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

Let the larger number to be x and the smaller number be y . Then, we have: x + y = 16 … … ( i ) And 1 x + 1 y = 1 3 ……(ii) ⇒ 3 ( x + y ) = xy ⇒ ( 3 × 16 ) = xy [From i ] ∴ xy = 48 … … . ( iii ) We know: ( x – y ) 2 = ( x + y ) 2 – 4 xy ⇒ ( x – y ) 2 = ( 16 ) 2 – 4 × 48 = 256 – 192 = 64 ⇔ x – y = 64 = 8 Since x is larger and y is smaller, we have: x - y = 8 … … … ( i v ) On adding i and i v , we get: ⇒ 2 x = 24 ⇒ x = 12 On substituting x = 12 in i , we get: 12 + y = 16 ⇒ y = ( 16 - 12 ) = 4 Hence, the required numbers are 12 and 4 .

There are two classrooms A and B . If 10 students are sent from A to B , the number of students in each room becomes the same. If 20 students are sent from B to A , the number of students in A becomes double the number of students in B . Find the number of students in each room.

Let the number of students in classroom A be x Let the number of students in classroom B be y . If 10 students are transferred from A to B , then we have: ⇒ x – 10 = y + 10 ⇒ x – y = 20 … . . ( i ) If 20 students are transferred from B to A , then we have: 2 ( y – 20 ) = x + 20 ⇒ 2 y – 40 = x + 20 ⇒ - x + 2 y = 60 … . . ( ii ) On adding i and i i , we get: ⇒ y = ( 20 + 60 ) = 80 On substituting y = 80 in i , we get: x – 80 = 20 ⇒ x = ( 20 + 80 ) = 100 Hence, the number of students in classroom A is 100 and the number of students in classroom B is 80 .

MEDIUM

10th CBSE

IMPORTANT

Earn 100

If $2$ is added to the numerator of a fraction, it reduces to $(1 / 2)$ and if $1$ is subtracted from the denominator, it reduces to $(1 / 3)$ . Find the fraction.

Important Questions on Linear Equations in Two Variables

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 21

The denominator of a fraction is greater than its numerator by

11

. If

8

is added to both its numerator and denominator, it becomes

3 / 4

. Find the fraction.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 22

Find a fraction which becomes

(\frac{1}{2})

when

1

is subtracted from the numerator and

2

is added to the denominator, and the fraction becomes

(\frac{1}{3})

when

7

is subtracted from the numerator and

2

is subtracted from the denominator.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 23

The sum of the numerator and denominator of a fraction is

4

more than twice the numerator. If the numerator and denominator are increased by

3

, they are in the ratio of

2 : 3

. Determine the fraction.

HARD

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 24

The sum of two numbers is

16

and the sum of their reciprocals is

\frac{1}{3}

. Find the numbers.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 25

There are two classrooms

A

and

B

. If

10

students are sent from

A

B

, the number of students in each room becomes the same. If

20

students are sent from

B

A

, the number of students in

A

becomes double the number of students in

B

. Find the number of students in each room.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 26

Taxi charges in a city consist of fixed charges and the remaining depending upon the distance travelled in kilometers. If a man travels

80 km

, he pays

₹ 1330

, and travelling

90 km

, he pays

₹ 1490

. Find the fixed charges and the rate per km.

MEDIUM

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 27

A part of monthly hostel charges in a college hostel are fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for

25

days, he has to pay

₹ 4500

, whereas a student B who takes food for

30

days, has to pay

₹ 5200

. Find the fixed charges per month and the cost of the food per day.

HARD

10th CBSE

IMPORTANT

Secondary School Mathematics for Class 10>Chapter 3 - Linear Equations in Two Variables>Exercise 3E>Q 28

A man invested an amount at

10 %

per annum and another amount at

8 %

per annum simple interest. Thus, he received

₹ 1350

as annual interest. Had he interchanged the amounts invested, he would have received

₹ 45

less as interest. What amounts did he invest at different rates?

If 2 is added to the numerator of a fraction, it reduces to (1/2) and if 1 is subtracted from the denominator, it reduces to (1/3). Find the fraction.

Important Questions on Linear Equations in Two Variables

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If $2$ is added to the numerator of a fraction, it reduces to $(1 / 2)$ and if $1$ is subtracted from the denominator, it reduces to $(1 / 3)$ . Find the fraction.