ratio compares parts; a rate compares quantities with different units.
A percent is an amount out of 100.

2. Choose a model and plan

A model can help show what the numbers mean. Use a number line to track positive and negative changes. Use a fraction bar to show equal parts of a whole. For a ratio that divides a total, count the total number of parts first.

To divide a total in a ratio, add the ratio numbers to find the total parts. Divide the known total by the number of parts to find the amount in one part. Then multiply by the number of parts in the group you need.

For a rate, keep the units visible. If someone travels 18 kilometres in 3 hours at a steady rate, divide 18 kilometres by 3 hours to get 6 kilometres per hour. To find the distance travelled at that rate over another time, multiply the rate by the time.

For a percent problem, find the percent amount before deciding what to do next. A discount is subtracted from the original amount. An increase is added to the starting amount. Check whether the question asks for the change or the amount after the change.

Choose a model that matches the quantities.
For a ratio sharing a total, find one part before finding a group’s share.
Keep units attached to rates and amounts.

3. Calculate and check

Write one calculation at a time when a problem has several steps. Give intermediate amounts a clear meaning. A discount amount and the price after the discount are not the same amount.

For positive and negative changes, follow the direction described in the problem. A rise in temperature is a positive change, while a fall is a negative change. Keep the values in the order the changes happen.

Check your answer against the story. The shares in a ratio should add back to the total. A discount should leave a smaller price. A temperature that rises and then falls should reflect both changes.

An estimate can help you spot an error. For example, 20% of an amount near 50 should be near 10. If your answer is much larger, review the calculation.

Show each important step and what it represents.
Use the story to check the direction and size of an answer.
Check that amounts and units fit the question.

4. Try the method

Before calculating, ask: What is the total? What are the parts? Are the units the same? Does the question ask for a share, a rate, a percent amount, or a final amount?

Here are three practice prompts. A temperature is

-4^\circ\text{C}

and rises by

7^\circ\text{C}

, then falls by

3^\circ\text{C}

. Find the final temperature. A group shares 42 counters in the ratio 4 to 3. Find the larger share. A price of CAD 60 is reduced by 15%. Find the reduced price.

For each prompt, write the calculations in a clear order. Add a label or unit to the answer, then check that it fits the situation.

Identify the total, parts, units, and requested amount.
Show enough work to make your reasoning clear.
Check the answer in the context of the problem.

Worked example

Track a changing temperature

At the start of a morning, the temperature is

-6^\circ\text{C}

. It rises by

9^\circ\text{C}

, then falls by

4^\circ\text{C}

. What is the final temperature?

Temperature changes

Start at −6, move right by 9 to 3, then left by 4 to −1. The positions mark the start, the temperature after the rise, and the final temperature.

Represent the changes
A rise is a positive change and a fall is a negative change. Start at $-6$ and include each change in the order it happens.
$-6+9-4$
Calculate
The rise takes the temperature from $-6$ to $3$ . The fall then takes it from $3$ to $-1$ .
$-6+9=3,\quad 3-4=-1$

Answer: The final temperature is

-1^\circ\text{C}

.

Check: The two changes together make a rise of

5^\circ\text{C}

. Starting at

-6^\circ\text{C}

and rising by 5 gives

-1^\circ\text{C}

.

Worked example

Divide a total using a ratio

A club has 56 paper badges. The badges are divided between the art group and the music group in the ratio 3 to 5. How many badges does the music group receive?

Music group’s share of the total

The whole is 8 equal parts. The music group receives 5 of them.

Count the equal parts
The art group has 3 parts and the music group has 5 parts. Together, the full set of badges is divided into 8 equal parts.
$3+5=8$
Find one part
Divide all 56 badges equally among the 8 parts. Each part represents 7 badges.
$56\div 8=7$
Find the music share
The music group receives 5 parts. Multiply the 7 badges in one part by 5.
$7\times 5=35$

Answer: The music group receives 35 badges.

Check: The art group receives 3 parts, or 21 badges. The shares total

21+35=56

badges.

Worked example

Find a percent discount

A backpack has an original price of CAD 80. A store offers a 15% discount. What is the price after the discount, before any other charges?

Discount as part of the original price

The bar is divided into 20 equal parts. Three parts represent the 15% discount, and the other 17 parts represent the remaining 85%.

Find the discount
A 15% discount means 15 out of every 100 of the original price. Use the decimal 0.15 to find that amount.
$0.15\times 80=12$
Subtract the discount
The discount lowers the original price. Subtract CAD 12 from CAD 80.
$80-12=68$

Answer: The price after the discount is CAD 68.

Check: The discount and reduced price add to the original price:

12+68=80

. The reduced price is also less than the original price.

Common mistakes and how to avoid them

Adding the two numbers in a ratio and treating the sum as a group’s share.

Correction: The sum counts all the equal parts. Divide the total by that number to find one part, then multiply by the number of parts in the group.

Subtracting 15 from CAD 80 to find a 15% discount.

Correction: A percent is not the same as a money amount. Find 15% of the original price, then subtract that discount.

Ignoring the negative sign in a temperature problem.

Correction: Use the sign to show the direction of the change. A rise adds a positive change; a fall subtracts.

Giving a rate without its units.

Correction: Include units such as kilometres per hour. The units explain what quantities the rate compares.

Lesson summary

Name the quantities, units, and amount the question asks for.
Choose a model or calculation that fits the meaning of the problem.
For a ratio sharing a total, find the amount in one part before finding a group’s share.
For a percent discount, find the discount amount and subtract it from the original amount.
Check that the answer’s size, direction, and units make sense.

Check your understanding

Question 1

A temperature is

-3^\circ\text{C}

. It rises by

8^\circ\text{C}

and then falls by

2^\circ\text{C}

. What is the final temperature?

$3^\circ\text{C}$
$-13^\circ\text{C}$
$7^\circ\text{C}$
$-1^\circ\text{C}$

Show answer and explanation

3^\circ\text{C}

The changes give

-3+8-2=3

. The final temperature is

3^\circ\text{C}

.

Question 2

A total of 45 beads is shared in the ratio 2 to 3. How many beads are in the larger share?

18 beads
27 beads
30 beads
15 beads

Show answer and explanation

27 beads

There are 5 parts, so one part is

45\div5=9

beads. The larger share has 3 parts:

3\times9=27

beads.

Question 3

A game costs CAD 50 and is reduced by 20%. What is the reduced price?

CAD 40
CAD 30
CAD 45
CAD 48

Show answer and explanation

CAD 40

The discount is

0.20\times50=10

dollars. Subtract it from the original price:

50-10=40

.

Key terms

Rational number: A number that can be written as a fraction. Whole numbers, fractions, decimals, and negative values are included.
Ratio: A comparison of quantities using equal-sized parts.
Rate: A comparison of quantities measured in different units.
Percent: An amount described as parts out of 100.
Unit: A label that tells what is being counted or measured, such as badges, hours, or degrees Celsius.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.1. 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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B2.1 · Solve multistep rational-number, ratio, rate, and percent problems

Ontario Grade 8 Mathematics

Ontario Grade 8 Mathematics | B2.1

What you will learn

1. Identify the quantities

2. Choose a model and plan

3. Calculate and check

4. Try the method

Track a changing temperature

Divide a total using a ratio

Find a percent discount

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through Grade 8 Math

About this lesson and its review