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Performing Arithmetic with Fractions

Adding and Subtracting
The simplest case when adding and subtracting fractions is the case when the
denominators of all fractions are the same. When adding fractions with like
denominators, simply add the numerators and keep the common denominator.

Example 1
To add the fractions and , add 1+3 to get 4 in the numerator, and retain the 5 in
the denominator throughout the solution . So we have .

The first step in adding or subtracting fractions with unlike denominators is to find a
common denominator of the fractions to be added. The easiest way to do this, especially
when adding two fractions , is to multiply each fraction by the denominator of the other.
For example, if you have two fractions, and , take and . The
resulting fractions are and , and the denominators are the same.

The final step when adding or subtracting fractions is simply to add (or subtract) the
numerators and keep the common denominator. Continuing with the above example we
have .

Example 2
Consider the fractions and . To add and , first multiply by and by
to get and . Then add the numerators to arrive at the final answer.


In other cases, it may be possible to find a multiple of the two denominators that is less
than the product of the two numbers . This is called the least common multiple or LCM.
It often occurs that the LCM is one of the denominators .

Example 3
In the fractions and , the denominators have an LCM of 4, since 2 · 2 = 4 and
4 ·1 = 4 . In this case, it would be simpler to leave the second fraction as it is and
multiply the first fraction by . So adding these two fractions we get

The reason we retain the denominator across and add only the numerator is that the
numerator represents how many pieces of the whole exist; while the denominator
represents how many pieces the whole is divided into . The example above is shown in
the figure below, where you can see that the above result is true.

Figure 1

Multiplying
Before multiplying with fractions, it is important to understand the basic concepts behind
multiplication with integers. As shown in the figure below, multiplication may be
interpreted as the area of a rectangle where the quantities multiplied are the length of the
sides and the resulting quantity is the value of the area inside. The lengths of the sides
below are 2 and 3. When these are multiplied together, the area inside is equal to 6 unit
squares. So we see that 2 · 3 = 6 .

Figure 2

The process is similar for multiplying fractions. In this case, we allow the denominator
to be the whole length of the side and the numerator to be the portion of the side
represented by the fraction.

Example 4
This time, lets multiply the fractionsand .
The multiplication is represented in figure 3 to the
right.

In this case, notice that the lengths of the sides, 2 and 3, are represented by the
denominators and the total area of the figure is the denominators multiplied together.
In this case 2 * 3 = 6 unit squares. Now, the lengths of the shaded sides, 1 and 2, are
represented by the numerators and the area of the shaded area is the numerators
multiplied together. In this case, 1 * 2 = 2 unit squares. So, the final result is two
unit squares out of six unit squares, represented by the fraction , where 2 represents
the shaded region and 6 represents the whole region.

This means that when multiplying fractions, a common denominator is not required.
Instead, multiply the denominators to get the total number of parts and multiply the
numerators to get the number of parts actually used.

In other words, if you have fractions and , multiplying them together results in the
fraction. As in the example above, .

Dividing
It is also important to understand the basic concept behind dividing before dividing
fractions. Consider the division problem 6 ÷3. The figure below shows how dividing six
figures into groups of three results in the answer of two.

6 men divided into two group of 3 men
each makes 2 groups
.

Figure 4

Now, when dividing with fractions, the result is similar.

Example 5
Consider the problem . Then you would divide the three units into groups of
half units, as shown in figure 5 below.

Each circle represents one of the 3 units, and
the three altogether are divided into 6 groups
of half a unit each.

Figure 5

In this case, we see that results in six total units.
Looking carefully, you may notice that is the same as . This is true in
all cases. Given the calculation , take the reciprocal of the second fraction (or a
denominator if the two fractions are expressed as a ratio ), and then multiply the two
resulting fractions together. The result in this case is .

Example 6
Calculate . Remember to simplify the fraction when you are finished.
Solution:

In a diagram, means that you have 2 pieces
divided into groups of 3, so you have three spaces for
pieces with only 2 filled, as shown to the right.

3 spaces and 2 are filled
Next, we divide each piece into groups of size each, resulting in 12 total pieces as
shown in the diagram below.

2 pieces divided into groups
of size one sixth results in
12 pieces

Since there are 12 pieces in 3 spaces or units, we see
that the resulting fraction is , which is the same as
the result we found in the calculation above .

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