Arithmetic Problems: Fractions


A fraction is a numerical quantity that is not whole. They are often encountered in weights and measurements. Examples include one-half (12), a third (13), and one-quarter (14).

A fraction is made up of two values: a numerator and a denominator. The numerator is the top value of the fraction and indicates how many pieces of a whole you have. The denominator is the bottom value of the fraction and indicates how many pieces make up the whole.

In the case of one-half (12), 2 pieces makes a whole (the denominator), and you have 1 piece (the numerator). In other words, you have 1 of 2 pieces, commonly referred to as one-half.

Simplifying Fractions

A fraction is simplified (or reduced) when there is no number that can be divided into both the numerator and denominator.

For example, in the fraction 24 (two-quarters) the numerator and the denominator can both be divided by 2, making the simplified fraction 12 (one-half).

In the example 1035 the top and the bottom values can both be divided by 5, making the simplified fraction 27.

Adding & Subtracting Fractions

To add or subtract fractions:

  1. First, find their least common denominator.
  2. Second, add (or subtract) the the numerator.
  3. Finally, simplify the answer.

In the example of 16 + 26 = 12, because the fractions 16 and 26 already have a common denominator (6), we simply add the numerators 1 and 2 together (making 3) and place the result over the common denominator (6), giving us 36. Finally, we simplify our answer of 36, giving us a final answer of 12.

In the example of 25 + 12 = 910, the denominators are different. So we must first find the least common denominator. We find the number 10 can be divided by both 5 and 2, so 10 is our least common denominator. Now we adjust the numerator in both fractions to their new common denominator. In the case of 25 we know 5 goes into 10 2 times, so we multiply 2 by 2, giving us 410. In the case of 12 we know 2 goes into 10 5 times, so we multiply 1 by 5, giving us 510. Now our problem becomes 410 + 510. Both of these fractions have a common denominator of 10, so now we add the numerators 4 and 5 together (giving us 9) and place the answer over our common denominator of 10, giving us an answer of 910. Since there is no number that factors into both 9 and 10, we don't need to simplify our answer.

Multiplying Fractions

To multiply fractions:

  1. First, multiply the numerators together.
  2. Second, multiply the denominators together.
  3. Finally, simplify the answer.

For example: 2512 = 210 which simplifies to 15.

Dividing Fractions

To divide fractions:

  1. First, get the reciprocal of one of the fraction, by flipping it.
  2. Second, multiply the numerators together.
  3. Third, multiply the denominators together.
  4. Finally, simplify the answer.

For example: 25 ÷ 12 = 2521 = 45