The sum of the two numbers is 16 . The sum of their reciprocals is 1 3 . Find the numbers.

Let the two numbers be x and ( 16 - x ) . Then, the reciprocals of the two numbers are 1 x and 1 16 - x respectively. By the given data, we can write, 1 x + 1 16 - x = 1 3 16 - x + x x ( 16 - x ) = 1 3 16 x ( 16 - x ) = 1 3 16 x - x 2 = 48 x 2 - 16 x + 48 = 0 Factorising by splitting the middle term, x 2 - 12 x - 4 x + 48 = 0 x ( x - 12 ) - 4 ( x - 12 ) = 0 ( x - 12 ) ( x - 4 ) = 0 ( x - 12 ) = 0 ⇒ x = 12 Also, ( x - 4 ) = 0 ⇒ x = 4 If x = 4 , then 16 - x = 16 - 4 = 12 . If x = 12 , then 16 - x = 16 - 12 = 4 . Thus, the two numbers are 4 and 12 .

The sum of a number and its reciprocal is 17 4 . Find the number.

Let x be a number, then its reciprocal will be 1 x . Sum of the number and its reciprocal is 17 4 . According to the given condition, we can write, x + 1 x = 17 4 x 2 + 1 x = 17 4 4 x 2 + 4 = 17 x 4 x 2 - 17 x + 4 = 0 Factorising the quadratic equation by splitting the middle term, we get, ⇒ 4 x 2 - x - 16 x + 4 = 0 ⇒ x ( 4 x - 1 ) - 4 ( 4 x - 1 ) = 0 ⇒ ( 4 x - 1 ) ( x - 4 ) = 0 ⇒ ( 4 x - 1 ) = 0 ⇒ x = 1 4 Also ⇒ ( x - 4 ) = 0 ⇒ x = 4 Thus, the number is 4 or 1 4 .

E M B I B E

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 17

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10th CBSE

IMPORTANT

Earn 100

Find two natural numbers which differ by $3$ and whose squares have the sum $117 .$

Important Questions on Quadratic Equations

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10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 18

The sum of the squares of three consecutive natural number is

149 .

Find the numbers.

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10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 19

The sum of the two numbers is $16$ . The sum of their reciprocals is $\frac{1}{3}$ . Find the numbers.

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10th CBSE

IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 20

Determine two consecutive multiples of

3

whose product is

270 .

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IMPORTANT

MATHEMATICS CLASS X>Chapter 4 - Quadratic Equations>EXERCISE 4.7>Q 21

The sum of a number and its reciprocal is $\frac{17}{4} .$ Find the number.

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IMPORTANT

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

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.

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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.

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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.

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10th CBSE

IMPORTANT

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

Two numbers differ by