  # Fractions

#### Chapter 56 5 Steps - 3 Clicks

# Fractions

### Introduction

A fraction is a number of the form $$\frac{a}{b}$$, where ‘a’ and ‘b’ are integers and b is not equal to zero since division by zero is undefined.

In essence, one can think a fraction as another way of writing a division problem.

Example: $$\frac{4}{5}$$ is also written as 4 ÷ 5.

### Methods

In general, the integer on the top of the fraction bar is called the numerator and the integer underneath the fraction bar is called the denominator.

Example: $$\frac{-5}{3}$$

Negative 5 over 3 is a fraction in which -5 is numerator and 3 is the denominator.

These types of numbers are also referred to as rational numbers.

Properties:

1. If both the numerator ‘a’ and ‘b’ are multiplied by the same non-zero integer, the resulting fraction will be equivalent to a/b.

Example:

$$\frac{-5}{3}$$

$$\frac{-5}{3}$$ = $$\frac{(-5)}{3}$$ * $$\frac{4}{4}$$ = $$\frac{-20}{12}$$

Here, $$\frac{-20}{12}$$ is equivalent to $$\frac{-5}{3}$$

$$\frac{-5}{3}$$ = $$\frac{(-5)}{3}$$ * $$\frac{(-1)}{(-1)}$$ = $$\frac{5}{-3}$$

2. If both the numerator and denominator have a common factor, then the numerator and denominator can be reduced into an equivalent fraction.

Example 1:

$$\frac{40}{72}$$

Numerator can be written as 8 * 5

Denominator can be written as 8 * 9

$$\frac{40}{72}$$ = $$\frac{(8 * 5)}{(8 * 9)}$$ = $$\frac{5}{9}$$

Example 2:

By prime factorization

$$\frac{16}{20}$$ = $$\frac{(2 * 2 * 2 * 2)}{(2 * 2 * 5)}$$ = $$\frac{4}{5}$$

Example 3:

By Greatest Common factor (GCF)

$$\frac{16}{20}$$ = $$\frac{(\frac{16}{4})}{(\frac{20}{4})}$$ = $$\frac{4}{5}$$

3. A fraction with a negative sign in either the numerator or denominator can be written with the negative sign in front of the fraction.

Example 1:

$$\frac{-5}{3}$$ = $$\frac{5}{-3}$$ = –$$\frac{5}{-3}$$

Example 2:

$$\frac{-5}{-3}$$ = $$\frac{5}{3}$$ (Since – x – = +)

Addition and Subtraction

Same Denominator

To add two fractions with the same denominator, add the numerator and keep the same denominator.

Example:

$$\frac{1}{10}$$ + $$\frac{2}{10}$$ = $$\frac{(1 + 2)}{10}$$ = $$\frac{3}{10}$$

To subtract two fractions with the same denominator, subtract the numerator and keep the same denominator.

Example:

$$\frac{5}{11}$$ – $$\frac{8}{11}$$ = $$\frac{(5 – 8)}{11}$$ = $$\frac{-3}{11}$$

Same method is applied for adding or subtracting more than two terms

Example:

$$\frac{1}{5}$$ + $$\frac{2}{5}$$ + $$\frac{3}{5}$$ = $$\frac{(1 + 2 + 3)}{5}$$ = $$\frac{6}{5}$$

Different denominator

To add two fractions with distinct denominators, first need to find a common denominator.

The common denominator is a common multiple of the two denominators, preferably the least common multiple (LCM).

Example:

$$\frac{2}{3}$$ + $$\frac{3}{4}$$

LCM of 3 and 4 is 12.

$$\frac{((4 * 2) + (3 * 3))}{12}$$

$$\frac{(8 + 9)}{12}$$

$$\frac{17}{12}$$

Similary, to subtract two fractions with distinct denominators, first need to find a common denominator.

Example:

$$\frac{1}{3}$$ – $$\frac{2}{5}$$

LCM of 3 and 5 is 15.

$$\frac{((1 * 5) – (2 * 3))}{15}$$

$$\frac{(5 – 6)}{15}$$

= –$$\frac{1}{15}$$

Same method is applied for adding or subtracting more than two terms

Example:

$$\frac{2}{3}$$ + $$\frac{1}{5}$$ – $$\frac{1}{2}$$

LCM of 3, 5, 2 is 30

$$\frac{((2 * 10) + (1 * 6) – (15 *1))}{30}$$

$$\frac{(20 + 6 – 15)}{30}$$

$$\frac{11}{30}$$

Multiplication

There is really nothing tricky about multiplying fractions.

When multiplying fractions all that is needed to do is multiply across each fraction’s numerator and each fraction’s denominator.

Example 1:

$$\frac{8}{3}$$ * $$\frac{7}{3}$$ = $$\frac{(8 * 7)}{(3 * 3)}$$ = $$\frac{56}{9}$$

Example 2:

$$\frac{4}{5}$$ * $$\frac{10}{12}$$ = $$\frac{(4 * 10)}{(5 * 12)}$$ = $$\frac{40}{60}$$ = $$\frac{4}{6}$$ = $$\frac{2}{3}$$

When an integer is multiplied to a fraction

Example 1:

2 * $$\frac{4}{5}$$ = $$\frac{2}{1}$$ * $$\frac{4}{5}$$ = $$\frac{8}{5}$$

Example 2:

$$\frac{3}{36}$$ * 3 = $$\frac{3}{36}$$ * $$\frac{3}{1}$$ = $$\frac{9}{36}$$ = $$\frac{1}{4}$$

Division

Dividing fractions are just like the multiplying fractions.

To divide two fractions, invert the second fraction (find the reciprocal) before multiplying the fractions.

$$\frac{a}{b}$$ ÷ $$\frac{c}{d}$$ = $$\frac{a}{b}$$ * $$\frac{c}{d}$$ = $$\frac{ad}{bc}$$

Example 1:

$$\frac{17}{8}$$ ÷ $$\frac{3}{4}$$ = $$\frac{17}{8}$$ * $$\frac{4}{3}$$ = $$\frac{(17 * 4)}{(8 * 3)}$$ = $$\frac{17}{6}$$

Example 2:

$$\frac{4}{3}$$ ÷ 3 = $$\frac{4}{3}$$ ÷ 3 = $$\frac{4}{3}$$ * $$\frac{1}{3}$$ = $$\frac{4}{9}$$

Complex fractions

A complex fraction is a fraction where the numerator, denominator or both contain a fraction.
ie. of the form $$\frac{\frac{a}{b}}{\frac{c}{d}}$$

Example 1:

$$\frac{7}{\frac{1}{4}}$$ = $$\frac{7}{1}$$ ÷ $$\frac{1}{4}$$ = $$\frac{7}{1}$$ * $$\frac{4}{1}$$ = 28

Example 2:

$$\frac{\frac{3}{10}}{\frac{7}{13}}$$ = $$\frac{3}{10}$$ ÷ $$\frac{7}{13}$$ = $$\frac{3}{10}$$ * $$\frac{13}{7}$$ = $$\frac{39}{70}$$

Mixed numbers

A mixed number (fraction) consists of an integer part and a fraction part.

Example:

$$4\frac{3}{8}$$

$$4\frac{3}{8}$$ = 4 + $$\frac{3}{8}$$ = $$\frac{4}{1}$$ + $$\frac{3}{8}$$

LCM of 1 and 8 is 8

$$\frac{(32 + 3)}{8}$$ = $$\frac{35}{8}$$

Here, 35/8 is a improper fraction.

Improper fraction

It is defined as the fraction in which numerator is greater or equal to denominator.

Compare fractions

Two or more than two terms of fractions are given and need to compare them to find which is smaller and which is bigger among the given terms.

In order to compare, first find the common denominator.

Example:

Compare $$\frac{3}{7}$$ and $$\frac{7}{12}$$

$$\frac{3}{7}$$ and $$\frac{7}{12}$$

$$\frac{(12)(3)}{(12)(7)}$$ and $$\frac{(7)(7)}{(12)(7)}$$

$$\frac{36}{84}$$ and $$\frac{49}{84}$$

As the denominators are same, now compare the numerators

Like $$\frac{49}{84}$$ is bigger

$$\frac{36}{84}$$ is smaller

$$\frac{36}{84}$$ < $$\frac{49}{84}$$

Fractions containing irrational numbers

The numbers of the form a/b may contain numbers in either ‘a’ or ‘b’ can be irrational numbers.

Example 1:

$$\frac{π}{2}$$ + $$\frac{π}{3}$$ = $$\frac{(3π +2π)}{6}$$ = $$\frac{5π}{6}$$

Example 2:

$$\frac{π}{2}$$ * $$\frac{π}{3}$$ = $$\frac{π^2}{6}$$

Example 3:

$$\frac{(\frac{π}{2})}{(\frac{π}{3})}$$ = $$\frac{π}{2}$$ * $$\frac{3}{π}$$ = $$\frac{3π}{2π}$$ = $$\frac{3}{2}$$

### Samples

1. Quantitative Comparison

Column A Column B
5 y
A. Column A is greater.
B. Column B is greater.
C. The two quantities are equal.
D. The relationship cannot be determined from the information given.

Solution:

$$\frac{x}{y}$$ = $$\frac{1}{5}$$

By cross multiplication,

5x = y

It is given that x is a positive integer. That could mean x is 1, 2, 3, ….

If x is 1, then y would equal 5. However, if x is an integer greater than or equal to 2 then y would be greater than 5. Since we don’t know x’s exact value, then we cannot determine the relationship between 5 and y.

Hence, option D is the right choice.

2. Solve:
$$\frac{ \frac{2}{3}}{\frac{1}{5}}$$ + $$\frac{9}{\frac{3}{2}}$$

A. 14
B. $$\frac{26}{3}$$
C. $$\frac{28}{2}$$
D. $$\frac{28}{3}$$

Solution:

To simplify the given fraction, simply invert the denominator and multiply by the numerator:

$$\frac{ \frac{2}{3}}{\frac{1}{5}}$$ + $$\frac{9}{\frac{3}{2}}$$

Multiply the denominator by the reciprocal of the denominator for each term:

$$\frac{2}{3}$$ * $$\frac{5}{1}$$ + $$\frac{9}{1}$$ * $$\frac{2}{3}$$

= $$\frac{10}{3}$$ + $$\frac{18}{3}$$

Since the denominator is same, add both the terms:

= $$\frac{28}{3}$$

Hence, option D is the right choice.

3. Which of the following fractions is greater than $$\frac{3}{5}$$ and less than $$\frac{6}{7}$$?

A. $$\frac{7}{8}$$
B. $$\frac{1}{3}$$
C. $$\frac{2}{3}$$
D. $$\frac{1}{2}$$

Solution:

$$\frac{3}{5}$$ = 0.6

$$\frac{6}{7}$$ = 0.85 (Taken only the first two digits after the decimal point)

Hence, the question is to find out a number which is greater than 0.6 and less than 0.85

The given choices are

$$\frac{1}{2}$$ = 0.5

$$\frac{2}{3}$$ = 0.66 (Taken only the first two digits after the decimal point)

$$\frac{1}{3}$$ = 0.33 (Taken only the first two digits after the decimal point)

$$\frac{7}{8}$$ = 0.87 (Taken only the first two digits after the decimal point)

Clearly, 0.66 = $$\frac{2}{3}$$ is the answer.

4. Which of the following are in descending order of their value?

A. $$\frac{7}{8}$$, $$\frac{4}{7}$$, $$\frac{3}{7}$$, $$\frac{2}{5}$$, $$\frac{1}{4}$$, $$\frac{1}{6}$$
B. None of these
C. $$\frac{1}{4}$$, $$\frac{2}{5}$$, $$\frac{4}{7}$$, $$\frac{1}{6}$$, $$\frac{3}{7}$$, $$\frac{7}{8}$$
D. $$\frac{1}{4}$$, $$\frac{2}{5}$$, $$\frac{4}{7}$$, $$\frac{5}{6}$$, $$\frac{6}{7}$$, $$\frac{7}{8}$$

Solution:

$$\frac{1}{4}$$ = 0.25 and $$\frac{2}{5}$$ = 0.4

Hence, $$\frac{2}{5}$$ > 1/4

Hence, $$\frac{1}{4}$$, $$\frac{2}{5}$$, $$\frac{4}{7}$$, $$\frac{5}{6}$$, $$\frac{6}{7}$$, $$\frac{7}{8}$$ is not in descending order.

Similarly, $$\frac{1}{4}$$, $$\frac{2}{5}$$, $$\frac{4}{7}$$, $$\frac{1}{6}$$, $$\frac{3}{7}$$, $$\frac{7}{8}$$ is also not in descending order.

$$\frac{7}{8}$$ = 0.8 (Taken only one digit after the decimal point)

$$\frac{4}{7}$$ = 0.5 (Taken only one digit after the decimal point)

$$\frac{3}{7}$$ = 0.42 (Taken only two digits after the decimal point)

$$\frac{2}{5}$$ = 0.4

$$\frac{1}{4}$$ = 0.2 (Taken only one digit after the decimal point)

$$\frac{1}{6}$$ = 0.1 (Taken only one digit after the decimal point)

Hence,

$$\frac{7}{8}$$, $$\frac{4}{7}$$, $$\frac{3}{7}$$, $$\frac{2}{5}$$, $$\frac{1}{4}$$, $$\frac{1}{6}$$ is in descending order.

5. When 0.232323….. is converted into a fraction, then the result is:

A. $$\frac{1}{5}$$
B. $$\frac{2}{9}$$
C. $$\frac{23}{99}$$
D. $$\frac{23}{100}$$

Solution:

0.232323… = $$\bar{23}$$ = $$\frac{23}{99}$$