1.02 Add and subtract fractions and mixed numbers

Suppose we want to find \dfrac{2}{7}+\dfrac{3}{7}.

Here we have a circle with 2 sevenths shaded and a circle with 3 sevenths shaded.

Notice that the parts of each circle are the same size.

Since the parts are the same size, we can place one circle on top of the other. Now, we can see 5 sevenths of the circle are shaded in.

So, we can conclude that \dfrac{2}{7}+\dfrac{3}{7}=\dfrac{5}{7}.

When the denominators are the same (called like denominators), we are adding quantities of the same size. This means we can add the number of shaded pieces, without needing to change the number of parts in the whole.

Mathematically, we are adding the numerators but keeping the same denominator.

Suppose we want to find \dfrac{3}{7}-\dfrac{2}{7}.

Using the same circles, we can take 2 sevenths away from 3 sevenths. The part that remains is 1 seventh of the circle.

So, we can conclude that \dfrac{3}{7}-\dfrac{2}{7}=\dfrac{1}{7}.

When the denominators are the same, we are subtracting quantities of the same size. Again, the number of shaded pieces is changing, but the number of parts in a whole stays the same.

Mathematically, we are subtracting the numerators but keeping the same denominator.

Examples

If the denominators of two fractions are different (called unlike denominators), then we are not adding quantities of the same size.

We can use area models to change fractions to equivalent fractions, so the denominators are the same. Then, we can add the equivalent fractions.

Consider \dfrac{2}{9}+\dfrac{3}{4}. These two fractions look like this:

If we tried to add the pieces together, how could we write the denominator?

We cannot place one circle on top of the other because the number of parts in a whole are different.

Before we can add these two fractions, we need to rewrite them with the same denominator. To do this, we create equivalent fractions.

The denominators here are 9 and 4, so the least common multiple of the denominators is 4\cdot9=36. This is sometimes called the least common denominator.

Now, we can rewrite the fractions with a denominator of 36.

The fractions can now be divided into the same number of parts. After shading the correct amount of pieces, the fractions look like this.

Since the sizes of the parts are the same, we can add the fractions together.

This shows that \dfrac{2}{9}+\dfrac{3}{4}=\dfrac{8}{36}+\dfrac{27}{36}=\dfrac{35}{36}

When the denominators are different, we create equivalent fractions with the same denominator, then add or subtract the numerators.

Examples

Mixed numbers have a whole number part and a fractional part. There are two methods for adding or subtracting mixed numbers:

1. Add or subtract the whole parts, then add or subtract the fractional parts.

   • If the fractional parts do not have the same denominator, we need to rewrite them with the same denominator before adding or subtracting.

2. Convert the mixed numbers into improper fractions, then add or subtract the improper fractions.

   • If the denominators are different, we need to rewrite them with the same denominator before adding or subtracting.

Examples

Example 3

Fill in the boxes to show the work for:

8 \dfrac {1}{3} - 6 \dfrac {3}{4}

a

Rewrite as improper fractions.

\dfrac {⬚}{3} - \dfrac {⬚}{4}

Worked Solution

Create a strategy

Multiply the whole number by the denominator, then add the numerator.

Apply the idea

\displaystyle 8 \dfrac {1}{3} - 6 \dfrac {3}{4}	\displaystyle =	\displaystyle \dfrac {8\cdot3+1}{3} - \dfrac {6\cdot4+3}{4}	Multiply the whole number by the denominator
	\displaystyle =	\displaystyle \dfrac {24+1}{3} - \dfrac {24+3}{4}	Evaluate the multiplication
	\displaystyle =	\displaystyle \dfrac {25}{3} - \dfrac {27}{4}	Evaluate the addition

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b

Rewrite with a common denominator.

\dfrac ⬚⬚ - \dfrac ⬚⬚

Worked Solution

Create a strategy

Find the least common multiple (LCM) of the denominators, and multiply each numerator and denominator by the quotient obtained by dividing the LCM by their respective denominators.

Apply the idea

The LCM of 3 and 4 is 12.

So we need to multiply both parts of \dfrac {25}{3} by 12 \div 3=4, and multiply both parts of \dfrac {27}{4} by 12 \div 4=3.

\displaystyle \dfrac {25}{3} - \dfrac {27}{4}	\displaystyle =	\displaystyle \dfrac {25 \cdot 4}{3 \cdot 4} - \dfrac {27 \cdot 3}{4 \cdot 3}	Multiply \dfrac {25}{3} by \dfrac {4}{4} and \dfrac {27}{4} by \dfrac {3}{3}
	\displaystyle =	\displaystyle \dfrac {100}{12} -\dfrac {81}{12}	Evaluate

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c

Evaluate the difference.

\dfrac {⬚}{⬚}

Worked Solution

Create a strategy

Evaluate the answer from part (b).

Apply the idea

\displaystyle \dfrac {100}{12} -\dfrac {81}{12}

\displaystyle =

\displaystyle \dfrac {19}{12}

Subtract the numerators

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Example 4

For the school fair, Aimee is making a batch of iced tea by mixing 4\dfrac{1}{2} liters of water with \dfrac{3}{8} of a liter of tea concentrate. What is the total volume of the iced tea mixture in liters?

Worked Solution

Create a strategy

Add the volume of water to the volume of tea concentrate.

Apply the idea

\displaystyle \text{Total volume}	\displaystyle =	\displaystyle 4\dfrac{1}{2} + \dfrac{3}{8}	Write the water and tea concentrate volumes
	\displaystyle =	\displaystyle \dfrac{9}{2} + \dfrac{3}{8}	Convert the mixed number to an improper fraction
	\displaystyle =	\displaystyle \dfrac{36}{8} + \dfrac{3}{8}	Find a common denominator
	\displaystyle =	\displaystyle \dfrac{39}{8}	Add the fractions
	\displaystyle =	\displaystyle 4\dfrac{7}{8}	Convert back to a mixed number

The total volume of the iced tea mixture is 4\dfrac{7}{8} liters.

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