# Fractions and decimals

Last updated: September 17, 2021

## What are fractions and decimals?

Fractions are numbers that address a piece of the entirety. An object or a gathering of items are separated into two halves; all the individual parts are fractions.

To understand the concept of fractions, let us take an example. Suppose it is your birthday, and your father has bought a pizza for you and your friends. When you open the box, you find the pizza is sliced into $$6$$ pieces, and there are $$6$$ friends at your house. There are $$6$$ individuals who can eat each slice of the pizza. That means every individual has $$\frac{1}{6}$$ or one-sixth of the pizza.

So, by the definition of a fraction, we can say $$1$$ piece out of the $$6$$ pieces of the pizza represents the fraction $$\frac{1}{6}$$.

### E1.5A:Use the language and notation of vulgar fraction and decimal fraction  in appropriate contexts.

Vulgar Fraction

A vulgar fraction is denoted as $$\frac{p}{q}$$. Here, $$p$$ and $$q$$ are the integers, and $$q$$ cannot be zero.

Some examples of vulgar fraction are $$\frac{7}{15}$$ and $$\frac{9}{23}$$. The upper part of a fraction is known as the numerator, and the lower part is known as the denominator.

A simple fraction is differentiated into three types:

1. Proper fraction: If the numerator of a fraction is less than the denominator, then the fraction is called a proper fraction. For example, $$\frac{5}{9}$$, $$\frac{6}{11}$$, and $$\frac{8}{17}$$.

2. Improper fraction: If the numerator of a fraction is greater than the denominator, then the fraction is called an improper fraction. For example, $$\frac{11}{7}$$, $$\frac{19}{9}$$, and $$\frac{15}{4}$$.

3. Mixed fraction: A mixed fraction comprises a whole number along with a proper fraction. For example, $$3\frac{1}{4}$$, $$4\frac{2}{5}$$, and $$9\frac{4}{7}$$.

Decimal Fraction

A decimal fraction is a fraction whose denominator is in the form of a power of 10.

An example of decimal fraction is $$\frac{7}{100}$$. Here, $$7$$ is the numerator, and $$100$$ is the denominator, which is written as $$10^{2}$$.

## Worked examples

Example 1: Calculate $$\frac{2}{9}$$ of $$36$$.

Step 1: First, divide $$36$$ into $$9$$ equal parts.

This means $$\frac{36}{9}=4$$.

Step 2: Multiply $$2$$ with the resultant $$4$$.

The final result is $$2 \times 4=8$$.

Example 2: Change $$5\frac{2}{3}$$ into an improper fraction.

Step 1: Separate the whole number and the proper fraction.

The whole number is $$5$$, and the proper fraction is $$\frac{2}{3}$$.

Step 2: Multiply and divide by $$3$$ to the whole number $$5$$.

$$5 \times \frac{3}{3}=\frac{15}{3}$$

Here, the improper fraction $$\frac{15}{3}$$ is equivalent to the whole number $$5$$.

Step 3: Add the improper fraction and proper fraction of the same denominator by adding numerators.

$$\frac{15}{3}+\frac{2}{3}=\frac{17}{3}$$

### E1.5B: Recognize equivalence and convert between these forms

Decimals

Decimal numbers are the numbers that include a decimal point in them. They have a whole part and a fractional part isolated by a decimal point.

The number on the left of the decimal point is known as a whole part. The numbers on the right of the decimal is known as a fractional part. Let us understand the place values of decimal numbers through the diagram given below.

In the above figure, a number $$39.57$$ is written. Here, $$39$$ is the whole part, and $$57$$ is the fractional part. Every digit has specified place values, as shown in the figure above. So, the place value of $$5$$ is $$5$$ tenths, and $$7$$ is $$7$$ hundredths.

## Worked examples

Example 1: Write the fraction $$3\frac{3}{10}$$ into decimal.

Step 1: First separate the whole number part and the proper fraction part.

Whole number part is $$3$$, and the fraction part is $$\frac{3}{10}$$.

Step 2: Decimal value of the proper fraction part.

$$\frac{3}{10}$$ means $$0.3$$

Step 3: Add the whole part and the decimal value part.

$$3+0.3=3.3$$

Example 2: Change the decimal $$0.052$$ into a fraction.

Step 1: Write the place value of every digit in the decimal.

It contains $$5$$ hundredths and $$2$$ thousandths.

Step 2: Change it into a fraction by multiplying the digit with its place value.

$$5 \times \frac{1}{100}=\frac{5}{100}$$ and $$2 \times \frac{1}{1000}=\frac{2}{1000}$$

Step 3: Add both values to find the resultant value

$$\frac{5}{100}+\frac{2}{1000}=\frac{50+2}{1000}=\frac{52}{1000}$$