Experts have designed these Class 7 Maths Notes and Chapter 8 Working with Fractions Class 7 Notes for effective learning.
Class 7 Maths Chapter 8 Notes Working with Fractions
Class 7 Maths Notes Chapter 8 – Class 7 Working with Fractions Notes
Fraction Class 7 Notes
A fraction is a number that is a part of the whole or a collection.
For example, we know that 1 hour has 60 minutes, so 1 minute is 60th part of an hour and is represented as a fraction \(\frac{1}{60}\) of an hour.
∴ A fraction is a number which can be written in the form \(\frac{a}{b}\) where both a and b are natural numbers.
a is called the numerator of the fraction \(\frac{a}{b}\) and b is called the denominator of the fraction \(\frac{a}{b}\).
For example, in the fraction \(\frac{4}{5}\), 4 is the numerator of this fraction and 5 is its denominator.
Some more examples of fractions are \(\frac{3}{7}\), \(\frac{11}{9}\), \(\frac{6}{13}\).
Multiplication of Fractions Class 7 Notes
(a) Multiplication of a Fraction by a Whole Number
To multiply a fraction by a whole number, we just multiply the numerator of the fraction by the whole number. The denominator remains the same.
Thus a fraction × a whole number \(=\frac{\text { Numerator of the fraction } \times \text { Whole number }}{\text { Denominator of the fraction }}\)
Now, simplify and reduce the product to its lowest term.
If the product is an improper fraction, change it into a mixed fraction.
For example, \(\frac{3}{5} \times 7=\frac{3 \times 7}{5}=\frac{21}{5}=4 \frac{1}{5}\)
(b) Multiplication of a Fraction by a Fraction
To multiply two or more fractions, we multiply the numerators of the given fractions to obtain the numerator of the product and multiply the denominators of the given fractions to obtain the denominator of the product.
∴ Product of fractions = \(\frac{\text { Product of their numerators }}{\text { Product of their denominators }}\)
That is \(\frac{a}{b} \times \frac{c}{d}=\frac{a \times c}{b \times d}\) and \(\frac{a}{b} \times \frac{c}{d} \times \frac{l}{m}=\frac{a \times c \times l}{b \times d \times m}\)
For example,
While multiplying, if any of the fractions is a mixed fraction, change it into an improper fraction and then multiply.
(c) Fractions with the Operator ‘of’
When a fraction is used with the word ‘of”, it acts as an operator (multiplication). That is ‘of’ represents multiplication.
For example,
(i) one third of 30 = \(\frac{1}{3}\) × 30
= \(\frac{1 \times 30}{3}\)
= 10
(ii) five-sevenths of 42 = \(\frac{5}{7}\) × 42
= \(\frac{5 \times 42}{7}\)
= 5 × 6
= 30
(iii) three-eighths of 80 = \(\frac{3}{8}\) × 80
= \(\frac{3 \times 80}{8}\)
= 3 × 10
= 30
(iv) two-fifths of \(\frac{6}{7}=\frac{2}{5} \times \frac{6}{7}=\frac{2 \times 6}{5 \times 7}=\frac{12}{35}\)
Question 1.
Multiply and reduce to the lowest form and convert into a mixed fraction:
(i) 7 × \(\frac{3}{5}\)
(ii) 11 × \(\frac{4}{7}\)
(iii) 20 × \(\frac{4}{5}\)
Solution:
Fractions obtained in (i) and (ii) above are already reduced to the lowest form, where a and b have no common factor.
Question 2.
Multiply and reduce to the lowest form (if possible):
(i) \(\frac{2}{7} \times \frac{7}{9}\)
(ii) \(\frac{3}{8} \times \frac{6}{4}\)
(iii) \(\frac{4}{5} \times \frac{12}{7}\)
Solution:
Division of Fractions Class 7 Notes
Reciprocal of a Fraction
(a) The reciprocal (or the multiplicative inverse) of a fraction is a new fraction in which the numerator and denominator are interchanged.
that is reciprocal of fraction \(\frac{a}{b}\) is \(\frac{b}{a}\).
For example
(i) Multiplicative inverse of \(\frac{9}{4}\) is \(\frac{4}{9}\)
(ii) Multiplicative inverse of \(2 \frac{3}{8}\left(=\frac{2 \times 8+3}{8}=\frac{19}{8}\right)\) is \(\frac{8}{19}\)
(b) The multiplicative inverse (or reciprocal) of a non-zero integer a (That is \(\frac{a}{1}\)) is \(\frac{1}{a}\).
The multiplicative inverse of 0 does not exist.
(c) Dividing a whole number by a fraction: To divide a whole number by a fraction, we multiply the whole number by the reciprocal of the fraction.
For example, \(6 \div \frac{7}{8}=6 \times \frac{8}{7}=\frac{6 \times 8}{7}=\frac{48}{7}\)
(d) Dividing a fraction by a natural number: To divide a fraction by a natural number, we multiply the fraction by the reciprocal of the natural number.
For example, \(\left(\frac{5}{7}\right) \div 9=\frac{5}{7} \times \frac{1}{9}=\frac{5 \times 1}{7 \times 9}=\frac{5}{63}\)
(e) Division of fractions: To divide a fraction \(\frac{a}{b}\) by a non-zero fraction \(\frac{c}{d}\), we multiply \(\frac{a}{b}\) with the reciprocal of \(\frac{c}{d}\). That is \(\frac{a}{b} \div \frac{c}{d}=\frac{a}{b} \times \frac{d}{c}=\frac{a d}{b c}\)
For applying (Division of fraction) (a), (c), (d), and (e); Convert a mixed fraction into an improper fraction.
Question 1.
Find:
(i) 14 ÷ \(\frac{5}{6}\)
(ii) 3 ÷ 2\(\frac{1}{3}\)
(iii) 5 ÷ 3\(\frac{4}{7}\)
Solution:
Question 2.
Find the reciprocal of each of the following fractions. Classify the reciprocals as proper fractions, improper fractions, and whole numbers.
(i) \(\frac{5}{8}\)
(ii) \(\frac{1}{11}\)
(iii) 1\(\frac{23}{49}\)
Solution:
We know that reciprocal of the fraction \(\frac{a}{b}\) is \(\frac{b}{a}\). (that is, interchange the numerator and the denominator)
(i) ∴ Reciprocal of \(\frac{5}{8}\) is \(\frac{8}{5}\)
This is an improper fraction because the numerator is greater than the denominator.
(ii) Reciprocal of \(\frac{1}{11}\) is \(\frac{11}{1}\) = 11.
(Interchanging the numerator and the denominator. This is a whole number.)
(iii) The given fraction is 1\(\frac{23}{49}\) (Mixed fraction)
= \(\frac{1 \times 49+23}{49}=\frac{49+23}{49}=\frac{72}{49}\)
Reciprocal of this fraction \(1 \frac{23}{49}\left(=\frac{72}{49}\right)\) is \(\frac{49}{72}\)
This is a proper fraction because the numerator is less than the denominator.
Question 3.
Find:
(i) \(\frac{6}{13}\) ÷ 7
(ii) 3\(\frac{1}{2}\) ÷ 4
(iii) 4\(\frac{3}{7}\) ÷ 7
Solution:
