Example 1 :
John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they have now is 124. We would like to find out how many marbles they had to start with.
Let 'x' be the number of marbles that john has.
Then, the number of marbles that Jivanthi has '45 - x'.
After losing 5 marbles,
number of marbles with John = x - 5
number of marbles with Jivanti = 45 - x - 5 = 40 - x
The product of number of marbles = 124
(x - 5)(40 - x) = 124
40x - x2 - 200 + 5x = 124
45x - x2 - 200 = 124
x2 - 45x + 124 + 200 = 0
x2 - 45x + 324 = 0
x2- 36x - 9x + 324 = 0
x(x - 36) - 9(x - 36) = 0
(x - 9)(x - 36) = 0
x - 9 = 0 or x - 36 = 0
x = 9 or x = 36
If x = 9,
45 - x = 45 - 9 = 36
If x = 36,
45 - x = 45 - 36 = 9
So, John is having 36 marbles, when Jivanti is having 9 marbles or Jivanti is having 36 marbles, when John is having 9 marbles.
Example 2 :
A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in rupees) was found to be 55 minus the number of toys produced in a day, the total cost of production was Rs. 750. We would like to find the number of toys produced on that day.
Let 'x' be the number of toys produced in a particular day
cost of production of one toy = 55 - x
Total cost = Number of toys produced on that day x cost of one toy
Total cost = x (55 - x)
750 = 55x - x2
x2 - 55x + 750 = 0
x2 - 30x - 25x + 750 = 0
x(x - 30) - 25(x - 30) = 0
(x - 30)(x - 25) = 0
x - 30 = 0 or x - 25 = 0
x = 30 or x = 25
So, the number of toys produced on that particular day is 30 or 25.
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