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

  • Last Updated : 13 Jan, 2021

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

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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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