Fractions - MathsTips.com

If an apple is divided into five equal parts, each part is said to be one fifth ( $\dfrac{1}{5}$ ) of the whole apple.

If out of these five equal parts, 2 parts are eaten, we say two-fifth ( $\dfrac{2}{5}$ ) of the apple is eaten or three-fifth ( $\dfrac{3}{5}$ ) of the apple is left.

The numbers $\dfrac{1}{5}, \dfrac{2}{5} \: and \: \dfrac{3}{5}$ used in the statement are fractions each of which indicates a part of the whole.

In the fraction $\dfrac{a}{b}$ , $a$ is called the numerator and $b$ is called the denominator of the fraction.

$\therefore$ $Fraction=\dfrac{numerator}{denominator}$

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.: $\dfrac{1}{10}, \dfrac{3}{100}, \dfrac{15}{1000}, \dfrac{8}{10^5}$

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

e.g.: $\dfrac{2}{5}, \dfrac{4}{7}, \dfrac{8}{19}, \dfrac{23}{1078}$

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

e.g.: $\dfrac{4}{5}, \dfrac{3}{7}, \dfrac{101}{235}$

4. Improper fractions: Denominator is less than its numerator

e.g.: $\dfrac{7}{5}, \dfrac{18}{13}, \dfrac{181}{60}$

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

e.g.: $2\dfrac{5}{7}, 10\dfrac{1}{9}$

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

e.g.: $\dfrac{4}{4}=1$

Example: Convert $3\dfrac{2}{7}$ into an improper fraction

Solution: $3\dfrac{2}{7}=\dfrac{3 \times 7+2}{7}=\dfrac{23}{7}$

Example: Convert $\dfrac{19}{5}$ into a mixed fraction (Divide 19 by 5)

Solution: On dividing 19 by 5 we have:

Quotient=3, Remainder=4, Divisor=5

$\therefore$ $\dfrac{19}{5} =3\dfrac{4}{5}$

Alternatively, $\dfrac{19}{5} =\dfrac{3 \times 5 +4}{5}=\dfrac{3 \times 5}{5} +\dfrac{4}{5}=3+\dfrac{4}{5}=3\dfrac{4}{5}$
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 $\dfrac{48}{60}$ to its lowest term.

Solution:

$\because$ H.C.F of terms 48 and 60 =12

$\therefore$ $\dfrac{48}{60}= \dfrac{48 \div 12}{60 \div 12}=\dfrac{4}{5}$

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

$\dfrac{48}{60}=\dfrac{2 \times 2 \times 2 \times 2 \times 3}{2 \times 2 \times 3 \times 5}$

$=\dfrac{2 \times 2}{5}=\dfrac{4}{5}$

Equivalent Fractions

Fractions having the same value are called equivalent fractions.

$\because$ $\dfrac{20}{25}=\dfrac{20 \div 5}{25 \div 5}= \dfrac{4}{5}$ and $\dfrac{28}{35}=\dfrac{28 \div 7}{35 \div 7}=\dfrac{4}{5}$

$\therefore$ the fractions $\dfrac{20}{25} \: and \: \dfrac{28}{35}$ 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 $\dfrac{3}{8}, \dfrac{(-10)}{17}, \dfrac{8}{(-15)}$ is a simple fraction.

Each of $\dfrac{5}{\dfrac{2}{3}}, \: \dfrac{1.4}{8}, \: \dfrac{\dfrac{9}{14}}{2\dfrac{3}{7}}$ 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 $\dfrac{3}{5}, \dfrac{3}{4}, \dfrac{7}{8} \: and \: \dfrac{9}{16}$ to like fractions.

Solution:

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

$\therefore$ $\dfrac{3}{4}=\dfrac{3 \times 20}{4 \times 20}=\dfrac{60}{80}$

$\dfrac{3}{5}=\dfrac{3 \times 16}{5 \times 16}=\dfrac{48}{60}$

$\dfrac{7}{8}=\dfrac{7 \times 10}{8 \times 10}= \dfrac{70}{80}$

$\dfrac{9}{16}=\dfrac{9 \times 5}{16 \times 5}=\dfrac{45}{80}$

$\therefore$ required like fractions are $\dfrac{60}{80}, \dfrac{48}{80}, \dfrac{70}{80}, \dfrac{45}{80}$

Comparing Fractions

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

Example: Compare the fractions: $\dfrac{2}{3}, \dfrac{3}{4}, \dfrac{5}{12} \: and \: \dfrac{9}{16}$

Solution:

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

$\dfrac{2}{3}=\dfrac{2 \times 16}{3 \times 16}=\dfrac{32}{48}$

$\dfrac{3}{4}=\dfrac{3 \times 12}{4 \times 12}=\dfrac{36}{48}$

$\dfrac{5}{12}=\dfrac{5 \times 4}{12 \times 4}=\dfrac{20}{48}$

$\dfrac{9}{16}=\dfrac{9 \times 3}{16 \times 3}= \dfrac{27}{48}$

$\therefore$ fractions in ascending order: $\dfrac{5}{12}, \dfrac{9}{16}, \dfrac{2}{3} \: and \dfrac{3}{4}$

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 $\dfrac{1}{2}$ and $\dfrac{3}{5}$

Solution:

$\dfrac{1}{2}, \dfrac{3}{5} =\dfrac{1}{2}, \dfrac{(1+3)}{(2+5)}, \dfrac{3}{5}=\dfrac{1}{2}, \dfrac{4}{7}, \dfrac{3}{5}$

$=\dfrac{1}{2}, \dfrac{(1+4)}{(2+7)}, \dfrac{4}{7}, \dfrac{(4+7)}{(7+5)}, \dfrac{3}{5}$

$=\dfrac{1}{2}, \dfrac{5}{9}, \dfrac{4}{7}, \dfrac{7}{12}, \dfrac{3}{5}$

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 $of$ 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: Bracket, Of, Division, Multiplication, Addition and Subtraction.

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

Example: Simplify: $(\dfrac{1}{3} + \dfrac{2}{9}) \: of \: \dfrac{\dfrac{8}{15}}{\dfrac{4}{9}} \times \dfrac{3}{4} -\dfrac{1}{2} +1$

Solution:

$\dfrac{(3+2)}{9} of \dfrac{\dfrac{8}{15}}{\dfrac{4}{9}} \times \dfrac{3}{4}-\dfrac{1}{2}+1$

$=\dfrac{5}{9} \: of \: \dfrac{\dfrac{8}{15}}{\dfrac{4}{9}} \times \dfrac{3}{4} -\dfrac{1}{2} +1$

$= \dfrac{8}{27} \times \dfrac{9}{4} \times \dfrac{3}{4}-\dfrac{1}{2} +1$

$= \dfrac{(8 \times 9 \times 3)}{(27 \times 4 \times 4)} -\dfrac{1}{2} +1$

$= \dfrac{1}{2} -\dfrac{1}{2} +1 =1$

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 $4 \times 12 =48 \: bananas$

$\therefore$ required fraction $=\dfrac{6}{48}=\dfrac{1}{8}$

Example: A man spent $\dfrac{2}{7}$ of his savings and still has Rs.1000 left with him. How much were his savings?

Solution:

The man spent $\dfrac{2}{7}$ of his money

Therefore he still has $1-\dfrac{2}{7}=\dfrac{5}{7}$ of his savings

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

Since $\dfrac{5}{7}$ of his savings =Rs 1000

$\therefore$ his savings $\dfrac{Rs. \: 1000}{\dfrac{5}{7}} =Rs. \: (1000 \times \dfrac{7}{5}) =Rs. \: 1400$

Example: $\dfrac{4}{7}$ of a pole is in the mud. When $\dfrac{1}{3}$ of it is pulled out, 250 cm of the pole is still in the mud. What is the full length of the pole?

Solution:

$\dfrac{4}{7}$ of the pole – $\dfrac{1}{3}$ of the pole =250 cm

$\therefore$ $(\dfrac{4}{7} -\dfrac{1}{3})$ of the pole =250 cm

$\therefore$ $\dfrac{5}{21}$ of the pole= 250 cm

Length of the pole $= \dfrac{250 \times 21}{5} \: cm = 1050 \: cm$

Exercise

Express the following improper fractions as mixed fractions: $i)\dfrac{25}{6} \: ii) \dfrac{38}{5}$
Express the following mixed fractions as improper fractions: $i) 2\dfrac{5}{48} \: ii) 12\dfrac{7}{11}$
Reduce the given fractions to lowest terms: $i)\dfrac{27}{36} \: ii) \dfrac{18}{45}$
True or false: $\dfrac{35}{49}, \dfrac{20}{28}, \dfrac{45}{63}, \dfrac{100}{140}$ are equivalent fractions.
Distinguish each of the following fractions as a simple fraction or a complex fraction:
- $i)\dfrac{0}{8} \: ii) \dfrac{5}{(-7)} \: iii) \dfrac{(-5\dfrac{2}{9})}{5}$
For each pair given below state whether it forms like fractions or unlike fractions:
- $i) \dfrac{8}{15} \: and \dfrac{8}{21} \: ii) \dfrac{4}{9} \: and \: \dfrac{9}{4}$
Find which fraction is greater:
- $i)\dfrac{3}{8} \: and \: \dfrac{4}{9} \: ii) -\dfrac{2}{7} \: and \: -\dfrac{3}{10}$
Insert two fractions between:
- $i)\dfrac{5}{9} \: and \: \dfrac{1}{4} \: ii) \dfrac{5}{6} \: and \: 1\dfrac{1}{5}$
Simplify:
1. $\dfrac{2}{3}-\dfrac{1}{5} +\dfrac{1}{10}$
2. $3\dfrac{3}{4} \times 1\dfrac{1}{5} \times \dfrac{20}{21}$
3. $\dfrac{7\dfrac{1}{5}}{1\dfrac{2}{3}} \times -\dfrac{5}{6}$
4. $\dfrac{\dfrac{1}{4}}{\dfrac{5}{8} -\dfrac{3}{5}}$
5. $(\dfrac{2}{9} +\dfrac{2}{3}) \times \dfrac{2\dfrac{1}{4}}{\dfrac{5}{6}} \times \dfrac{5}{12}$
6. $\dfrac{0}{\dfrac{8}{11}}$
7. $\dfrac{2}{3} \times \dfrac{1\dfrac{1}{4}}{\dfrac{3}{7}} \: of \: 2\dfrac{5}{8}$
Subtract $(\dfrac{2}{7} -\dfrac{5}{21})$ from the sum of $\dfrac{3}{4}, \dfrac{5}{7} \: and \: \dfrac{7}{12}$
A student bought $4\dfrac{1}{3} \: m$ of yellow ribbon, $6\dfrac{1}{6} \: m$ of red ribbon and $3\dfrac{2}{9} \: m$ of blue ribbon for decorating a room. How many metres of ribbon did he buy?
A man spends $\dfrac{2}{5}$ of his salary on food, $\dfrac{3}{10}$ on house rent and $\dfrac{1}{8}$ on other expenses. What fraction of his salary is still left with him?
In a business , Ram and Deepak invest $\dfrac{3}{5} \: and \dfrac{2}{5}$ 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 $\dfrac{2}{3}$ of them . How many problems were still left with her?
Shyam bought a refrigerator for Rs 5000. He paid $\dfrac{1}{10}$ of the price in cash and the rest in 12 equal monthly installments. How much had he to pay each month?
In a school $\dfrac{4}{5}$ of the children are boys. If the number of girls is 200, find the number of boys.
If $\dfrac{4}{5}$ of an estate is worth Rs 42,000, find the worth of whole estate. Also find the value of $\dfrac{3}{7}$ of it.
After going $\dfrac{3}{4}$ 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 $\dfrac{3}{5}$ of his journey was still left. What was the length of the whole journey?