# Class 10 RD Sharma Solutions – Chapter 3 Pair of Linear Equations in Two Variables – Exercise 3.6 | Set 1

### Question 1. 5 pens and 6 pencils together cost ₹ 9 and 3 pens and 2 pencils cost ₹ 5. Find the cost of 1 pen and 1 pencil.

**Solution:**

Let the cost of 1 pen be Rs. x

and the cost of 1 pencil be Rs. y

Now, 5 pens and 6 pencils cost Rs. 9

=> 5x + 6y = 9 (1)

And, 3 pens and 2 pencils cost Rs. 5

=> 3x + 2y = 5 (2)

Multiplying (1) by 1 and (2) by 3 and subtracting them we get:

5x+6y-9x-6y = 9-15

=> -4x = 6

=> x = 3/2

Putting x = 3/2 in equation (1), we get:

6y = 9-15/2

=> 6y = 3/2

=> y = 1/4

So, the cost of one pen is Rs. 3/2 and cost of one pencil is Rs. 1/4

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

**Solution:**

Let the cost of 1 audio cassette be Rs. x

and cost of 1 video cassette be Rs. y

Now, cost of 7 audio cassette and 3 video cassette is Rs. 1100

=> 7x + 3y= 1110 (1)

And, cost of 5 audio cassette and 4 video cassette is Rs. 1350

=> 5x + 4y = 1350 (2)

Multiplying (1) by 4 and (2) by 3 and subtracting them we get:

28x+12y-15x-12y = 4440-4050

=> 13x = 390

=> x = 390/13 = 30

Putting x=30 in equation (1), we get:

210+3y = 1110

=> 3y = 900

=> y = 900/3 =300

So, the cost of one audio cassette is Rs. 30 and cost of one video cassette is Rs. 300

### Question 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 the number of pencils be y

x + y = 40 (1)

Now, if she has 5 more pencils and 5 less pens, then number of pencils would become 4 times the number of pens

number of pens now = x – 5

and number of pencils now = y + 5

(y + 5) = 4 (x – 5)

=> y + 5 = 4x – 20

=> 4x – y = 25 (2)

Adding (1) and (2), we get:

5x = 65

=> x = 13

Putting x=3 in equation (1), we get:

13 + y = 40

=> y = 40 – 13 = 27

So, original number of pens were 13 and pencils were 27

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

**Solution:**

Let the cost of 1 table be Rs. x

and the cost of 1 chair be Rs. y

Now, 4 tables and 3 chairs together cost Rs 2250

4x + 3y = 2250 (1)

And, 3 tables and 4 chairs cost Rs. 1950

3x + 4y= 1950 (2)

Multiplying (1) by 3 and (2) by 4 and subtracting them, we get:

12x+9y-12x-16y = 6750-7800

=> -7y = -1050

=> y = 1050/7 = 150

Putting y = 150 in equation (1), we get:

4x + 450 = 2250

=> 4x = 1800

=> x = 1800/4 = 450

So, the cost of 1 table and 2 chairs will be:

450 + 2(150) = Rs. 750

### Question 5. 3 bags and 4 pens together cost ₹ 257 whereas 4 bags and 3 pens together cost ₹ 324. Find the total cost of 1 bag and 10 pens.

**Solution:**

Let the cost of 1 bag be Rs. x

and the cost of 1 pen be Rs. y

Now, 3 bags and 4 pens together cost Rs 257

3x + 4y = 257 (1)

And, 4 bags and 3 pens together cost Rs. 324

4x + 3y = 324 (2)

Multiplying (1) by 3 and (2) by 4 and subtracting them, we get:

9x+12y-16x-12y = 771-1296

=> -7x = -525

=> x = 525/7 = 75

Putting x = 75 in equation (1), we get:

225 + 4y = 257

=> 4y = 32

=> y = 32/4 = 8

So, the cost of 1 bag and 10 pens will be:

75 + 10(8) = Rs. 155

### Question 6. 5 books and 7 pens together cost ₹ 79 whereas 7 books and 5 pens together cost ₹ 77. Find the cost of 1 book and 2 pens.

**Solution:**

Let the cost of 1 book be Rs. x

and the cost of 1 pen be Rs. y

Now, 5 books and 7 pens together cost Rs. 79

5x + 7y = 79 (1)

And, 7 books and 5 pens together cost Rs. 77

7x + 5y = 77 (2)

Multiplying (1) by 7 and (2) by 5 and subtracting them, we get:

35x+49y-35x-25y = 553-385

=> 24y = 168

=> y = 168/24 = 7

Putting y=7 in equation (2), we get:

7x + 35 = 77

=> 7x = 42

=> x = 42/7 = 6

So, the cost of 1 book and 2 pens will be:

6 + 2(7) = Rs. 20

### Question 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 profit of 25% on the table and 10% on the chair she would have got ₹ 1065. Find the cost price of each.

**Solution:**

Let the cost price of the table be Rs. x

and the cost price of the chair be Rs. y

The selling price of the table, when it is sold at a profit of 10% will be Rs. [x+(10/100)x]

The selling price of the chair, when it is sold at a profit of 25% will be Rs. [y+(25/100)y]

Their sum total is Rs. 1050

=> [x+(10/100)x] + [y+(25/100)y] = 1050 (1)

Had she taken 25% profit on table, selling price of it would be Rs. [x+(25/100)x]

Had she taken 10% profit on chair, selling price of it would be Rs. [y+(10/100)y]

And their sum total would be Rs. 1065

=> [x+(25/100)x] + [y+(10/100)y] = 1065 (2)

Simplifying equation (1) and (2) we get:

110x + 125y = 105000 (3)

125x + 110y = 106500 (4)

Multiplying equation (3) by 25 and equation (4) by 22 and subtracting them, we get:

2750x + 3125y – 2750x – 2420y = 2625000-2343000

=> 705y = 282000

=> y = 282000/705 = 400

Putting y = 400 in equation (3), we get:

110x + 50000 = 105000

=> 110x = 55000

=> x = 55000/110 = 500

So, the cost price table is Rs. 500 and chair is Rs. 400

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

**Solution:**

Let the amount of investments in schemes A and B be Rs. x and Rs. y, respectively.

Now she has received Rs. 1860 in total

Interest on Rs. x at the rate of 8% per annum on scheme A + Interest on Rs. y at the rate of 9% per annum on scheme B = Total amount received

=> (x*8*1)/100 + (y*9*1)/100 = 1860 [because simple interest = (principal*rate*time)/100]

=> 8x+9y = 186000 (1)

Now, if she has interchanged the investment amount, her profit would be 720 more

=> she has invested Rs. x in Scheme B and Rs. y in Scheme A

Interest on Rs. x at the rate of 9% per annum on scheme A + Interest on Rs. y at the rate of 8% per annum on scheme B = Total amount received

=> (x*9*1)/100 + (y*8*1)/100 = 1860+20

=> 9x+8y = 188000 (2)

Multiplying equation (1) by 9 and equation (2) by 8 and subtracting them, we get:

72x+81y-72x-64y = 1000(1674-1504)

=> 17y = 1000(170)

=> y = 10000

Putting y = Rs. 10000 in equation (1), we get:

8x + 90000 = 186000

=> 8x = 96000

=> x = 12000

So, she invested Rs. 12000 in Scheme A and Rs.10000 in Scheme B

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

**Solution:**

Let the cost of 1 bat be Rs. x

and the cost of 1 ball be Rs. y

Now, 7 bats and 6 balls costs Rs. 3800

7x + 6y = 3800 (1)

Also, 3 bats and 5 balls costs Rs. 1750

3x + 5y = 1750 (2)

Multiplying (1) by 5 and (2) by 6 and subtracting them, we get:

35x + 30y – 18x – 30y = 19000-10500

=> 17x = 8500

=> x = 8500/17 = 500

Putting x = Rs. 500 in equation (2), we get:

1500 + 5y = 1750

=> 5y = 250

=> y = 50

So, the cost of a bat is Rs. 500 and the cost of a ball is Rs. 50

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

**Solution:**

Let the fixed charge for the book be Rs. x

and let the extra charge for each day be Rs. y

For seven days charge is Rs. 27, extra charge will be applicable after 3 days (for 4 days)

x + 4y = 27 (1)

For five days charge is Rs. 21, extra charge will be applicable after 3 days (for 2 days)

x + 2y = 21 (2)

Subtracting both equation, we get:

2y = 6

=> y = 3

Putting y= Rs. 3 in equation (2), we get:

x + 6 = 21

=> x = 15

So, the fixed charge amount is Rs. 15 and charges for each extra day is Rs. 3

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