Fractions - formulas and examples

Fractions

The terms we use for fractions are the "numerator" and the "denominator". The numerator is the number of parts we have and the denominator is the total number of parts that make up the whole.

• The numerator is read with cardinal numbers: one, two, three, ten, twenty-four, etc.
• The denominator is read with fractions: Half, third, quarter, etc.

Adding

Finding the sum of fractions: When calculating fractions, a distinction is made between sums with equal and unequal denominators.

Sum of fractions with the same denominator:

To add fractions with the same denominator, you must add the numerators and use the same denominator.

Example:

Calculation: 3/4 2/4

As the two fractions have the same denominator, we must keep the same denominator, namely 4, and add the numerators.

4 2 = 5

And the result of the sum of the fractions is: 3/4 2/4 = 5/4

Sum of fractions with different denominators:

To add fractions with different denominators, you must first find a common denominator: This is the smallest common multiple of the denominators you have. Then we multiply the numerators by the number we multiplied the denominator by. Finally, we add the numerators that we have obtained and keep the same denominator.

Example:

Calculation: 2/3 4/5

The first thing to do is to find a common denominator between 3 and 5. To do this, we calculate the least common multiple between the two numbers.

3 * 5 = 15

So 15 is the common denominator of the two fractions.

Now we need to multiply each numerator by the number we multiplied the denominator by. To do this, we divide the lowest common multiple by the initial denominator and multiply the result by the numerator of this subset.

For the first fraction:

15 / 3 = 5

5 x 2 = 10

10 is therefore the numerator of the first fraction.

For the second fraction:

15 / 5 = 3

3 x 4 =12

12 is therefore the numerator of the second fraction.

2/3 4/5 = 10/15 12/15

Now we just have to add the numerators:

10 12 = 22

And the result of the sum of the fractions is 22/15

Subtract

Finding the difference between fractions: The denominator is also decisive when subtracting fractions:

If the fraction has the same denominator:

Write the denominator that the fractions have in the last fractional part. Subtract the numerators and write the solution in the last fractional part.

Example:

7/3 - 2/3 = 5/3

If the fractions have different denominators:

Determine the least common multiple of the denominators of the fractions. Start creating the new replacement fractions with the lowest common multiple as the denominator of these new fractions. The second fraction should have the same denominator as the other fractions. Subtract the numerators and write the solutions in the last fraction part.

Example:

2/3 - 5/3 becomes 10/15 - 9/15 = 1/15

Multiplying

To multiply fractions, only the following steps need to be followed:

1. Simplify the fractions: any numerator can be simplified with any denominator.
2. Multiply fractions in a line: Multiply the denominators to get the final denominator and multiply the numerators to get the final numerator.

Example:

4/8 * 15/9

First we should simplify the fractions so that they are easier to multiply afterwards. So to simplify the whole thing, we will break each number down into its main factors.

4 = 2 * 2

8 = 2 * 2 * 2 * 2 * 2

15 = 3 * 5

9 = 3 * 3

And we replace each number in the fractions with its principal factors.

4/8 * 15/9 = 2 * 2/2 * 2 * 2 * 3 *5/3 *3

Now we simplify it and delete the numerators and denominators that are equal. And we are left with the result of the multiplication, which is 5/6.

How to multiply fractions by a whole number:

Multiplying a fraction by a whole number is very simple, we just make the whole number a fraction by putting 1 as the denominator.

Example:

3/6 * 7

The fraction 3/6 can be simplified, as we saw in the previous example, by breaking it down into its prime factors and shortening it to 1/2.

We turn the whole number 7 into a fraction by placing a 1 as the denominator: 7/1.

Now we multiply the denominators: 2 * 1 = 2.

We multiply the numerators: 1 * 7 = 7

This gives us the fraction 7/2.

3/6 * 7 = 3/3 * 2 * 7/1 = 7/2

Dividing

Reverse and multiply:

• Step 1: Reverse the second fraction. In other words, swap the numerator for the denominator.
• Step 2: Simplify each numerator with any denominator.
• Step 3: Multiply the values.

Example:

12/5 : 6/4

• Step 1: We exchange the second fraction: 6/4. This becomes 4/6.
• Step 2: We simplify the numerators with the denominators.

The numerators are:

12 = 2 * 2 * 2 * 3 * 3

4 = 2 * 2

The denominators are:

5 = 5

6 = 2 * 3

We can simplify a 2 and a 3 from both numerator and denominator and call this process "cross multiplication" if one numerator shows a common factor with the other denominator.

And then we multiply:

12/5 * 6/4 = 12/5 * 4/6 = 2 * 2 * 2/5 * 2 * 2/2 * 3 = 8/5

Another method: multiply crosswise

This method involves multiplying the numerator of the first fraction by the denominator of the second fraction and then entering the answer in the numerator of the resulting fraction. Next, we multiply the denominator of the first fraction by the numerator of the second fraction and then write the solution in the denominator of the resulting fraction.