## Chapter 3 Pair of Linear Equations in Two Variables R.D. Sharma Solutions for Class 10th Math Exercise 3.6

**Exercise 3.6**

In each of the following systems of equations determine whether the system has a unique solution, no solution or infinitely many solutions. In case there is a unique solution, find it:

1. 5 pens and 6 pencils together cost Rs 9 and 3 pens and 2 pencils cost Rs 5. Find the cost of1 pen and 1 pencil.

**Solution**

2. 7 audio cassettes and 3 video cassettes cost Rs 1110, while 5 audio cassettes and 4 video cassettes cost Rs 1350. Find the cost of an audio cassette and a video cassette.

**Solution**

Let the cost of a audio cassette be Rs x and that of a video cassette be . Rs y Then,

Hence, cost of one audio cassette = Rs 30 and cost of one video cassette = Rs 300 .

3. Reena has pens and pencils which together are 40 in number. If she has 5 more pencils and 5 less pens, then number of pencils would become 4 times the number of pens. Find the original number of pens and pencils.

**Solution**

Let the number of pens be x and that of pencil be y. then,

4. 4 tables and 3 chairs, together, cost Rs 2,250 and 3 tables and 4 chairs cost Rs 1950. Find the cost of 2 chairs and 1 table.

**Solution**

Let the cost of a table be Rs x and that of a chairs be Rs y Then,

Cost of 2 chairs = 300 Rs and cost of 1 table = 450

∴ The cost of 2 chairs and 1 table 300 + 450 + 750

5. 3 bags and 4 pens together cost Rs 257 whereas 4 bags and 3 pens together cost R 324. Find the total cost of 1 bag and 10 pens.

**Solution**

Hence, the total cost of 1 bag and 10 pens 75+ 80 = 155.

6. 5 books and 7 pens together cost Rs 79 whereas 7 books and 5 pens together cost Rs 77. Find the total cost of 1 book and 2 pens.

**Solution**

Let the cost of a book be Rs x and that of a pen be Rs y. Then,

Hence, the total cost of 1 book and 2 pens 6 + 14 = 20

7. Jamila sold a table and a chair for 1050 , thereby making a profit of 10% on a table and 25% on the chair . If she had taken a profit of 25% on the table and 10% on the chair she would have got Rs 1065. Find the cost price of each .

**Solution**

Let the CP of the table be Rs x and that of the chair be Rs y.

8. Susan nvested certain amount of money in two shemes A and B, which offer interest at the rate of 8% per annum and 9% per annum, respectively . She received Rs 1860 as annual interest . However, had she interchanged the amount of investment in the two schemes, she would have received Rs 20 more as annual interest . How much money did she invest in each scheme ?

**Solution**

Let the money invested in Scheme A be Rs x and that in Scheme B be Rs y.

So, the money invested in scheme A = Rs 12,000 and in scheme B = Rs 10,000.

9.The coach of a cricket team buys 7 bats and 6 balls for Rs 3800. Later, he buys 3 bats and 5 balls for Rs 1750. Find the cost of each bat and each ball.

**Solution**

x = 50

Hence, cost of 1 bat = x = 500

Hence, cost of 1 ball = x = 50

10. A lending library has a fixed charge for the first three days and additional charge for each day thereafter. Saritha paid Rs 27 for a book kept for seven days, while Susy paid Rs 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

**Solution**

To find:

(1) the fixed charge

(2) The charge for each day

Let the fixed charge be Rs x

And the extra charge per day be Rs y.

According to the given conditions,

x + 4y = 27

x + 4y - 27 = 0 ...(i)

x + 2y = 21

x + 2y - 21 = 0 ...(ii)

Subtracting equation 1 and 2 we get

2y = 6

y = 3

Substituting the value of y in equation 1 we get

x + 4(3) - 27 = 0

x + 12 - 27 = 0

x - 15 = 0

x = 15

Hence the fixed charge is x = Rs 15 and the charge of each day y = Rs 3

11. The cost of 4 pens and 4 pencils boxes is 100 . Three times the cost of a pen is 15 more than the cost of a pencil box . Form the pair of linear equations for the above situation. Find the cost of a pen and a pencil box .

**Solution**

Let the cost of 1 pen be x and that of 1 pencil box be y.

Now,

Cost of 4 pens + Cost of 4 pencil boxes = Rs 100 (given)

⇒ 4x + 4y = 100

⇒ x + y = 25 ...(i)

Also,

3 × Cost of a pen = Cost of a pencil box + 15

3x = y + 15

⇒ 3x - y = 15 ...(ii)

Adding (i) and (ii), we get

4x = 40

⇒ x = 10

Putting x = 10 in (i), we get

10 + y = 25

⇒ y = 15

Thus, the cost of a pen = 10 and that of a pencil box = 15.

12. One says, "Give me a hundred, friend! I shall then become twice as rich as you." The other replies, "If you give me ten, I shall be six times as rich as you." Tell me what is the amount of their respective capital ?

**Solution**

To find:

(1) Total amount of A.

(2) Total amount of B.

Suppose A has Rs x and B has Rs y

According to the given conditions,

x + 100 = 2(y − 100)

x + 100 = 2y − 200

x − 2y = −300 ....(1)

and

y + 10 = 6(x − 10)

y + 10 = 6x − 60

6x − y = 70 ....(2)

Multiplying equation (2) by 2 we get

12x − 2y = 140 ....(3)

Subtracting (1) from (3), we get

11x = 440

x = 40

Substituting the value of x in equation (1), we get

40 - 2y = -300

-2y = -340

y = 170

Hence A has x = Rs40 and B has y = Rs170

13. A and B each have a certain number of mangoes. A says to B, "if you give 30 of your mangoes, I will have twice as many as left with you." B replies, "if you give me 10, I will have thrice as many as left with you." How many mangoes does each have?

**Solution**

To find:

(1) Total mangoes of A.

(2) Total mangoes of B.

Suppose A has x mangoes and B has y mangoes,

According to the given conditions,

x + 30 = 2(y-30)

x + 30 = 2y - 60

x - 2y + 30 + 60 = 0

x - 2y + 90 = 0

y + 10 = 3(x-10)

y + 0 = 3x - 30

y - 3x + 10 + 30 = 0

y - 3x + 40 = 0

Multiplying eq. 1 by (3),

3x + 6y + 270 = 0 .... (3) and

Now adding eq.2 and eq.3

5y = 310

y = 310/5

y = 62

x - 2 × 62 + 90 = 0

x - 124 + 90 = 0

x - 34 = 0

x = 34

Hence A has 34 mangoes and B has 62 mangoes .

14. Vijay had some bananas, and he divided them into two lots A and B . He solds first lot at the rate of Rs 2 for 3 bananas and the second lot at the rate of Rs 1 per banana and got a total of 400 . If he had sold the first lot at the rate of 1 per banana and the second lot at the rate of Rs 4 per five bananas, his total collection would have been Rs 460 . Find the total number of bananas he had.

**Solution**

Let the bananas in lot A be x and that in lot B be y.

Vijay sold 3 bananas for Rs 2 in lot A.

So, the cost of x bananas in lot A = 2/3 x

∴ 2/3 x + y = 400

⇒ 2x + 32x + 3y = 1200n ...(i)

Now, if he sells the first lot at the rate of Rs 1 per banana and second for Rs 4 for 5 bananas, then

x + 4/5 y = Rs 460

⇒ 5x + 4y = 2300 ...(ii)

Solving (i) and (ii), we get

x = 300 and y = 200

So, the total number ob bananas = x+y = 300 + 200 = 500

15. On selling a T.V. at 5% gain and a fridge at 10% gain, a shopkeeper gains Rs 2000. But if he sells the T.V. at 10% gain the fridge at 5% loss. He gains Rs 1500 on the transaction. Find the actual prices of T.V. and fridge

**Solution**

Given:

(i) On selling of a T.V. at 5% gain and a fridge at 10% gain, shopkeeper gain Rs.2000.

(ii) Selling T.V. at 10% gain and fridge at 5% loss. He gains Rs. 1500.

To find: Actual price of T.V. and fridge.