The simplifying fractions calculator allows you to quickly simplify proper and improper fractions. The output of the calculator is represented either by a mixed number or by a proper fraction in its simplest form.

## Directions for use

- To reduce a fraction using this fraction simplifier, simply enter the numerator and the denominator of the given fraction and press “Calculate.”
- If the input fraction is proper, the calculator will return the simplest form of the fraction as an answer.
- If the input fraction is improper, a mixed number in its simplest form will be returned as an answer. The calculator will also demonstrate the detailed solution.

## Definitions

### Fraction

A fraction is defined as a part, or a proportion, of a whole. The whole can be represented by any number, value, or even an object. For example, if “the whole” is represented by a whole pie, then cutting this pie into 6 pieces will create 6 fractions, where each piece will represent one-sixth, or \$\frac{1}{6}\$ of the whole pie.

Any fraction consists of two parts – the numerator and the denominator, separated by a horizontal line, called the fractional bar. The denominator is positioned under the fractional bar, and describes the total number of parts the whole was divided into. In the fraction described above the denominator is 6, and the pie was cut into 6 pieces. The numerator is positioned above the fractional bar, and describes the number of parts we are interested in. In the example above, the numerator was 1, since we were talking about 1 of the 6 pieces. If we wanted to take 2 pieces, the resulting fraction would be \$\frac{2}{6}\$.

Fractions can also be written with the help of a diagonal line. For example, 1/3 and \$\frac{1}{3}\$ describe the same fraction.

### Proper and improper fractions

A fraction is called proper, if its denominator is larger than its numerator.

\$\frac{1}{3}\$, \$\frac{2}{50}\$, \$\frac{56}{125}\$ are the examples of proper fractions.

Similarly, a fraction is called improper if its numerator is larger than its denominator. For example, \$\frac{33}{15}\$, \$\frac{17}{8}\$, \$\frac{3}{2}\$ are all improper fractions.

Any improper fraction can be written as a mixed number – a number that consists of a whole number and a proper fraction, for example, \$5 \frac{1}{3}\$, \$12 \frac{132}{256}\$.

### Simplest form of a fraction

A fraction is in its simplest form, if its numerator and denominator do not have any common factors, apart from 1. For example, \$\frac{1}{3}\$ is a fraction in its simplest form, but \$\frac{4}{6}\$ is not. 4 and 6 have another common factor – 2, therefore, this fraction has not been written in its simplest form.

## Calculation algorithms

### Simplifying a proper fraction

To simplify a fraction, follow the steps below:

- Find the greatest common factor (GCF) of the numerator and the denominator of the fraction.
- Divide both the numerator and the denominator of the fraction by the GCF.
- The resulting fraction will be in its simplest form.

For example, let’s simplify the following fraction: \$\frac{70}{236}\$.

- All factors of 70 are: 1, 2, 5, 7, 10, 14, 35, 70.
- All factors of 236 are: 1, 2, 4, 59, 118, 236.

The greatest common factor of 70 and 236 is: 2.

- \$\frac{70}{2} = 35\$
- \$\frac{236}{2} = 118\$
- \$\frac{70}{236} = \frac{35}{118}\$

Answer: \$\frac{70}{236} = \frac{35}{118}\$

### Converting an improper fraction to a mixed number

To perform an improper fraction to mixed number conversion, carry out the following steps:

- Check if the fraction can be simplified, by identifying if there any common factors. If yes, simplify the fraction by dividing both the numerator and the denominator by the GCF.
- To find the whole number part of the final mixed number, divide the numerator by the denominator and write down only the whole number of the division result.
- Write down the proper fraction part of the mixed number, using the remainder of the division from step 2 as the numerator and the denominator of the original (simplified) fraction.

For example, let’s simplify the reciprocal of the previous fraction: \$\frac{236}{70}\$.

First, let’s simplify the given fraction, by dividing the numerator and the denominator by the GCF.

- All factors of 236 are: 1, 2, 4, 59, 118, 236.
- All factors of 70 are: 1, 2, 5, 7, 10, 14, 35, 70.

The greatest common factor of 70 and 236 is: 2.

- \$\frac{236}{2} = 118\$
- \$\frac{70}{2} = 35\$
- \$\frac{236}{70} = \frac{118}{35}\$

Now let’s divide the numerator of the resulting fraction by the denominator of the resulting fraction, and write down the whole number of the division:

$$\frac{118}{35} = 3 + the\ remainder\ of\ 13$$

The proper fraction part of the mixed number will have the remainder of the division as the numerator, so, the numerator is 13. The denominator will be the same as in the original fraction, so, the denominator is 35.

The resulting mixed number is \$3\frac{13}{35}\$.

Answer: \$\frac{236}{70} = 3\frac{13}{35}\$

## Calculation example

Fractions are commonly used in recipes, and very often you would need to convert improper fractions to mixed numbers when you want to adjust a recipe to a larger number of people.

Imagine, you want to bake some cupcakes for a party. The recipe states that the given ingredients will provide enough cupcakes for 4 people. You, however, have invited 12 guests. If the recipe says you need \$\frac{3}{4}\$ cups of flour for the cupcakes for 4 people, how much flour will you need to adjust the recipe to feed 12 guests?

### Solution

To adjust the amount of flour, you need to multiply the given amount \$\frac{3}{4}\$ by 3, since \$\frac{12}{4}\$ = 3, and you will need 3 times as much flour:

$$\frac{3}{4} × 3 = \frac{9}{4}$$

To figure out, how many cups of flour you need, you have to convert the improper fraction \$\frac{9}{4}\$ to a mixed number. Let’s follow the steps described above.

Check, if the fraction can be simplified.

- The factors of 9 are: 1, 3, 9.
- The factors of 4 are: 1, 2, 4.

The greatest common factor is 1, therefore, this fraction cannot be simplified.

To find the whole number part of the mixed number, divide the numerator by the denominator:

$$\frac{9}{4} = 2 + the\ remainder\ of\ 1$$

The proper fraction part of the mixed number will have the remainder of the division in step 2 as the numerator, so, the numerator is 1. The denominator will be the same as in the original fraction, so, the denominator is 4.

The resulting mixed number is \$2\frac{1}{4}\$.

**Answer**

To adjust the recipe for 12 people you will need to triple the ingredients.

$$\frac{3}{4} × 3 = \frac{9}{4} = 2\frac{1}{4}$$

You will need 2 and one quarter cups of flour.