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B2.12 · Solve problems with ratios, percents, and rates

Learn to solve problems with ratios, percents, and rates through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Number

Grade 6 Mathematics — B2.12

Ratios, percents, and rates help describe how amounts compare. A ratio can compare two kinds of items. A percent compares an amount with 100 equal parts. A rate compares amounts measured in different units. In this lesson, you will use simple drawings, tables, and multiplication to solve problems. All numbers in the examples are hypothetical; they are given for practice, not presented as real measurements or offers.

What you will learn

  • Explain what a ratio compares and use it to solve a problem.
  • Understand percent as a number of parts out of 100.
  • Use a rate to compare two kinds of measures, such as kilometres and hours.
  • Choose a sensible model and check whether an answer fits the question.

1. Ratios compare amounts

A ratio tells how one amount compares with another. For example, imagine a bag with 2 red counters and 3 blue counters. The ratio of red counters to blue counters is 2 to 3. The order matters: blue to red would be 3 to 2.
The total number of counters is 5, but the ratio of red to blue is not 2 to 5. That would compare red counters with all counters. Always read the question to see which amounts are being compared.
Ratios can describe equal groups. If a recipe uses 2 scoops of one ingredient for every 3 scoops of another, doubling both amounts keeps the same comparison: 4 scoops to 6 scoops. Each group has the same mix. To solve a problem, first identify the two amounts, then decide whether the comparison stays the same.
2:3=4:62:3=4:6
  • A ratio compares amounts in a stated order.
  • Read carefully to tell whether the comparison is part-to-part or part-to-whole.
  • Multiplying both amounts in a ratio by the same number keeps the comparison the same.

2. Percents describe parts out of 100

Percent means “out of 100.” The symbol % means percent. If 25 out of 100 squares are shaded, then 25% are shaded. The whole set is the 100 squares.
A percent can also describe part of a different-sized whole. For example, 25% of 20 means one quarter of 20, because 25 is one quarter of 100. Split 20 into 4 equal groups. Each group has 5, so 25% of 20 is 5.
To find 10% of a whole number, you can split the number into 10 equal groups. To find other percents, use a familiar part, such as 50% for half or 25% for one quarter. Check that your answer makes sense: 25% of a positive amount should be less than the whole amount.
25%=25100=1425\%=\frac{25}{100}=\frac{1}{4}
  • Percent means out of 100.
  • A percent is a part of a whole.
  • Use equal groups or a familiar fraction to find a percent of an amount.

3. Rates compare different units

A rate compares amounts measured in different units. A rate might compare kilometres with hours or pages with minutes. The units tell what is being compared.
Suppose a walker travels 12 kilometres in 3 hours at a steady pace. Think of the trip as 3 equal one-hour parts. Divide 12 kilometres equally among those 3 hours. The walker travels 4 kilometres in each hour. This is a rate of 4 kilometres per hour.
The word per means “for each.” In a rate, keep the units with the number. The number 4 alone does not tell us whether it means kilometres per hour, pages per minute, or something else. A table can help: list the total amount and the matching time, then find the amount for one equal time part when the problem allows it.
12 km÷3 h=4 km/h12\text{ km}\div3\text{ h}=4\text{ km/h}
  • A rate compares amounts with different units.
  • Per means for each.
  • Keep units in the answer and use only information stated in the problem.

4. Choose a model and check

Before calculating, name the whole or the two amounts being compared. For a ratio, state the order. For a percent, identify the whole. For a rate, identify both units.
Then choose a model. Equal groups work well for many ratio and percent questions. A table can organize a rate. A labelled bar can show parts of a whole. Write the calculation only after the model makes clear what the numbers mean.
Finally, check the answer in words. Does the ratio answer compare the requested amounts? Is the percent answer a sensible part of the whole? Does the rate include both units? A correct calculation with the wrong comparison or missing units does not fully answer the problem.
  • Identify what is being compared before calculating.
  • Use a model that matches the question.
  • Check the meaning and units of the final answer.

Rate table for the cyclist example

TimeDistance
1 hour6 kilometres
2 hours12 kilometres
3 hours18 kilometres

Worked example

Use a ratio to find a matching amount

A craft mix uses 2 cups of blue beads for every 3 cups of white beads. How many cups of white beads are needed when 6 cups of blue beads are used?
  1. Name the comparison
    The given ratio is blue beads to white beads. Keep that order: 2 cups of blue for 3 cups of white.
    CAD2:3CAD 2:3
  2. Find the scale-up
    The blue amount grows from 2 cups to 6 cups. Since 6 is 3 groups of 2, multiply the white amount by 3 as well to keep the same mix.
    2×3=62\times3=6
  3. Calculate and answer
    Three groups of 3 cups of white beads make 9 cups. The matching amount is 9 cups of white beads.
    3×3=93\times3=9
Answer: 9 cups of white beads.
Check: The ratio 6 to 9 has the same mix as 2 to 3 because both amounts were multiplied by 3.

Worked example

Find a percent of an amount

A class has 24 library books set aside for a display. One quarter of the books are adventure stories. What percent of the books are adventure stories, and how many books is that?
One quarter of 24 books
One quarter of 24 booksThe whole bar represents 24 books divided into 4 equal groups. One group is 6 books, or 25%.1/4 = 25%books

The whole bar represents 24 books divided into 4 equal groups. One group is 6 books, or 25%.

  1. Connect the fraction and percent
    One quarter means 1 of 4 equal parts. Since 100 split into 4 equal parts gives 25 in each part, one quarter is 25%.
    14=25%\frac{1}{4}=25\%
  2. Find one quarter of the books
    Split the 24 books into 4 equal groups. Each group has 6 books, so one quarter of the books is 6.
    24÷4=624\div4=6
  3. State both results
    Adventure stories make up 25% of the display books. That is 6 books. 25% of 24=6
Answer: 25%; 6 adventure books.
Check: Four equal groups of 6 make 24, so one group is one quarter of the whole.

Worked example

Solve a rate problem

For a practice problem, assume a cyclist travels 18 kilometres in 3 hours at a steady rate. How many kilometres does the cyclist travel in 1 hour?
  1. Identify the units
    The problem compares distance in kilometres with time in hours. It asks for the distance for 1 hour. 18 km in 3 h\text{ h}
  2. Split into equal time parts
    At a steady rate, the 3 hours match 3 equal one-hour parts. Divide the total distance equally among those parts.
    18÷3=618\div3=6
  3. Include the units
    Each one-hour part matches 6 kilometres. The rate is 6 kilometres per hour.
    6 km/h6\text{ km/h}
Answer: 6 kilometres in 1 hour, or 6 kilometres per hour.
Check: Three one-hour parts at 6 kilometres each total 18 kilometres.

Common mistakes and how to avoid them

Reversing the order of a ratio.
Correction: Write the quantities in the order asked. Red to blue and blue to red are different comparisons.
Treating a percent as a number of items without finding the whole.
Correction: First identify the whole amount. Then find the stated percent or fraction of that amount.
Giving a rate without units.
Correction: Write what the number measures, such as kilometres per hour, so the comparison is clear.

Lesson summary

  • A ratio compares amounts in a stated order.
  • A percent describes a part out of 100.
  • A rate compares amounts with different units.
  • Models and equal groups can help show why a calculation works.
  • Check that your answer matches the question and includes needed units.

Check your understanding

Question 1

A fruit bowl has 3 pears and 5 apples. What is the ratio of pears to apples?
  1. 3 to 5
  2. 5 to 3
  3. 3 to 8
  4. correctIndex
Show answer and explanation
3 to 5
The question asks for pears to apples, so keep that order. There are 3 pears and 5 apples.

Question 2

What is 50% of 18?
  1. 9
  2. 18
  3. 5
  4. correctIndex
Show answer and explanation
9
50% means half. Half of 18 is 9.

Question 3

A reader finishes 20 pages in 4 minutes at a steady rate. How many pages per minute is that?
  1. 5 pages per minute
  2. 16 pages per minute
  3. 80 pages per minute
  4. correctIndex
Show answer and explanation
5 pages per minute
Divide 20 pages equally across 4 minutes. That gives 5 pages for each minute.

Key terms

Ratio
A comparison of two amounts in a stated order.
Percent
A way to describe an amount as parts out of 100.
Rate
A comparison of amounts measured in different units.
Per
For each; it is used to name the units in a rate.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.12. It is a study resource, not an official curriculum publication.

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