Add and subtract fractions in context

Ontario Grade 8 Mathematics

Number

Use equal-sized parts to solve situations involving fractions

Fractions describe equal parts of a whole or a unit. For example, a jug can be divided into equal parts to show how full it is. The denominator tells how many equal parts make one whole. The numerator tells how many of those parts are counted. In

\frac{3}{8}

, the jug is thought of as eight equal parts, and three parts are counted. When we combine or compare amounts, the fractions must refer to the same kind of whole or unit. We will use that idea to add and subtract fractions in situations.

What you will learn

Explain what the numerator and denominator show.
Add and subtract fractions with the same denominator.
Rewrite fractions as equivalent fractions to make denominators match.
Choose an operation from a situation and explain what the answer means.

See the parts before calculating

Picture a strip divided into eight equal sections. Shade three sections. The shaded amount is

\frac{3}{8}

. Now shade two more sections of the same strip. Five of the eight equal sections are shaded, so the total is

\frac{5}{8}

. This is a concrete way to see why the numerators are added while the denominator stays the same.

The denominator names the size of each part. The numerator counts those parts. When two fractions have the same denominator, their parts are the same size. We can count the parts together or find how many remain after taking some away.

\frac{a}{n}+\frac{b}{n}=\frac{a+b}{n}

The denominator tells the number of equal parts in one whole.
The numerator counts how many parts are being considered.
For equal denominators, add or subtract the numerators and keep the denominator.

Make equal-sized parts

Thirds and fourths are different-sized parts. One third is not the same size as one fourth. So, before adding or subtracting fractions with different denominators, rewrite them as equivalent fractions with a common denominator. Equivalent fractions name the same amount, even though they use different numbers of parts.

A common denominator is a denominator that both fractions can use. For thirds and fourths, twelve is a useful choice. One third is the same amount as four twelfths, and one fourth is the same amount as three twelfths. Now the parts are all twelfths, so they can be counted together.

To make an equivalent fraction, multiply the numerator and denominator by the same number. This keeps the amount unchanged. After calculating, simplify when possible. Simplifying means writing the same amount with smaller numbers. For example, six eighths equals three fourths.

\frac{a}{b}=\frac{a\times k}{b\times k}

Do not add or subtract denominators.
Use equivalent fractions to make the parts the same size.
Simplify a result when its numerator and denominator can both be divided by the same number.

Let the situation guide the operation

First identify what each fraction measures. Fractions being combined should describe the same kind of whole or matching unit. For example, parts of the same jug capacity can be combined. A length in metres can be combined with another length in metres.

Use addition when amounts are put together or when a question asks for a total. Use subtraction when an amount is taken away, when asking what remains, or when comparing how much more one amount is than another.

Include the unit in your answer. Then check whether the answer fits the story. When you add positive amounts, the result should be larger than either amount. When you take some away, the result should be smaller than the starting amount.

Use the story to decide whether to add or subtract.
Check that the fractions describe matching units and wholes.
State what the answer measures and check its size.

A useful routine

Read the question and name the unit. Decide whether the story describes joining amounts or taking one amount from another. If the denominators match, count the numerators. If they do not match, rewrite the fractions as equivalent fractions with a common denominator first.

Calculate, simplify if possible, and state the result in the context. An answer can be greater than one whole. For example, five fourths is one whole and one fourth. The amount is still a count of fourth-sized parts. Check that this answer fits what the situation describes.

Keep track of the whole and the unit.
Make denominators match before combining fractions.
Explain the result in words, not just as a number.

Worked example

Filling a water jug

A jug contains

\frac{3}{8}

of its capacity in water. Mina pours in another

\frac{2}{8}

of the jug's capacity. What fraction of the jug's capacity is filled now?

Jug capacity in eighths

Five of eight equal parts are filled after combining three eighths and two eighths.

Choose the operation
The water already in the jug and the new water are put together, so add. Both amounts describe parts of the same jug.
Count eighths
Both fractions count eighths, so the parts are already the same size. Add the numerators and keep the denominator.
$\frac{3}{8}+\frac{2}{8}=\frac{5}{8}$
State and check the amount
The jug is filled to five eighths of its capacity. This is less than one whole jug, which makes sense because five of the eight equal parts are filled.

Answer: The jug is filled to

\frac{5}{8}

of its capacity.

Check:

\frac{5}{8}<1

Worked example

Ribbon left after cutting

A ribbon is

\frac{5}{6}

of a metre long. A piece that is

\frac{1}{3}

of a metre long is cut off. How much ribbon remains?

Ribbon remaining

Three of six equal parts remain after two sixths are cut from five sixths of a metre.

Choose the operation
A piece is cut off the original length, so subtract. Both measurements are in metres.
Rename thirds as sixths
The starting length is counted in sixths. Rewrite one third as an equivalent fraction in sixths so both amounts use equal-sized parts.
$\frac{1}{3}=\frac{2}{6}$
Subtract and simplify
Subtract two sixths from five sixths. Three sixths remain. Divide both numbers by three to write the same amount more simply.
$\frac{5}{6}-\frac{2}{6}=\frac{3}{6}=\frac{1}{2}$
State and check the result
The remaining ribbon is one half of a metre. It is shorter than the original ribbon, as expected after a piece was cut off.

Answer: The ribbon remaining is

\frac{1}{2}

metre.

Check:

\frac{1}{2}<\frac{5}{6}

Worked example

Combining two garden paths

One garden path is

\frac{2}{3}

of a metre long. A second path is

\frac{3}{4}

of a metre long. What is their total length?

Choose addition
The question asks for the total length, so add. Both lengths are in metres, but their denominators differ. Make the parts the same size before adding.
Rewrite using twelfths
Twelve can be split into equal groups of thirds and fourths. Two thirds becomes eight twelfths, and three fourths becomes nine twelfths.
$\frac{2}{3}=\frac{8}{12},\quad \frac{3}{4}=\frac{9}{12}$
Add and interpret
Add the counts of twelfths. The total is greater than one metre, so the result can be written as more than one whole metre.
$\frac{8}{12}+\frac{9}{12}=\frac{17}{12}$

Answer: The total length is

\frac{17}{12}

metres.

Check:

\frac{17}{12}>1

Common mistakes and how to avoid them

Adding or subtracting both the numerators and denominators.

Correction: If denominators match, keep that denominator and work with the numerators. If they differ, rewrite equivalent fractions first.

Combining fractions that describe different units or unrelated wholes.

Correction: Check what each fraction measures. Combine amounts only when their units and reference whole make sense together.

Changing a fraction's value while making an equivalent fraction.

Correction: Multiply the numerator and denominator by the same number. This changes the way the amount is divided, not the amount itself.

Leaving out what the answer measures.

Correction: Include the unit and explain what the fraction means in the situation.

Lesson summary

A fraction counts equal-sized parts of a whole or unit.
For matching denominators, add or subtract numerators and keep the denominator.
For different denominators, rewrite equivalent fractions with a common denominator before calculating.
Use the context to choose an operation, include the unit, and check whether the answer makes sense.

Check your understanding

Question 1

A container has

\frac{2}{7}

of its capacity filled. Another

\frac{3}{7}

is added. What fraction is filled?

$\frac{5}{7}$
$\frac{5}{14}$
$\frac{6}{7}$
$\frac{1}{7}$

Show answer and explanation

\frac{5}{7}

Both fractions count sevenths. Add the numerators and keep the denominator:

\frac{2}{7}+\frac{3}{7}=\frac{5}{7}

. The result is five sevenths of the capacity.

Question 2

A walk is

\frac{3}{4}

of a kilometre long. A person has completed

\frac{1}{2}

of a kilometre. How much farther is left?

$\frac{1}{4}$ kilometre
$\frac{2}{2}$ kilometre
$\frac{1}{6}$ kilometre
$\frac{5}{4}$ kilometres

Show answer and explanation

\frac{1}{4}

kilometre

Subtract the completed distance from the total. Rewrite one half as two fourths, then subtract:

\frac{3}{4}-\frac{2}{4}=\frac{1}{4}

kilometre.

Question 3

A recipe uses

\frac{1}{3}

of a cup of one ingredient and

\frac{1}{6}

of a cup of another. How much is used altogether?

$\frac{1}{2}$ cup
$\frac{2}{9}$ cup
$\frac{2}{6}$ cup
$\frac{1}{9}$ cup

Show answer and explanation

\frac{1}{2}

cup

Rewrite one third as two sixths. Then add:

\frac{2}{6}+\frac{1}{6}=\frac{3}{6}=\frac{1}{2}

cup.

Key terms

Denominator: The bottom number in a fraction. It tells how many equal parts make one whole.
Numerator: The top number in a fraction. It tells how many parts are counted.
Equivalent fractions: Fractions that use different numbers but name the same amount.
Common denominator: A denominator that lets fractions be counted in equal-sized parts.
Simplify: Write a fraction with smaller numerator and denominator while keeping the same amount.

Continue through Grade 8 Math

View the complete Grade 8 Math Ontario Grade 8 Mathematics curriculum and lessons

About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 8 Mathematics, expectation B2.5. 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.

Official curriculum reference

Report a correction or ask a question

B2.5 · Add and subtract fractions in context

Ontario Grade 8 Mathematics

Use equal-sized parts to solve situations involving fractions

What you will learn

See the parts before calculating

Make equal-sized parts

Let the situation guide the operation

A useful routine

Filling a water jug

Ribbon left after cutting

Combining two garden paths

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