# How to calculate fractions

## What is a fraction?

A fraction is defined as a part or a section of a whole number or quantity. A fraction consists of two numbers that are written one above the other and are separated by a horizontal line. The number present above the line, is called the ‘numerator’ whereas the number present below the horizontal line, is the denominator. A denominator represents the total number of equal parts into which something needs to be divided. The numerator depicts how many of these equal parts are to be considered.

**Vulgar Fraction**

Any number that is written in the form of one integer over another integer, both separated by a horizontal line, is referred to as a ‘vulgar fraction’.

Some examples of vulgar fractions are:

$$\frac{23}{41}$$, $$\frac{21}{59}$$, $$\frac{71}{23}$$

When the value of the numerator is less than that of the denominator, it is known as a ‘proper fraction’ whereas all the rest are referred to as ‘improper fractions’.

**Decimal Fraction**

A decimal fraction is defined as the fraction whose denominator is $$10$$ or a multiple of $$10$$, such as $$100$$, $$1000$$, etc.

These are written in the form of a decimal without any denominator. Some examples of decimal fractions are:

$$\frac{9}{10}$$, $$\frac{21}{100}$$, $$\frac{73}{1000}$$

These are written by using a decimal point, as stated below.

$$\frac{9}{10}$$ is a decimal fraction and is also written as $$0.9$$.

$$\frac{21}{100}$$ is a decimal fraction and is also written as $$0.21$$.

$$\frac{73}{1000}$$ is a decimal fraction and is also written as $$0.073$$.

Consider the example of a fruit basket that contains different fruits like $$4$$ apples, $$5$$ guavas, $$2$$ oranges, and $$3$$ bananas. The fraction of guava among these is expressed as $$\frac{5}{14}$$, as there are $$5$$ guava among a total of $$14$$ fruits. The fraction obtained, that is $$\frac{5}{14}$$, is an example of a ‘vulgar fraction’.

**E1.5A: Use the language and notation of simple and decimal fractions and percentages in appropriate contexts**

In a given fraction, the numerator and denominator can be a whole number, natural number, or real number. A fraction can be negative as well as positive. There are four mathematical operations which are addition, subtraction, multiplication, and division. If there are two fractions, different mathematical operations can be performed on them.

For example, if a pizza is divided into $$4$$ equal parts, and $$\frac{1}{4}$$ of the pizza is eaten. What fraction of the pizza is left?

**Worked examples**

**Example 1:** Evaluate $$2\frac{2}{3}$$ of $$\frac{1}{4}$$.

**Step 1: Convert the mixed fraction $$2\frac{2}{3}$$ to a simple fraction.**

$$2\frac{2}{3}=\frac{8}{3}$$

**Step 2: Multiply the fraction $$\frac{8}{3}$$ with $$\frac{1}{4}.**

$$\frac{8}{3} \times \frac{1}{4}=\frac{2}{3}$$

$$\frac{2}{3}$$

**Example 2: **If Sara eats 10 chocolates from a box containing 48 chocolates, what fraction of chocolates is remaining in the box?

**Step 1: Find out the remaining chocolates in the box by subtracting $$10$$ from $$48$$.**

$$48-10=38$$

**Step 2: Express this as a fraction by writing the number of remaining chocolates in the numerator and the total number of chocolates in the denominator.**

$$\frac{38}{48}$$

**Step 3: Simplify the fraction.**

$$\frac{38}{48}=\frac{19}{24}$$

The fractional value of remaining chocolates is $$\frac{19}{24}$$

**E1.5B: Recognise equivalence and convert between these forms**

If a value repeats infinitely after the decimal point in a regular interval, it can be converted into a fraction. For example, suppose a man wants to buy a medicine whose cost is $$$20$$ per $$3$$ tablets. If he wants to buy only one tablet, the cost of the same will be $$\frac{20}{3}$$, that is $$6.66666\cdots$$ or $$6.\bar{6}$$.

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**Worked examples**

**Example 1:** Convert $$0.\bar{3}$$ into a simplified fraction.

**Step 1: Expand $$0.\bar{3}$$ into the normal form.**

$$0.\bar{3}=0.33333\cdots$$

**Step 2: Let the decimal obtained be equal to x.**

$$x=0.3333\cdots$$

**Step 3: Multiply the above equation by $$10$$.**

$$10x=3.3333\cdots$$

**Step 4: Subtract $$x=0.33333\cdots$$ from $$10x=3.3333\cdots$$.**

$$10x-x=3.3333\cdots-0.3333\cdots$$

$$9x=3$$

**Step 5: Solve the above equation for $$x$$.**

$$x=\frac{1}{3}$$

If a value does not repeat infinitely after a decimal point in a regular interval, it cannot be converted into a fraction.