**Problem 1 :**

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.

Then, we have

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 2 :**

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.

Then, we have

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.

Then, we have

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

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

5 x - 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).

4(1) - 5(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 3 :**

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.

Then,

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

Then,

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) - 2(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

Add 1/9 to each side.

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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