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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