DoAssignment.ca

DoAssignment study guide

B2.9 · Multiply whole numbers by proper fractions

Learn to multiply whole numbers by proper fractions through clear examples and targeted practice.

Ontario Grade 6 Mathematics

Number

Count equal fraction parts across groups

A fraction names equal parts of a whole. In 25\frac{2}{5}, the bottom number, 5, tells how many equal parts make one whole. The top number, 2, tells how many of those parts are counted. A proper fraction has a top number smaller than its bottom number, so it is less than one whole. Multiplying a whole number by a proper fraction means finding that many groups of the fraction. We can count the fraction parts in all the groups, then write the total as a fraction.

What you will learn

  • Explain what a whole number times a proper fraction means.
  • Use equal groups or a fraction bar to model the multiplication.
  • Multiply the whole number by the numerator and keep the denominator.
  • Check that the answer matches the number of fraction parts counted.

Begin with equal groups

Suppose there are three equal groups, and each group is one fifth. Picture three same-size strips. Divide each strip into five equal pieces and shade one piece in each strip. There are three shaded pieces altogether. Each shaded piece is still one fifth, so the total is three fifths.
This model helps us read the multiplication. The whole number tells how many groups to count. The fraction tells how much is in each group. Here, there are three groups of one fifth.
A numerator is the top number in a fraction. It tells how many equal parts are counted. A denominator is the bottom number. It tells how many equal parts make one whole. Equal parts must be the same size so the denominator means the same thing for every part.
  • The whole number tells how many groups there are.
  • The fraction tells the amount in each group.
  • The denominator names the size of each counted part.

Connect the model to a rule

If each group has two fifths and there are three groups, we count two fifths three times. That is six fifth-sized parts in all. The parts are still fifths, so we write six over five.
The numerator tells how many parts are in one group. Multiply it by the number of groups to count all the parts. Keep the denominator because the size of each part has not changed.
This gives a useful rule for a whole number multiplied by a proper fraction. The answer may have a numerator as large as, or larger than, the denominator. It still records the total number of equal parts counted. In this lesson, we will read that answer as a fraction without changing its form.
n×ab=n×abn\times\frac{a}{b}=\frac{n\times a}{b}
  • Multiply the whole number by the numerator.
  • Keep the denominator unchanged.
  • The answer counts all the parts from all the groups.

Read a fraction-bar model

A fraction bar can show how many equal parts are counted in a total. For example, if a bar is divided into five equal parts and three are shaded, the bar shows three fifths. Each shaded piece is one fifth.
When groups are combined, count the pieces from every group. A bar showing the total can make the count easy to see, as long as it has the correct number of equal parts and a clear label. The figure in the first example shows three fifths: three equal pieces out of five.
Before writing an answer, ask: How many groups are there? How many parts are in each group? What size is each part? These questions help keep the numerator and denominator in the right places.
  • Make every part equal in size.
  • Count the parts from all groups.
  • Label the total with the correct denominator.

Use the rule and check the meaning

For a multiplication question, first name the number of groups and the fraction amount in each group. Then multiply the whole number by the numerator. Keep the denominator. This counts all the parts while keeping their size the same.
Check your answer against the groups. If there are four groups with two tenths in each, there should be eight tenths altogether. The numerator counts those eight parts, and the denominator still says each part is one tenth.
Try to explain the answer in words. Saying “eight tenths” connects the written fraction to the equal parts in the model. The explanation is a useful check that the calculation answers the original question.
  • Match the whole number to the number of groups.
  • Match the numerator to the parts in each group.
  • Use the same denominator for the total.

Worked example

Three groups of one fifth

Find 3×153\times\frac{1}{5}. Think of three equal groups, with one fifth in each group.
Three fifths
Three fifthsThree equal fifth-sized parts are counted out of five equal parts in one whole.3/5 = 60%equal parts

Three equal fifth-sized parts are counted out of five equal parts in one whole.

  1. Count the groups
    There are three groups. Each group contributes one fifth, so we count one fifth three times.
    15+15+15\frac{1}{5}+\frac{1}{5}+\frac{1}{5}
  2. Count the fifths
    There are three pieces, and each piece is one fifth. Multiply three by the numerator, 1, and keep the denominator, 5.
    3×15=3×15=353\times\frac{1}{5}=\frac{3\times1}{5}=\frac{3}{5}
Answer: The product is 35\frac{3}{5}, or three fifths.
Check: The model has three groups with one fifth in each, so it shows three fifths altogether.

Worked example

Four groups of two tenths

Find 4×2104\times\frac{2}{10}. Each of four groups contains two tenths.
Eight tenths
Eight tenthsEight equal tenth-sized parts are counted out of ten equal parts in one whole.8/10 = 80%equal parts

Eight equal tenth-sized parts are counted out of ten equal parts in one whole.

  1. Name the parts in each group
    Each group has two tenths. The whole number, 4, tells us to count that amount four times.
    210+210+210+210\frac{2}{10}+\frac{2}{10}+\frac{2}{10}+\frac{2}{10}
  2. Find the total parts
    Four groups contribute eight tenth-sized parts. Since each part is still one tenth, the denominator stays 10.
    4×210=4×210=8104\times\frac{2}{10}=\frac{4\times2}{10}=\frac{8}{10}
Answer: The product is 810\frac{8}{10}, or eight tenths.
Check: Four groups with two parts in each give eight parts. Each part is one tenth.

Worked example

Six groups of one sixth

Find 6×166\times\frac{1}{6}. Imagine six equal groups, with one sixth in each group.
Six sixths
Six sixthsAll six equal sixth-sized parts are counted in one whole.6/6 = 100%equal parts

All six equal sixth-sized parts are counted in one whole.

  1. Count one sixth in each group
    There are six groups, and each group contributes one sixth. We count six one-sixth parts altogether.
    16+16+16+16+16+16\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}
  2. Write the product
    Six groups of one sixth give six sixths. The numerator is six counted parts, and the denominator stays six because each part is one sixth.
    6×16=6×16=666\times\frac{1}{6}=\frac{6\times1}{6}=\frac{6}{6}
Answer: The product is 66\frac{6}{6}, or six sixths.
Check: There are six parts, and six equal sixths make one whole.

Common mistakes and how to avoid them

Multiplying the whole number by the denominator.
Correction: Multiply the whole number by the numerator to count all the parts. Keep the denominator to show the size of each part.
Changing the denominator when counting more groups.
Correction: The groups contain parts of the same size. If each part is one tenth, the total is still counted in tenths.
Thinking the numerator in the answer must be smaller than the denominator.
Correction: The starting fraction is proper, but the total counts parts from several groups. The numerator can be as large as the denominator.

Lesson summary

  • A whole number times a proper fraction means that many groups of the fraction.
  • Multiply the whole number by the numerator to count all the parts.
  • Keep the denominator because the size of each part stays the same.
  • Check that the fraction names the total parts and their size.

Check your understanding

Question 1

What is 5×275\times\frac{2}{7}?
  1. 107\frac{10}{7}
  2. 710\frac{7}{10}
  3. 1035\frac{10}{35}
  4. 77\frac{7}{7}
Show answer and explanation
107\frac{10}{7}
Five groups of two sevenths give ten seventh-sized parts. Multiply 5 by 2 and keep 7 as the denominator.

Question 2

Which multiplication sentence represents four groups of one sixth?
  1. 4×164\times\frac{1}{6}
  2. 6×146\times\frac{1}{4}
  3. 4×614\times\frac{6}{1}
  4. 4+164+\frac{1}{6}
Show answer and explanation
4×164\times\frac{1}{6}
The whole number 4 tells how many groups there are. The fraction in each group is one sixth.

Key terms

Numerator
The top number in a fraction. It tells how many equal parts are counted.
Denominator
The bottom number in a fraction. It tells how many equal parts make one whole.
Proper fraction
A fraction with a numerator smaller than its denominator.
Product
The answer to a multiplication question.
Equal groups
Groups that contain the same amount.

Continue through Grade 6 Math

View the complete Grade 6 Math Ontario Grade 6 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 6 Mathematics, expectation B2.9. 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.

Official curriculum reference

Report a correction or ask a question