The sum of the two numbers is 18 . The sum of their reciprocals is 1 4 . Find the numbers.

Let the required numbers be x and 18 - x Sum of reciprocals is 1 4 ⇒ 1 x + 1 18 - x = 1 4 ⇒ 18 - x + x x ( 18 - x ) = 1 4 ⇒ 18 × 4 = 18 x - x 2 ⇒ 72 = 18 x - x 2 ⇒ x 2 - 18 x + 72 = 0 Factorizing the quadratic equation by splitting the middle term, ⇒ x ( x - 12 ) - 6 ( x - 12 ) = 0 ⇒ ( x - 6 ) ( x - 12 ) = 0 ⇒ x - 6 = 0 ⇒ x = 6 or ⇒ x - 12 = 0 ⇒ x = 12 When x = 12 ⇒ 18 - x = 18 - 12 = 6 When x = 6 ⇒ 18 - x = 18 - 6 = 12 Thus, the required two numbers are 6 and 12 .

The sum of two number a and b is 15 , and the sum of their reciprocals 1 a and 1 b is 3 10 . Find the numbers a and b .

Let a + b = 15 ⇒ b = 15 - a … . ( 1 ) Sum of reciprocals of the two numbers is 3 10 . By the given data, ⇒ 1 a + 1 b = 3 10 ⇒ a + b a b = 3 10 ⇒ 10 a + 10 b = 3 a b … . . ( 2 ) Substituting equation ( 1 ) in ( 2 ) ⇒ 10 a + 10 ( 15 - a ) = 3 a ( 15 - a ) ⇒ 10 a + 150 - 10 a = 45 a - 3 a 2 ⇒ 3 a 2 - 45 a + 150 = 0 ⇒ 3 ( a 2 - 15 a + 50 ) = 0 ⇒ ( a 2 - 15 a + 50 ) = 0 ⇒ a 2 - 10 a - 5 a + 50 = 0 ⇒ a ( a - 10 ) - 5 ( a - 10 ) = 0 ⇒ ( a - 10 ) ( a - 5 ) = 0 ⇒ ( a - 10 ) = 0 ⇒ a = 10 Also, ⇒ ( a - 5 ) = 0 ⇒ a = 5 When a = 10 ⇒ b = 15 - a = 15 - 10 = 5 When a = 5 ⇒ b = 15 - a = 15 - 5 = 10 Thus, the value of a and b are either 5 and 10 or 10 and 5 .

E M B I B E

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 22

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10th CBSE

IMPORTANT

Earn 100

A two-digit number is such that the product of its digits is $8$ . When $18$ is subtracted from the number, the digits interchange their places. Find the number.

Important Questions on Quadratic Equations

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10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 23

A two-digit number is such that the product of its digits is

12

. When

36

is added to the number, the digits interchange their places. Find the number.

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 24

A two digit number is such that the product of its digits is $16$ . When $54$ is subtracted from the number, the digits interchange their places. Find the number.

MEDIUM

10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 25

Two numbers differ by