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Fractions

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Fractions are expressed as the ratio of two numbers and are primarily used for comparison between parts and the whole. It can be a part of an object or a group of objects. They are better understood when seen. A fraction is a number expressed in the form of $\frac{a}{b}$, where the numerator 'a' signifies the number of parts it has, and the denominator 'b' is the number of parts the whole is divided into.
Fractions
There are three types of fractions:
  • Proper fractions
  • Improper fractions
  • Mixed fractions.

Proper Fraction

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If the numerator value is less than the denominator value, then the fraction is a proper fraction. The value of the proper fraction will always be less than 1.

Example:
$\frac{6}{7}$, $\frac{2}{5}$, $\frac{5}{9}$

Improper Fraction

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If the numerator value is greater than the denominator value, then the fraction is a improper fraction. The value of this fraction will always be greater than 1.

Example:
$\frac{7}{6}$, $\frac{9}{4}$, $\frac{8}{3}$

Mixed Fraction

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A mixed fraction is a whole number and a fraction combined or we can say that it is the sum of a non zero integer and a fraction.
Mixed fraction can be negative also.
Example: 1$\frac{1}{6}$, 2$\frac{3}{4}$, 5$\frac{9}{5}$

How to Simplify Fractions?

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A fraction can be simplified when its numerator and denominator have no common factors. Given below are the simple steps to simplify a fraction:
  1. List the prime factors of numerator and denominator.
  2. Find the common factors to both numerator and the denominator.
  3. Cancel out the common terms in both the numerator as well as denominator.

Operations With Fractions

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All mathematical operations namely addition, subtraction, multiplication and division can be performed on the fractions.

Addition of Fractions

1. Adding Fractions with Same Denominator: Add numerators keeping the denominator same.
Example: $\frac{2}{3}$ + $\frac{5}{3}$ = $\frac{7}{3}$

2. Adding Fractions with Different Denominator: When adding fractions, make sure the fractions being added have the same denominator. If not, find the lowest common divisor and put each in its equivalent form. Then, add the numerators of the fraction and simplify if possible.
Example: $\frac{3}{5}$ + $\frac{1}{4}$ = $\frac{12}{20}$ + $\frac{5}{20}$ = $\frac{17}{20}$

Subtraction of Fractions

1. Subtracting Fractions with Same Denominator: Subtract numerators keeping the denominator same.
Example: $\frac{8}{7}$ - $\frac{5}{7}$ = $\frac{3}{7}$

2. Subtracting Fractions with Different Denominator: When subtracting fractions, make sure the fractions being subtracted have the same denominator. If not, find the lowest common divisor and put each in its equivalent form. Then, simply subtract the numerators of the fraction.
Example: $\frac{5}{7}$ - $\frac{1}{3}$ = $\frac{15}{21}$ - $\frac{14}{21}$ = $\frac{1}{21}$

Multiplying Fractions

When multiplying fractions, multiply the numerators together and then, multiply the denominators together. Simplify the result.
Example: $\frac{5}{7}$ $\times$ $\frac{1}{3}$ = $\frac{5}{21}$

Dividing Fractions

Divide two fractions by taking the reciprocal of the second fraction, multiply and then simplify.
Example: $\frac{3}{5}$ / $\frac{2}{3}$ = $\frac{3}{5}$ $\times$ $\frac{3}{2}$ = $\frac{9}{10}$

Fraction Word Problems

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Given below are some of the word problems in fraction.

Solved Examples

Question 1: David and sam bought a large pizza to share. David ate $\frac{4}{7}$ of the pizza. What fraction of pizza is left for sam?
Solution:
David ate $\frac{4}{7}$ of the pizza.
So, sam is left with $\frac{3}{7}$ of the pizza.
Therefore, Sam gets to eat $\frac{3}{7}$ portion of pizza.
Verify: $\frac{4}{7}$ + $\frac{3}{7}$ = $\frac{7}{7}$
$\Rightarrow$ 1 = 1
Therefore, the answer is verified.

Question 2: In an exam, Melody finished $\frac{7}{10}$ of the math problems while Christine did $\frac{2}{5}$ of her math problems. Who did a greater fraction of math problems?
Solution:
Given: Melody finished $\frac{7}{10}$ of the math problems  
In terms of percentage, it is 70%.

Christine finished $\frac{2}{5}$ of the math problems  
In terms of percentage, it is 40%.

Compare the whole number parts which is in percentages.

As 70 is greater than 40 (70 > 40), Melody completed greater fraction of her math problems.

Question 3: Simplify: $\frac{30}{45}$
Solution:
All prime factors for $\frac{30}{45}$
$\frac{30}{45}$ = $\frac{2 \times 3 \times 5}{3 \times 3 \times 5}$

Cancelling the common factors, we get
$\frac{2 \times \not{3} \times \not{5}}{3 \times \not{3} \times \not{5}}$

As above, the only factor remaining in the numerator is 2 and the only factor remaining in denominator is 3. It cannot be simplified further.
Therefore, the simplified answer is $\frac{2}{3}$.