If an apple is divided into five equal parts, each part is said to be one fifth () of the whole apple.

If out of these five equal parts, 2 parts are eaten, we say two-fifth () of the apple is eaten or three-fifth () of the apple is left.

The numbers used in the statement are fractions each of which indicates a part of the whole.

In the fraction , is called the **numerator** and is called the **denominator** of the fraction.

- Classification of Fractions
- Reducing fraction to lowest terms
- Equivalent Fractions
- Simple and Complex Fractions
- Like and Unlike Fractions
- Comparing Fractions
- Insert fraction between two given fractions
- Operations on fractions
- Problems Involving Fractions
- Exercise

## Classification of Fractions

1. **Decimal fractions:** Denominator is 10 or higher power of 10.

e.g.:

2. **Vulgar fractions:** Denominator is other than 10, 100, 1000, etc.

e.g.:

3. **Proper fractions:** Denominator is greater than it’s numerator.

e.g.:

4. **Improper fractions:** Denominator is less than its numerator

e.g.:

5. **Mixed fractions:** Consists of an integer and a proper fraction

e.g.:

If the numerator is equal to the denominator, the fraction is equal to unity

e.g.:

**Example: Convert into an improper fraction**

**Solution:**

**Example: Convert into a mixed fraction (Divide 19 by 5)**

**Solution:** On dividing 19 by 5 we have:

Quotient=3, Remainder=4, Divisor=5

Alternatively,

**Note:**

- The value of a fraction remains the same if its numerator and denominator both are either multiplied or divided by the same non-zero number
- A fraction must always be expressed in its lowest term

## Reducing Fraction to Lowest Term

First of all find H.C.F of both the terms (numerator and denominator) of the given fraction. Then divide each term by this H.C.F.

**Example: Reduce to its lowest term.**

**Solution:**

H.C.F of terms 48 and 60 =12

**Alternative method:** Resolve both numerator and denominator into prime factors ,then cancel out the common factors of both numerator and denominator.

## Equivalent Fractions

Fractions having the same value are called equivalent fractions.

and

the fractions are equivalent.

## Simple and Complex Fractions

A fraction whose numerator and denominator both are integers is called a simple fraction, whereas a fraction, whose numerator or denominator or both are not integers, is called a complex fraction.

Example: Each of is a simple fraction.

Each of is a complex fraction.

## Like and Unlike Fractions

Fractions having the same denominators are called like fractions ; whereas the fractions with different denominators are called unlike fractions.

Converting unlike fractions into like fractions:

**Example: Change to like fractions.**

**Solution:**

L.C.M of the denominators 4, 5, 8 and 16 is 80

required like fractions are

## Comparing Fractions

Convert all the given fractions into like fractions. Then the fraction with the greater numerator is greater.

Example: Compare the fractions:

Solution:

L.C.M of the denominators 3, 4, 12 and 16=48

fractions in ascending order:

**Alternate method:** Convert all the given fractions into fractions of equal numerators. The fraction which has a smaller denominator is greater.

## Insert Fraction between two Fractions

Add numerators of the given fractions to get the numerator of the required fraction . Similarly add their denominators to get denominator of the required fraction. Then simplify if required.

**Example: Insert three fractions between and **

**Solution:**

## Operations on fractions:

**Addition / Subtraction**- For like fractions add or subtract their numerators ,keeping the denominator same.
- For unlike fractions ,first of all change them into like fractions and then do the addition or subtraction as above.

**Multiplication**- To multiply a fraction with an integer ;multiply its numerator with the integer
- To multiply two or more fractions ;multiply their numerators together and their denominators separately together

**Division:**- To divide one quantity by some other quantity (fraction or integer), multiply the first by the reciprocal of the second.

**Using of:**- The word between any two fractions, is to be used as multiplication.

**Using BODMAS:**- The word BODMAS is the abbreviation formed by taking the initial letters of six operations:
**B**racket,**O**f,**D**ivision,**M**ultiplication,**A**ddition and**S**ubtraction.

- The word BODMAS is the abbreviation formed by taking the initial letters of six operations:

According to the rule of BODMAS, working must be done in the order corresponding to the letters appearing in the word.

**Example: Simplify: **

**Solution:**

## Problems Involving Fractions

**Example: What fraction is 6 bananas of four dozen bananas?**

**Solution:**

Here 6 bananas are to be compared with 4 dozen that is

required fraction

**Example: A man spent of his savings and still has Rs.1000 left with him. How much were his savings?**

**Solution:**

The man spent of his money

Therefore he still has of his savings

**Note: In fractions the whole quantity is always taken as 1**

Since of his savings =Rs 1000

his savings

**Example: of a pole is in the mud. When of it is pulled out, 250 cm of the pole is still in the mud. What is the full length of the pole?**

**Solution:**

of the pole – of the pole =250 cm

of the pole =250 cm

of the pole= 250 cm

Length of the pole

## Exercise

- Express the following improper fractions as mixed fractions:
- Express the following mixed fractions as improper fractions:
- Reduce the given fractions to lowest terms:
- True or false: are equivalent fractions.
- Distinguish each of the following fractions as a simple fraction or a complex fraction:
- For each pair given below state whether it forms like fractions or unlike fractions:
- Find which fraction is greater:
- Insert two fractions between:
- Simplify:
- Subtract from the sum of
- A student bought of yellow ribbon, of red ribbon and of blue ribbon for decorating a room. How many metres of ribbon did he buy?
- A man spends of his salary on food, on house rent and on other expenses. What fraction of his salary is still left with him?
- In a business , Ram and Deepak invest of the total investment. If Rs 40,000 is the total investment, calculate the amount invested by each.
- Geeta had 30 problems for homework. She worked out of them . How many problems were still left with her?
- Shyam bought a refrigerator for Rs 5000. He paid of the price in cash and the rest in 12 equal monthly installments. How much had he to pay each month?
- In a school of the children are boys. If the number of girls is 200, find the number of boys.
- If of an estate is worth Rs 42,000, find the worth of whole estate. Also find the value of of it.
- After going of my journey, I find that I have covered 16 Km. how much journey is still left?
- When Krishna travelled 25 km, he found that of his journey was still left. What was the length of the whole journey?

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