Class 10 RD Sharma Solutions – Chapter 3 Pair of Linear Equations in Two Variables – Exercise 3.6 | Set 2

Question 11. The cost of 4 pens and 4 pencil 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 pencil box.

Solution:

Let the cost of a pen be Rs. x 

and the cost of a pencil box be Rs. y.

Now, the cost of 4 pens and 4 pencil boxes is Rs 100.

     4x + 4y = 100 

=> x + y = 25     (1)

And, three times the cost of a pen is ₹ 15 more than the cost of a pencil box

     3x = y + 15

=> 3x – y = 15     (2)

On adding Equation (1) and (2), we get:

     4x = 40 

=> x = 10

Putting x = Rs. 10 in equation (1), we get:

     10 + y = 25

=> y = 25 – 10 = 15

So, the cost of a pen and a pencil box are ₹ 10 and ₹ 15, respectively.

Question 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:

Let the amount of first person be Rs. x

and amount of second one be Rs. y

Now, upon giving Rs. 100 to first from second, first person will have twice the amount of the second

     x + 100 = 2 (y- 100)

=> x + 100 = 2y – 200

=> x – 2y = -200 – 100

=> x – 2y = -300     (1)

Also, if Rs. 10 is given to second from first, it will have six times the amount of first

     6(x – 10) = (y + 10)

=> 6x – 60 = y + 10

=> 6x – y = 10 + 60

=> 6x – y = 70     (2)

Multiplying (i) by 1 and (ii) by 2 and subtracting them, we get:

     x – 2y – 12x + 2y = -300-140

=> -11x = -440

=> x = 440/11 = 44

Putting x = Rs. 44 in equation (1), we get:

     44 – 2y = -300

=> 340 = 2y

=> y = 170

Hence first person has money Rs. 40 and second person has Rs. 17

Question 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 left with you.” How many mangoes does each have?

Solution:

Let A has x mangoes 

and B has y mangoes

According to the first condition,

     x + 30 = 2 (y – 30)

=> x + 30 = 2y – 60

=> x – 2y = -60 – 30

=> x – 2y = -90     (1)

and according to the second condition

     3 (x – 10) = (y + 10)

=> 3x – 30 = y + 10

=> 3x – y = 10 + 30

=> 3x – y = 40     (2)

Multiplying equation (1) with 1 and equation (2) with 2 and subtracting them, we get:

     x – 2y – 6x + 2y = -90 -80

=> -5x = -170

=> x = 34

Putting x = 34 in equation (1), we get:

     34 – 2y = -90

=> 2y = 124

=> y = 62

So, A has 34 mangoes and B has 62 mangoes

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

Solution:

Let the number of bananas in lots A and B be x and y, respectively.

Now,

Cost of the first lot at the rate of ₹ 2 for 3 bananas + Cost of the second lot at the rate of ₹ 1 per banana = Amount received (Rs. 400)

=> (2/3) x + y = 400

=> 2x + 3y= 1200     (1)

Also,

Cost of the first lot at the rate of ₹ 1 per banana + Cost of the second lot at the rate of ₹ 4 for 5 bananas = Amount received (Rs. 460)

=> x + (4/5) y = 460

=> 5x + 4y = 2300     (2)

On multiplying in the Equation (1) by 4 and Equation (2) by 3 and subtracting them, we get:

     8x + 12y -15x – 12y = 4800-6900

=> -7x = -2100

=> x = 300

Putting x = 300 in equation (1), we get:

     600 + 3y = 1200

=> 3y = 600

=> y = 200

So, total numbers of bananas he had was (300+200) = 500

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

Solution:

Let the price of T.V. be Rs. x

and price of Fridge be Rs. y

Now, on selling a T.V. at 5% gain and a fridge at 10% gain, gain is Rs. 2000

     (x*5)/100 + (y*10)/100 = 2000     

=> x + 2y = 40000     (1)

Also, if he sells the T.V. at 10% gain and the fridge at 5% loss, gain is Rs. 1500

     (x*10)/100 – (y*5)/100 = 1500

=> 2x – y = 30000     (2)

Multiplying equation (1) by 2 and subtracting it from equation (2), we get:

     2x + 4y – 2x + y = 80000 – 30000

=> 5y = 50000

=> y = 10000

Putting y = Rs. 10000 in Equation (2), we get:

    2x – 10000 = 30000

=> 2x = 40000

=> x = 20000

So, the cost of T.V. is Rs. 20000 and cost of fridge is Rs. 10000

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