RD Chapter 3 Pair of Linear Equations in Two Variables Ex 3.6 Solutions
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. [NCERT Exemplar]
Answer - 11 : -
Let the cost of a pen be ₹ x and the cost of a pencil box be ₹ y.
Then, by given condition,
4x + 4y = 100 => x + y = 25 …(i)
and 3x = y + 15
=> 3x – y = 15 …(ii)
On adding Eqs. (i) and (ii), we get
4x = 40 => x = 10
By substituting x = 10, in Eq. (i) we get
y = 25 – 10 = 15
Hence, 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?
Answer - 12 : -
Let the amount of first person = ₹ x
and amount of second = ₹ y
According to the first condition,
x + 100 = 2 (y- 100)
=> x + 100 = 2y – 200
=> x – 2y = -200 – 100
=> x – 2y = -300 …….(i)
According to the second condition,
6(x – 10) = (y + 10)
6x – 60 = y + 10
6x – y = 10 + 60
6x – y = 70 ….(ii)
Multiplying (i) by 1 and (ii) by 2, we get
Hence first person has money = ₹ 40 and second person has = ₹ 170
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 ?
Answer - 13 : -
Let A has mangoes = x
and B has mangoes = y
According to the first condition,
x + 30 = 2 (y – 30)
x + 30 = 2y – 60
x – 2y = -60 – 30
=> x – 2y = -90 ….(i)
and according to the second condition
3 (x – 10) = (y + 10)
=> 3x – 30 = y + 10
=> 3x – y = 10 + 30
=> 3x – y = 40 ….(ii)
From (i) x = -90 + 2y Substituting in (i)
3 (-90 + 2y) – y = 40
– 270 + 6y – y = 40
=> 5y = 40 + 270 = 310
=> y = 62
and x = – 90 + 2y = – 90 + 2 x 62 = 124 – 90 = 34
A has mangoes = 34
and B has mangoes = 62
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. [NCERT Exemplar]
Answer - 14 : -
Let the number of bananas in lots A and B be x and y, respectively.
Case I:
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
=> (2/3) x + y = 400
=> 2x + 3y= 1200 …(i)
Case II:
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
=> x + (4/5) y = 460
=> 5x + 4y = 2300 …(ii)
On multiplying in the Eq. (i) by 4 and Eq. (ii) by 3 and then subtracting them, we get,
Now, putting the value of x in Eq. (i), we get,
2 x 300 + 3y = 1200
=> 600 + 3y = 1200
=> 3y = 1200 – 600
=> 3y = 600
=> y = 200
Total number of bananas = Number of bananas in lot A + Number of bananas in lot B
= x + y
= 300 + 200 = 500
Hence, he had 500 bananas.
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.
Answer - 15 : -
Let the price of T.V. = ₹ x
and price of Fridge = ₹ y
According to first condition,
y/10 = 2000 – 1000 = 1000
=> y = 10 x 1000 = 10000
Hence price of T.V. = ₹ 20000 and of fridge = ₹ 10000