UNIT 2
FRACTIONS AND DECIMALS
(A) Main Concepts and Results
• A fraction is either a proper fraction or an improper fraction.
• A proper fraction is a number representing a part of a whole. This
whole may be a single object or a group of objects. An improper
fraction is a number in which numerator is greater than denominator.
• A mixed fraction is a combination of a natural number and a proper
fraction.
• Two fractions are multiplied by multiplying their numerators and
denominators separately and writing the product as
product of numerators 2 3 2×3 6
. For example, 5 × 4 = 5 × 4 = 20 .
product of denominators
1 1
• A fraction acts as an operator ‘of ’. For example, of 3 is × 3 = 1.
3 3
• The product of two proper fractions is less than each of the fractions,
1 1 1 1 1 1
For example, × = and is less than both and .
2 3 6 6 2 3
• The product of a proper and an improper fraction is less than the
improper fraction and greater than the proper fraction. For example,
1 3 3 3 3 1
× = and is less than but greater than .
2 2 4 4 2 2
• The product of two improper fractions is greater than the two fractions.
3 7 21 21 3 7
For example, × = and is greater than both and .
2 4 8 8 2 4
15-04-2018
, UNIT 2
• The reciprocal of a non-zero fraction is obtained by interchanging
3 2
its numerator and denominator. For example, reciprocal of is .
2 3
• While dividing a whole number by a fraction, we multiply the whole
1 2
number with the reciprocal of that fraction. For example, 3 ÷ =3× .
2 1
• While dividing a fraction by a natural number, we multiply the fraction
1 1 1
by the reciprocal of the natural number. For example, ÷2= × .
4 4 2
• While dividing one fraction by another fraction, we multiply the first
1 1 1 3
fraction by the reciprocal of the other. For example, ÷ = × .
2 3 2 1
• 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
number of digits equal to sum obtained from its rightmost place. For
example, 1.2 × 1.24 = 1.488.
• 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 many
zeros (0) are the right of one. For example, 1.33 × 10 = 13.3.
• To divide a decimal number by a natural number, we first take the
decimal number as natural number and divide by the given natural
number. Then place the decimal point in the quotient as in the decimal
1.2
number. For example, = 0.3
4
• To divide a decimal number by 10, 100 or 1000, shift the decimal
point in the decimal number to the left by as many places as there
1.34
are zeros over 1, to get the quotient. For example, = 0.0134
100
• While dividing one decimal number by another, first shift the decimal
points to the right by equal number of places in both, to convert the
divisor to a natural number and then divide. For example
1.44 14.4
= = 1.2.
1.2 12
FRACTIONS AND DECIMALS 27
15-04-2018
, MATHEMATICS
(B) Solved Examples
In Examples 1 to 11, there are four options, out of which one is correct.
Write the correct one.
3
Example 1: Savita is dividing 1 kg of sweets equally among her
4
seven friends. How much does each friend receive?
3 1 1 3
(a) kg (b) kg (c) kg (d) kg
4 4 2 28
Solution: Correct answer is (b)
3
Example 2: If of a number is 12, the number is
4
(a) 9 (b) 16 (c) 18 (d) 32
Solution: Correct answer is (b)
2 5
Example 3: Product of fractions and is
7 9
2×5 2+5 2×9 2×5
(a) (b) (c) (d)
7+9 2+9 5×7 7×9
Solution: Correct answer is (d)
Example 4: Given that 0 < p < q < r < s and p, q, r, s are integers,
which of the following is the smallest?
p +q p +s q +s r +s
(a) (b) (c) (d) p + q
r +s q +r p +r
Solution: Correct answer is (a)
Example 5: The next number of the pattern
60, 30, 15, _______ is
15 15
(a) 10 (b) 5 (c) (d)
4 2
Solution: Correct answer is (d)
28 EXEMPLAR PROBLEMS
15-04-2018
FRACTIONS AND DECIMALS
(A) Main Concepts and Results
• A fraction is either a proper fraction or an improper fraction.
• A proper fraction is a number representing a part of a whole. This
whole may be a single object or a group of objects. An improper
fraction is a number in which numerator is greater than denominator.
• A mixed fraction is a combination of a natural number and a proper
fraction.
• Two fractions are multiplied by multiplying their numerators and
denominators separately and writing the product as
product of numerators 2 3 2×3 6
. For example, 5 × 4 = 5 × 4 = 20 .
product of denominators
1 1
• A fraction acts as an operator ‘of ’. For example, of 3 is × 3 = 1.
3 3
• The product of two proper fractions is less than each of the fractions,
1 1 1 1 1 1
For example, × = and is less than both and .
2 3 6 6 2 3
• The product of a proper and an improper fraction is less than the
improper fraction and greater than the proper fraction. For example,
1 3 3 3 3 1
× = and is less than but greater than .
2 2 4 4 2 2
• The product of two improper fractions is greater than the two fractions.
3 7 21 21 3 7
For example, × = and is greater than both and .
2 4 8 8 2 4
15-04-2018
, UNIT 2
• The reciprocal of a non-zero fraction is obtained by interchanging
3 2
its numerator and denominator. For example, reciprocal of is .
2 3
• While dividing a whole number by a fraction, we multiply the whole
1 2
number with the reciprocal of that fraction. For example, 3 ÷ =3× .
2 1
• While dividing a fraction by a natural number, we multiply the fraction
1 1 1
by the reciprocal of the natural number. For example, ÷2= × .
4 4 2
• While dividing one fraction by another fraction, we multiply the first
1 1 1 3
fraction by the reciprocal of the other. For example, ÷ = × .
2 3 2 1
• 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
number of digits equal to sum obtained from its rightmost place. For
example, 1.2 × 1.24 = 1.488.
• 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 many
zeros (0) are the right of one. For example, 1.33 × 10 = 13.3.
• To divide a decimal number by a natural number, we first take the
decimal number as natural number and divide by the given natural
number. Then place the decimal point in the quotient as in the decimal
1.2
number. For example, = 0.3
4
• To divide a decimal number by 10, 100 or 1000, shift the decimal
point in the decimal number to the left by as many places as there
1.34
are zeros over 1, to get the quotient. For example, = 0.0134
100
• While dividing one decimal number by another, first shift the decimal
points to the right by equal number of places in both, to convert the
divisor to a natural number and then divide. For example
1.44 14.4
= = 1.2.
1.2 12
FRACTIONS AND DECIMALS 27
15-04-2018
, MATHEMATICS
(B) Solved Examples
In Examples 1 to 11, there are four options, out of which one is correct.
Write the correct one.
3
Example 1: Savita is dividing 1 kg of sweets equally among her
4
seven friends. How much does each friend receive?
3 1 1 3
(a) kg (b) kg (c) kg (d) kg
4 4 2 28
Solution: Correct answer is (b)
3
Example 2: If of a number is 12, the number is
4
(a) 9 (b) 16 (c) 18 (d) 32
Solution: Correct answer is (b)
2 5
Example 3: Product of fractions and is
7 9
2×5 2+5 2×9 2×5
(a) (b) (c) (d)
7+9 2+9 5×7 7×9
Solution: Correct answer is (d)
Example 4: Given that 0 < p < q < r < s and p, q, r, s are integers,
which of the following is the smallest?
p +q p +s q +s r +s
(a) (b) (c) (d) p + q
r +s q +r p +r
Solution: Correct answer is (a)
Example 5: The next number of the pattern
60, 30, 15, _______ is
15 15
(a) 10 (b) 5 (c) (d)
4 2
Solution: Correct answer is (d)
28 EXEMPLAR PROBLEMS
15-04-2018
