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Fractions
Understanding Fractions
Look at the circle
It is divided into 4 equal parts.
1 part out of the 4 is coloured.
We say that \(1 \over 4\)of the circle is coloured.
\(1 \over 4\) is an example of a fraction.
A fraction represents a part of a whole that is a divided into equal parts
Look at the rectangle.
3 parts out of 5 are coloured.
\(3 \over 5\) of the rectangle is coloured.
A fraction has two parts.
Numerator shows the number of equal parts of a whole that are taken
Denominator shows the number of equal parts the whole is divided into
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Fractions
Like and unlike fractions
Gary’s mother bought two similar cakes.
She cut first cake in 8 equal pieces and took 1 piece out
She took of the first cak
She cut second cake in 8 equal pieces and took 3 pieces out
Since, both cakes were cut into equal sized pieces
Denominator is same for both fractions. We call such fractions like fractions
Like Fractions result in equal sized pieces of the whole. Like fractions are the fractions which have same denominators
Anam’s mother bought two similar cakes.
She cut first cake in 6 equal pieces and took 1 piece ou
She took of the first cake.
She cut second cake in 8 equal pieces and took 1 piece out
She took of the second cake.
Since, both cakes were cut into different sized pieces. Denominator is different for both fraction. We call such fractions unlike fractions.
Proper Improper and Mixed fractions :
Proper fractions: A fraction whose numerator is less than its denominator is called a proper fraction.
For example: etc are proper fractions.
Improper fractions: A fraction whose numerator is greater than or equal to its denominator is called an improper fraction.
For example: etc are improper fractions.
Mixed fractions: A fraction which is a combination of a whole number and a proper fraction is called a mixed fraction. In a mixed fraction, the whole number part is known as integral part and the fraction is known as fractional part.
For example: etc. are mixed fractions. -
Fractions
Equivalent fractions
Look at the circle.
It is divided into 2 equal parts.
\(1 \over 2\) of the circle is coloured
Now look at this circle.
It is divided into 4 equal parts.
\(2 \over 4\)of the circle is coloured.
You can see that the area coloured in both circles is same.
This means \(1 \over 2\)and \(2 \over 4\) are equal.We call such fractions equivalent fractions.
Equivalent fractions are fractions which have different numerators and denominators but have same value.
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Fractions
Equivalent fractions
Look at the circle.
It is divided into 2 equal parts.
\(1 \over 2\) of the circle is coloured
Now look at this circle.
It is divided into 4 equal parts.
\(2 \over 4\)of the circle is coloured.
You can see that the area coloured in both circles is same.
This means \(1 \over 2\)and \(2 \over 4\) are equal.
Equivalent fractions are fractions which have different numerators and denominators but have same value.
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Fractions
Equivalent fractions
Look at the circle.
It is divided into 2 equal parts.
\(1 \over 2\) of the circle is coloured
Now look at this circle.
It is divided into 4 equal parts.
\(2 \over 4\)of the circle is coloured.
You can see that the area coloured in both circles is same.
This means \(1 \over 2\)and \(2 \over 4\) are equal.
Equivalent fractions are fractions which have different numerators and denominators but have same value.
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Fractions
Equivalent fractions
Look at the circle.
It is divided into 2 equal parts.
\(1 \over 2\) of the circle is coloured
Now look at this circle.
It is divided into 4 equal parts.
\(2 \over 4\)of the circle is coloured.
You can see that the area coloured in both circles is same.
This means \(1 \over 2\)and \(2 \over 4\) are equal.
Equivalent fractions are fractions which have different numerators and denominators but have same value.
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Fractions
Equivalent fractions
Look at the circle.
It is divided into 2 equal parts.
\(1 \over 2\) of the circle is coloured
Now look at this circle.
It is divided into 4 equal parts.
\(2 \over 4\)of the circle is coloured.
You can see that the area coloured in both circles is same.
This means \(1 \over 2\)and \(2 \over 4\) are equal.
Equivalent fractions are fractions which have different numerators and denominators but have same value.
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Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Like Fractions
To add or subtract like fractions, add or subtract the numerators and divide by the common denominator
Example: Find the sum of \({5 \over 36},{8 \over 36},{15 \over 36}\)
Solution: \(\frac{5}{36}+\frac{8}{36}+\frac{15}{36}=\frac{5+8+15}{36}=\frac{28}{36}=\frac{7}{9}\)
Example: Subtract \(1 {6 \over 18}\) from \(2 {5 \over 18}\)
Solution:\(1 {6 \over 18}=\) \(\frac{1\times 18+6}{18}=\frac{18+6}{18}=\frac{24}{18}\)
and \(2\frac{5}{18}=\frac{2\times 18+5}{18}=\frac{36+5}{18}=\frac{41}{18}\)
Now \(2\frac{5}{18}-1\frac{6}{18}=\frac{41}{18}-\frac{24}{18}=\frac{41-24}{18}=\frac{17}{18}\)
Addition and Subtraction of Unlike Fractions
To add or subtract unlike fractions, convert the unlike fractions into like fractions and then proceed as before
Example: Find the difference of each of the following:
Solution: (a) LCM of 30 and 15 = 2 × 3 × 5 = 30
\(\begin{align} & \frac{9}{30}=\frac{9\times 1}{30\times 1}=\frac{9}{30}\text{and }\frac{5}{15}=\frac{5\times 2}{15\times 2}=\frac{10}{30} \\ & \therefore \frac{5}{15}-\frac{9}{30}=\frac{10}{30}-\frac{9}{30}=\frac{10-9}{30}=\frac{1}{30} \\ \end{align}\)
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Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Like Fractions
To add or subtract like fractions, add or subtract the numerators and divide by the common denominator
Example: Find the sum of \({5 \over 36},{8 \over 36},{15 \over 36}\)
Solution: \(\frac{5}{36}+\frac{8}{36}+\frac{15}{36}=\frac{5+8+15}{36}=\frac{28}{36}=\frac{7}{9}\)
Example: Subtract \(1 {6 \over 18}\) from \(2 {5 \over 18}\)
Solution:\(1 {6 \over 18}=\) \(\frac{1\times 18+6}{18}=\frac{18+6}{18}=\frac{24}{18}\)
and \(2\frac{5}{18}=\frac{2\times 18+5}{18}=\frac{36+5}{18}=\frac{41}{18}\)
Now \(2\frac{5}{18}-1\frac{6}{18}=\frac{41}{18}-\frac{24}{18}=\frac{41-24}{18}=\frac{17}{18}\)
Addition and Subtraction of Unlike FractionsTo add or subtract unlike fractions, convert the unlike fractions into like fractions and then proceed as before
Example: Find the difference of each of the following:
(a) \(9\over 30\) and \(5\over 15\) (b) \(13{5\over 4}\) and \(12{1\over 5}\)
Solution: (a) LCM of 30 and 15 = 2 × 3 × 5 = 30
\(\begin{align} & \frac{9}{30}=\frac{9\times 1}{30\times 1}=\frac{9}{30}\text{and }\frac{5}{15}=\frac{5\times 2}{15\times 2}=\frac{10}{30} \\ & \therefore \frac{5}{15}-\frac{9}{30}=\frac{10}{30}-\frac{9}{30}=\frac{10-9}{30}=\frac{1}{30} \\ \end{align} \)
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Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Like Fractions
To add or subtract like fractions, add or subtract the numerators and divide by the common denominator
Example: Find the sum of \({5 \over 36},{8 \over 36},{15 \over 36}\)
Solution: \(\frac{5}{36}+\frac{8}{36}+\frac{15}{36}=\frac{5+8+15}{36}=\frac{28}{36}=\frac{7}{9}\)
Example: Subtract \(1 {6 \over 18}\) from \(2 {5 \over 18}\)
Solution:\(1 {6 \over 18}=\) \(\frac{1\times 18+6}{18}=\frac{18+6}{18}=\frac{24}{18}\)
and \(2\frac{5}{18}=\frac{2\times 18+5}{18}=\frac{36+5}{18}=\frac{41}{18}\)
Now \(2\frac{5}{18}-1\frac{6}{18}=\frac{41}{18}-\frac{24}{18}=\frac{41-24}{18}=\frac{17}{18}\)
Addition and Subtraction of Unlike Fractions
To add or subtract unlike fractions, convert the unlike fractions into like fractions and then proceed as before
Example: Find the difference of each of the following:
(a) \(9\over 30\) and \(5\over 15\) (b) \(13{5\over 4}\) and \(12{1\over 5}\)
Solution: (a) LCM of 30 and 15 = 2 × 3 × 5 = 30
\(\begin{align} & \frac{9}{30}=\frac{9\times 1}{30\times 1}=\frac{9}{30}\text{and }\frac{5}{15}=\frac{5\times 2}{15\times 2}=\frac{10}{30} \\ & \therefore \frac{5}{15}-\frac{9}{30}=\frac{10}{30}-\frac{9}{30}=\frac{10-9}{30}=\frac{1}{30} \\ \end{align} \)
\(\begin{align} & (b)13\frac{5}{4}=\frac{13\times 4+5}{4}=\frac{52+5}{4}=\frac{57}{4} \\ & \text{and 12}\frac{1}{5}=\frac{12\times 5+1}{5}=\frac{60+1}{5}=\frac{61}{5} \\ & \text{LCM of 4 and 5 }=\text{ 4 }\times \text{ 5 }=\text{ 2}0 \\ & \text{Now},\text{ }\frac{61}{5}=\frac{61\times 4}{5\times 4}=\frac{244}{20}\text{and }\frac{57}{4}=\frac{57\times 5}{5\times 5}=\frac{285}{20} \\ & \therefore 13\frac{5}{4}-\text{12}\frac{1}{5}=\frac{57}{4}-\frac{61}{5}=\frac{285}{20}-\frac{244}{20}=\frac{285-244}{20}=\frac{41}{20} \\ \end{align} \)
Example: Rupa had one stick of 2\(1 \over 5\) m length and another of 3 \(1 \over 3\)m length. How much length of stick did Rupa have in all?
Solution:
\(\begin{align} & \text{Length of one stick }=2\frac{1}{5}\text{m}=\frac{11}{5}\text{m} \\ & \text{Length of another stick }=3\frac{1}{3}\text{m}=\frac{10}{3}\text{m} \\ & \text{Total length of two sticks }=\left( \frac{11}{5}+\frac{10}{3} \right)\text{m} \\ & \text{=}\left( \frac{11\times 3}{5\times 3}+\frac{10\times 5}{3\times 5} \right)\text{m}(\because \text{ LCM of 5},\text{ and 3}=15) \\ & =\left( \frac{33}{15}+\frac{50}{15} \right)\text{m=}\frac{83}{15}\text{m} \\ \end{align} \)
So, Rupa had 5\(8 \over 15\) m rope in all.
Example: Sindhu got ₹50 from her mother. She spent ₹10\(1\over 2\) and saved the rest. How much money did she save?Solution: Sindhu got money from her mother = ₹50
Sindhu spent money = ₹\(10\frac{1}{2}=\) ₹\(\frac{21}{2}\)
Sindhu saved money = ₹\(\left( \frac{50}{1}-\frac{21}{2} \right)\)
=₹\(\left( \frac{50\times 2}{1\times 2}-\frac{21\times 1}{2\times 1} \right)\)
=₹\(\left( \frac{100}{2}-\frac{21}{2} \right)\)=₹\(\left( \frac{100-21}{2} \right)\)
=₹\(79 \over 2\)=₹\(39\frac{1}{2}\)
So she saved ₹ \(39\frac{1}{2}\)
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Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Like Fractions
To add or subtract like fractions, add or subtract the numerators and divide by the common denominator
Example: Find the sum of \({5 \over 36},{8 \over 36},{15 \over 36}\)
Solution: \(\frac{5}{36}+\frac{8}{36}+\frac{15}{36}=\frac{5+8+15}{36}=\frac{28}{36}=\frac{7}{9}\)
Example: Subtract \(1 {6 \over 18}\) from \(2 {5 \over 18}\)
Solution:\(1 {6 \over 18}=\) \(\frac{1\times 18+6}{18}=\frac{18+6}{18}=\frac{24}{18}\)
and \(2\frac{5}{18}=\frac{2\times 18+5}{18}=\frac{36+5}{18}=\frac{41}{18}\)
Now \(2\frac{5}{18}-1\frac{6}{18}=\frac{41}{18}-\frac{24}{18}=\frac{41-24}{18}=\frac{17}{18}\)
Addition and Subtraction of Unlike Fractions
To add or subtract unlike fractions, convert the unlike fractions into like fractions and then proceed as before
Example: Find the difference of each of the following:
(a) \(9\over 30\) and \(5\over 15\) (b) \(13{5\over 4}\) and \(12{1\over 5}\)
Solution: (a) LCM of 30 and 15 = 2 × 3 × 5 = 30
\(\begin{align} & \frac{9}{30}=\frac{9\times 1}{30\times 1}=\frac{9}{30}\text{and }\frac{5}{15}=\frac{5\times 2}{15\times 2}=\frac{10}{30} \\ & \therefore \frac{5}{15}-\frac{9}{30}=\frac{10}{30}-\frac{9}{30}=\frac{10-9}{30}=\frac{1}{30} \\ \end{align} \)
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Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Like Fractions
To add or subtract like fractions, add or subtract the numerators and divide by the common denominator
Example: Find the sum of \({5 \over 36},{8 \over 36},{15 \over 36}\)
Solution: \(\frac{5}{36}+\frac{8}{36}+\frac{15}{36}=\frac{5+8+15}{36}=\frac{28}{36}=\frac{7}{9}\)
Example: Subtract \(1 {6 \over 18}\) from \(2 {5 \over 18}\)
Solution:\(1 {6 \over 18}=\) \(\frac{1\times 18+6}{18}=\frac{18+6}{18}=\frac{24}{18}\)
and \(2\frac{5}{18}=\frac{2\times 18+5}{18}=\frac{36+5}{18}=\frac{41}{18}\)
Now \(2\frac{5}{18}-1\frac{6}{18}=\frac{41}{18}-\frac{24}{18}=\frac{41-24}{18}=\frac{17}{18}\)
Addition and Subtraction of Unlike Fractions
To add or subtract unlike fractions, convert the unlike fractions into like fractions and then proceed as before
Example: Find the difference of each of the following:
(a) \(9\over 30\) and \(5\over 15\) (b) \(13{5\over 4}\) and \(12{1\over 5}\)
Solution: (a) LCM of 30 and 15 = 2 × 3 × 5 = 30
\(\begin{align} & \frac{9}{30}=\frac{9\times 1}{30\times 1}=\frac{9}{30}\text{and }\frac{5}{15}=\frac{5\times 2}{15\times 2}=\frac{10}{30} \\ & \therefore \frac{5}{15}-\frac{9}{30}=\frac{10}{30}-\frac{9}{30}=\frac{10-9}{30}=\frac{1}{30} \\ \end{align} \)
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Fractions
Addition and Subtraction of Fractions
Addition and Subtraction of Like Fractions
To add or subtract like fractions, add or subtract the numerators and divide by the common denominator
Example: Find the sum of \({5 \over 36},{8 \over 36},{15 \over 36}\)
Solution: \(\frac{5}{36}+\frac{8}{36}+\frac{15}{36}=\frac{5+8+15}{36}=\frac{28}{36}=\frac{7}{9}\)
Example: Subtract \(1 {6 \over 18}\) from \(2 {5 \over 18}\)
Solution:\(1 {6 \over 18}=\) \(\frac{1\times 18+6}{18}=\frac{18+6}{18}=\frac{24}{18}\)
and \(2\frac{5}{18}=\frac{2\times 18+5}{18}=\frac{36+5}{18}=\frac{41}{18}\)
Now \(2\frac{5}{18}-1\frac{6}{18}=\frac{41}{18}-\frac{24}{18}=\frac{41-24}{18}=\frac{17}{18}\)
Addition and Subtraction of Unlike Fractions
To add or subtract unlike fractions, convert the unlike fractions into like fractions and then proceed as before
Example: Find the difference of each of the following:
(a) \(9\over 30\) and \(5\over 15\) (b) \(13{5\over 4}\) and \(12{1\over 5}\)
Solution: (a) LCM of 30 and 15 = 2 × 3 × 5 = 30
\(\begin{align} & \frac{9}{30}=\frac{9\times 1}{30\times 1}=\frac{9}{30}\text{and }\frac{5}{15}=\frac{5\times 2}{15\times 2}=\frac{10}{30} \\ & \therefore \frac{5}{15}-\frac{9}{30}=\frac{10}{30}-\frac{9}{30}=\frac{10-9}{30}=\frac{1}{30} \\ \end{align} \)
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Fractions
Multiplication of Fractions
Multiplication of a Fraction by a Whole Number Multiplication of a Fraction by Another Fraction To multiply a fraction by a whole number, multiply the numerator of the given fraction by the whole number.
The denominator of the product will be the denominator of the given fraction.
Fraction × Whole number =\(\frac{\text{Numerator }\times \text{ Whole number}}{\text{Denominator }}\)
Example: Multiply:
(a)\(2\over 4\) by 8 (b) \(3{2\over 7}\) by 3
Solution: (a) \(\frac{2}{4}\times 8=\frac{2\times 8}{4}=\frac{16}{4}=4\)
(b)\( 3\frac{2}{7}\times 3=\frac{23}{7}=\frac{23\times 3}{7}=\frac{69}{7}=9\frac{6}{7}\)
Multiplication of a Fraction by Another Fraction
When two fractions are multiplied, numerator of one fraction is multiplied by the numerator of another fraction, while denominator is multiplied by the denominator. Product of numerators becomes the numerator of new fraction, while the ciao of denominators becomes the denominator of new fraction
Fraction × Fraction =\(\frac{\text{Product of the Numerators}}{\text{Product of the Denominator }}\)
Example: Multiply:
(a) \( 2\frac{3}{6}\)by \(\frac{3}{8}\) (b)\(2\frac{3}{5}\) by \(3\frac{3}{4}\)
Solution:
(a) \( 2\frac{3}{6}\times \frac{3}{8}=\frac{15}{6}\times \frac{3}{8}=\frac{15\times 3}{6\times 8}=\frac{15}{16}\)
(b)\(2\frac{3}{5}\times 3\frac{3}{4}=\frac{13}{5}\times \frac{15}{4}=\frac{13\times 15}{5\times 4}=\frac{13\times 3}{4}=\frac{39}{4}=9\frac{3}{4}\)
Example: Find \(3\over 6\) of \(7 {4 \over 5}\)
Solution:\(\frac{3}{6}\times \frac{39}{5}=\frac{3\times 39}{6\times 5}=\frac{39}{10}=3\frac{9}{10}\)
Example: How many paise are there in \(2 \over 5\) of a rupee?
Solution: We know that, 1 rupee = 100 paise
\(2 \over 5\) of a rupee= \(2 \over 5\) of 100 paise
\(\begin{align} & =\frac{2}{5}\times 100\text{paise}=\frac{2\times 100}{5}\text{paise} \\ & \text{=2}\times \text{20paise} \\ & =40\text{paise} \\ \end{align} \)
Example: Sandhya had 120 metres of cloth. She sold \(5\over 8\) of it. How many metres of cloth did she sell?
Solution: Total length of cloth = 120 metres
The length of the cloth sold=
\(\begin{align} & =\frac{5}{8}\text{of 12}0\text{ metres} \\ & \text{=}\frac{5}{8}\text{ }\times \text{ 12}0\text{metres} \\ & =\frac{5\times 120}{8}\text{metres} \\ & =75\text{metres} \\ \end{align}\)
So, he sold 75 metres of cloth.
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Fractions
Reciprocal of a Number
Two numbers are reciprocal (multiplicative inverse) of each other, if their product is 1.
For Example: \({2\over 5}\times{5\over 2}=1\)
\(2\over 5\) and \(5\over 2\) are the reciprocals of each other.
We can find the reciprocal of the given fraction by interchanging the numerator and denominator.
Example: Find the reciprocal of the following :
(a) 4 (b)\(23 \over 56\) (c)\(4 {6 \over 7}\)
(a) Reciprocal of 4 is \(1 \over 4\)
(b) Reciprocal of \(23 \over 56\) is \(56 \over 23\)
(C) \(4 {6 \over 7}={{4 \times 7+6}\over 7}={{28+6}\over 7}={{34}\over 7}\)
Reciprocal of \(4{6 \over 7}\) i.e., \(34 \over 7\) is \(7 \over 34\)
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Fractions
Division of Fractions
Division of a Fraction by a Whole Number
To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number.
Fraction ÷ Whole number = Fraction × Reciprocal of whole number
Example: Divide \(4 \over 5\) by 8
Solution: Reciprocal of 8 = \(1 \over 8\)
\(\therefore \frac{4}{5}\div 8=\frac{4}{5}\times \frac{1}{8}=\frac{4\times 1}{5\times 8}=\frac{1\times 1}{5\times 2}=\frac{1}{10}\)
Example: Divide \(5 {5 \over 8}\) by 40
Solution: \(5\frac{5}{8}=\frac{5\times 8+5}{8}=\frac{40+5}{8}=\frac{45}{8}\)
Reciprocal of 40 = \(1\over 40\)
\(5\frac{5}{8}\div 40=\frac{45}{8}\times \frac{1}{40}=\frac{45\times 1}{8\times 40}=\frac{9\times 1}{8\times 40}=\frac{9}{64}\)
Division of a Fraction by Another Fraction
To divide a fraction by another fraction, multiply the dividend by the reciprocal of the divisor
Example: Divide:
(a) \(6 \over 9\) by \(3 \over 8\) (b) \(14 {2 \over 4}\) by \(5 {3 \over 8}\)
Solution: (a) Reciprocal of \({3 \over 8}={8 \over 3}\)
\(\therefore \frac{6}{9}\div \frac{3}{8}=\frac{6}{9}\times \frac{8}{3}=\frac{6\times 8}{9\times 3}=\frac{16}{9}=1\frac{7}{9}\)
\(\begin{align} & \left( \text{b} \right)~~~\therefore 14\frac{2}{4}=\frac{14\times 5+2}{4}=\frac{56+2}{4}=\frac{58}{4} \\ & \text{and }5\frac{3}{8}=\frac{5\times 8+3}{8}=\frac{40+3}{8}=\frac{43}{8} \\ & \text{Reciprocal of }5\frac{3}{8}i.e.,\frac{43}{8}=\frac{8}{43} \\ & \therefore 14\frac{2}{4}\div 5\frac{3}{8}=\frac{58}{4}\div \frac{43}{8}=\frac{58}{4}\times \frac{8}{43}=\frac{58\times 8}{4\times 43}=\frac{116}{43}=2\frac{30}{43} \\ \end{align}\)Example: How many one fourths make one-half?
Solution: To find the number of one-fourths in one-half, we need to divide \(1\over 2\) by \(1\over 4\)
\(\therefore \frac{1}{2}\div \frac{1}{4}=\frac{1}{2}\times \frac{4}{1}=\frac{1\times 4}{2\times 1}=2\times 1=2\)
So, 2 one-fourths make one-half
Example: A can holds 3\(1\over 2\) litres of milk. How many mugs can be filled with milk if each mug can hold \(1 \over 2\) litre of milk?
Solution:
\(\begin{align} & \text{Amount of milk in the can}=\text{3}\frac{1}{2}\text{litres}=\frac{7}{2}\text{litres} \\ & \text{Capacity of mug }=\frac{1}{2}\text{litres} \\ & \text{Number of mugs that can be filled }=\frac{7}{2}\div \frac{1}{2} \\ & =\frac{7}{2}\times 2=\frac{7\times 2}{2}=7 \\ \end{align} \)
So, 6 mugs can be filled up.
Example:The product of two numbers is 4 \(3 \over 4\), If one of them is 2 \(3 \over 8\) , find the other number
Solution:
\(\begin{align} & \text{The product of the two numbers }=4\frac{3}{4}=\frac{19}{4} \\ & \text{One number }=2\frac{3}{8}=\frac{19}{8} \\ & \text{Other number }=\frac{19}{4}\div \frac{19}{8} \\ & =\frac{19}{4}\times \frac{8}{19}=2 \\ \end{align}\)