Students can go through AP 7th Class Maths Notes Chapter 2 Fractions and Decimals to understand and remember the concepts easily.
Class 7 Maths Chapter 2 Notes Fractions and Decimals
→ Fraction : A fraction is a number representing a part of a whole.
Ex : 1) \(\frac{3}{7}\)
2) \(\frac{2}{19}\)
→ In the fraction \(\frac{\mathrm{a}}{\mathrm{~b}}\), a is called Numerator and b is called Denominator
Ex : In the fraction \(\frac{9}{11}\)
Numerator = 9 ; Denominator = 11
→ Proper Fraction : A fraction whose numerator is less than the denominator is called proper fraction.
Ex : \(\frac{7}{11}\), \(\frac{4}{19}\), \(\frac{2}{3}\), etc.
→ Improper Fraction : A fraction whose numerator in more than or equal to the denominator is called an improper fraction.
Ex : \(\frac{7}{5}\), \(\frac{14}{9}\), \(\frac{121}{12}\), etc.
→ Mixed Fraction : A combination of a whole number and a proper fraction is called a mixed fraction.
Ex : 1\(\frac{1}{2}\), 2\(\frac{3}{4}\), 17\(\frac{1}{4}\), etc.
→ Equivalent Fraction : A fraction (or) ’ fractions obtained by multiplying (or dividing) its numerator and denominator by the same non – zero number, are called equivalent fraction.
Ex : 1) Equivalent fraction of \(\frac{2}{3}\) are.
\(\frac{2}{3}\) × \(\frac{2}{2}\) = \(\frac{4}{6}\) ; \(\frac{2}{3}\) × \(\frac{3}{3}\) = \(\frac{6}{9}\) ; \(\frac{2}{3}\) × \(\frac{8}{8}\) = \(\frac{16}{24}\)
2) Equivalent fraction of \(\frac{100}{72}\) are.
\(\frac{100 \div 2}{72 \div 2}\) = \(\frac{50}{36}\) ; \(\frac{50 \div 2}{36 \div 2}\) = \(\frac{25}{18}\)
Note : If \(\frac{\mathrm{a}}{\mathrm{~b}}\) and \(\frac{\mathrm{c}}{\mathrm{~d}}\) are two equivalent fractions, then
a × d = b × c ; \(\frac{\mathrm{a}}{\mathrm{~b}}\) = \(\frac{\mathrm{c}}{\mathrm{~d}}\) ; a × d = b × c
→ Multiplication of a fraction by a whole number: To multiply a whole number with a proper or an improper fraction, we multiply the whole number with the numerator of the fraction, keeping the denominator same.
Ex : 1) 2 × \(\frac{5}{3}\) = \(\frac{10}{3}\)
2) \(\frac{9}{4}\) × 3 = \(\frac{27}{4}\)
→ Multiplication of a fraction by a fraction : We multiply two fractions as \(\frac{\text { Product of Numerators }}{\text { Product of Denominators }}\)
Ex :
1) Multiply \(\frac{5}{6}\) and \(\frac{2}{9}\)
= \(\frac{5}{6}\) × \(\frac{2}{9}\) = \(\frac{5 \times 2}{6 \times 9}\) = \(\frac{5 \times 1}{3 \times 9}\) = \(\frac{5}{27}\)
2) Multiply \(\frac{7}{4}\) and \(\frac{3}{8}\) = \(\frac{7}{4}\) × \(\frac{3}{8}\) = \(\frac{21}{32}\)
→ ‘Of’ represents multiplication while calculating the product in fractions.
Ex:
1) \(\frac{1}{2}\) of 30 = \(\frac{1}{2}\) × 30 = 15
2) \(\frac{3}{4}\) of 1 min = \(\frac{3}{4}\) × 1 min = \(\frac{3}{4}\) × 60 sec
= 3 × 15 sec = 45 sec
→ When two proper fractions are multiplied, the product is less than each of the fractions. (Or) The value of the product of two proper fractions is smaller than each of the two fractions.
Ex : 1) \(\frac{4}{17}\) × \(\frac{2}{3}\) = \(\frac{8}{51}\)
2) \(\frac{4}{13}\) × \(\frac{1}{2}\) = \(\frac{4}{26}\) = \(\frac{2}{13}\)
→ The product of two improper fractions is greater than each of the two fractions (or) The product of two improper fractions is more than each of the two fractions.
Ex :1) \(\frac{17}{4}\) × \(\frac{3}{2}\) = \(\frac{51}{8}\)
2) \(\frac{13}{4}\) × \(\frac{2}{3}\) = \(\frac{26}{12}\) = \(\frac{13}{6}\)
→ Division of whole number by a fraction :
In 1 ÷ \(\frac{1}{2}\) we divide a whole into a number of equal parts such that each part is half of the whole.
Similarly 1 ÷ \(\frac{1}{4}\) we divide a whole into a number of 4 equal parts such each part is one fourth of the whole.
→ Reciprocal of a fraction: The non-zero numbers whose product with each other is 1 are called reciprocals of each other.
Ex:
1) Reciprocal of \(\frac{15}{91}\) is \(\frac{91}{15}\) ⇒ \(\frac{15}{91}\) × \(\frac{91}{15}\) = 1
2) Reciprocal of 21 is \(\frac{1}{21}\) ⇒ 21 × \(\frac{1}{21}\) = 1
3) Reciprocal of \(\frac{a}{b}\) can be obtained by inverting it
Reciprocal of \(\frac{a}{b}\) = \(\frac{b}{a}\)
→ Reciprocal of improper fraction is a proper fraction.
Ex : 1) Reciprocal of \(\frac{17}{4}\) is \(\frac{4}{17}\)
2) Reciprocal of \(\frac{4}{3}\) is \(\frac{3}{4}\)
→ Reciprocal of proper fraction is an improper fraction.
Ex : 1) Reciprocal of \(\frac{4}{7}\) is \(\frac{7}{4}\)
2) Reciprocal of \(\frac{3}{14}\) is \(\frac{14}{3}\)
→ Division of whole number by a fraction : To divide a whole number by any fraction, multiply that whole number by the reciprocal of that fraction.
Ex : 1) Find 7 ÷ \(\frac{9}{4}\)
Solution:
7 ÷ \(\frac{9}{4}\) = 7 × \(\frac{4}{9}\) = \(\frac{28}{9}\)
2) Find 3 ÷ \(\frac{4}{9}\)
Solution:
3 ÷ \(\frac{4}{9}\) = \(\frac{3}{1}\) × \(\frac{9}{4}\) = \(\frac{27}{4}\)
→ While dividing a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then solve it.
Ex : 1) 4 ÷ 2\(\frac{1}{5}\)
Solution:
4 ÷ 2\(\frac{1}{5}\) = 4 ÷ 2\(\frac{11}{5}\) = 4 × \(\frac{5}{11}\) = \(\frac{20}{11}\)
2) 14 ÷ 1\(\frac{1}{2}\)
Solution:
14 ÷ 1\(\frac{1}{2}\) = 14 ÷ \(\frac{3}{2}\) = 14 × \(\frac{2}{3}\) = \(\frac{28}{3}\)
→ Division of a fraction by Another fraction : When we divide a fraction by other, the reciprocal of second fraction is to be multiplied with first fraction.
Ex : 1) \(\frac{7}{4}\) ÷ \(\frac{9}{2}\)
Solution:
\(\frac{7}{4}\) ÷ \(\frac{9}{2}\) = \(\frac{7}{4}\) × \(\frac{2}{9}\) = \(\frac{7}{2}\) × \(\frac{1}{9}\) = \(\frac{7}{18}\)
2) \(\frac{4}{3}\) ÷ \(\frac{3}{4}\)
Solution:
\(\frac{4}{3}\) ÷ \(\frac{3}{4}\) = \(\frac{4}{3}\) × \(\frac{4}{3}\) = \(\frac{16}{9}\)
3) 1\(\frac{1}{2}\) ÷ \(\frac{1}{2}\)
Solution:
1\(\frac{1}{2}\) ÷ \(\frac{1}{2}\) = \(\frac{3}{2}\) ÷ \(\frac{1}{2}\) = \(\frac{3}{2}\) × \(\frac{2}{1}\) = 3
→ We also learnt how to multiply two decimal numbers. While multiplying two decimal numbers, first multiply them as whole numbers. Count the number of digits to the right of the decimal point in both the decimal numbers. Add the number of digits counted. Put the decimal point in the product by counting the digits from its rightmost place. The count should be the sum obtained earlier.
For example, 0.5 × 0.7 = 0.35
→ To multiply a decimal number by 10, 100 or 1000, we move the decimal point in the number to the right by as many places as there are zeros over 1.
Thus 0.53 × 10 = 5.3,
0.53 × 100 = 53,
0.53 × 1000 – 530
→ We have seen how to divide decimal numbers.
a) To divide a decimal number by a whole number, we first divide them as whole numbers. Then place the decimal point in the quotient as in the decimal number.
For example, 8.4 ÷ 4 = 2.1
Note that here we consider only those divisions in which the remainder is zero.
b) To divide a decimal number by 10, 100 or 1000, shift the digits in the decimal number to the left by as many places as there are zeros over 1, to get the quotient.
So, 23.9 ÷ 10 = 2.39,
23.9 ÷ 100 = 0.239,
23.9 ÷ 1000 = 0.0239
c) While dividing two decimal numbers, first shift the decimal point to the right by equal number of places in both, to convert the divisor to a whole number. Then divide.
Thus, 2.4 ÷ 0.2 = 24 ÷ 2 = 12.
→ ‘0’ has no reciprocal.
→ Division by zero is not defined
→ \(\frac{a}{b}\) ÷ \(\frac{b}{a}\) = 1, b ≠ 0
→ Reciprocal of \(\frac{\mathrm{a}}{\mathrm{~b}}\) = \(\frac{b}{a}\)
→ Reciprocal of \(\frac{1}{b}\) = b
→ The product of a fraction and its reciprocal is always 1.
→ No fraction contains ‘0’ in its denominator.
→ Zero divided by any number gives quotient as ‘0’.
→ Zero divided by zero is not defined.