SOLVING LINEAR EQUATIONS WORD PROBLEMS

Problem 1 :

10 students of class X took part in a mathematics quiz. If the number of girls is 4 more than the number of boys, find the number of boys and girls who took part in the quiz.

Solution :

Let x and y be no. of boys and no. of girls respectively.

From the given information,

x + y = 10 ----(1)

y = x + 4 ----(2)

Substitute y = x + 4 in (1).

x + (x + 4) = 10 ----(2)

x + x + 4 = 10

2x + 4 = 10

Subtract 4 from both sides.

2x = 6

Divide both sides by 2.

x = 3

Substitute x = 3 in (2).

y = 3 + 4

= 7

Therefore,

number of boys = 3

number of girls = 7

Problem 2 :

5 pencils and 7 pens together cost $50, whereas 7 pencils and 5 pens together cost $46. Find the cost of one pencil and that of one pen.

Solution :

Let x and y be the cost of one pencil and one pen respectively.

From the given information,

5x + 7y = 50 ----(1)

7x + 5y = 46 ----(2)

7x(1) - 5x(2) :

24y = 120

Divide both sides by 24.

y = 5

Substitute y = 5 in (1).

7x + 5(5) = 46

7x + 25 = 46

Subtract 25 from both sides.

7x = 21

Divide both sides by 7.

x = 3

Therefore,

cost of one pencil = $3

cost of one pen = $5

Problem 3 :

On selling the product x at 5% gain and the product y at 10% gain, a store owner gains $2000. But if he sells the product x at 10% gain and the product y at 5% loss, he gains $1500 on the transaction. Find the actual prices of the product x and the product y. 

Solution :

Let x and y be the selling prices of the products x and y respectively.

Given : On selling the product x at 5% gain and the product y at 10% gain, the store owner gains $2000.

0.05x +  0.1y = 2,000

To get rid of the decimal, multiply each side by 100. 

5x + 10y = 200,000

Divide each side by 5.

x + 2y = 40,000 -----(1)

Given : If x is sold at 10% gain and y at 5% loss, the gain is $1500. 

0.1x - 0.05y = 1,500

To get rid of the decimal, multiply each side by 100. 

10x - 5y = 150,000

Divide each side by 5.

2x - y = 30,000 -----(2)

In order to eliminate y in (1) and (2), add (1) and 2 times of (2).

(1) + 2(2) : 

5x = 100,000

Divide each side by 5.

x = 20,000

Substitute 20,000 for x in (2).

(2)-----> 2(20,000) - y = 30,000

40,000 - y = 30,000

Subtract 40,000 from each side. 

-y = -10,000

Multiply each side by (-1).

y = 10,000

So, the actual prices of x and y are $20,000 and $10,000 respectively.

Problem 4 :

Two numbers are in the ratio 5 : 6. If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5. Find the numbers.

Solution :

Let x and y be the two numbers.

Given : Two number are in the ratio 5 : 6.

x : y = 5 : 6

x/y = 5/6

6x = 5y

6x - 5y = 0 ----(1)

Given : If 8 is subtracted from each of the numbers, the ratio becomes 4 : 5.

(x - 8) : (y - 8) = 4 : 5

(x - 8)/(y - 8) = 4/5

5(x - 8) = 4(y - 8)

5x - 40 = 4y - 32

5x - 4y = -32 + 40

5x - 4y = 8 ----(2)

In order to eliminate y in (1) and (2), subtract 5 times (2) from 4 times (1).

4x(1) - 5x(2) :

x = 40

Substitute 40 for x in (2). 

(2)----> 5(40) - 4y = 8

200 - 4y = 8

Subtract 200 from each side. 

- 4y = -192

Divide each side by (-4).

y = 48

So, the required numbers are 40 and 48.

Problem 5 :

4 Indians and 4 Chinese can do a piece of work in 3 days. While 2 Indians and 5 Chinese can finish it in 4 days. How long will it take for 1 Indian to do it? How long will it take for 1 Chinese to do it ?

Solution :

Let x and y be the number of days taken by each Indian and each Chinese.

Then,

Work done by 1 Indian in 1 day = 1/x

Work done by 1 Chinese in 1 day = 1/y

Given : 4 Indians and 4 Chinese can complete the work in 3 days.

Work done by 4 Indians and 4 Chinese in 1 day = 1/3

That is, 

4/x + 4/y = 1/3

Let a = 1/x and b = 1/y.

4a + 4b = 1/3 ----(1)

Given : 2 Indians and 5 Chinese can finish it in 4 days.

Work done by 2 Indians and 5 Chinese in 1 day  =  1 / 4

That is, 

2/x + 5/y = 1/4

2a + 5b = 1/4 ----(2)

In order to eliminate a in (1) and (2), subtract 2 times (2) from (1).

(1) - 2x(2) :

-6b = 1/3 - 1/2

-6b = 2/6 - 3/6

-6b = (2 - 3)/6

-6b = -1/6

b = 1/36

Substitute 1/36 for b in (1).

(1)----> 4a + 4(1/36) = 1/3

4a + 1/9 = 1/3

Subtract 1/9 from both sides.

4a = 1/3 - 1/9

4a = 3/9 - 1/9

4a = (3 - 1)/9

4a = 2/9

a = 1/18

Find the values of x and y from the values of a and b.

a = 1/18

1/x = 1/18

x = 18

b = 1/36

1/y = 1/36

y = 36

So, one Indian will take 18 days to complete the work and one Chinese will take 36 days to complete the same work.  

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