Example 1:
Represent the following situations mathematically:
(i) John and Jivanti together have 45 marbles. Both of them lost 5 marbles each, and the product of the number of marbles they now have is 124. We would like to find out how many marbles they had to start with.
(ii) 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. On a particular day, the total cost of production was ₹750. We would like to find out the number of toys produced on that day.
Solution:
(i) Let us say the number of marbles John has = x
Therefore, the number of marbles Jivanti has = 45 – x
After losing 5 marbles each,
Number of marbles John has = x – 5
Number of marble Jivanti has = 45 – x – 5 = 40 – x
Given that the product of their marbles is 124.
∴ (x – 5)(40 – x) = 124
⇒ x2 – 45x + 324 = 0
⇒ x2 – 36x – 9x + 324 = 0
⇒ x(x – 36) -9(x – 36) = 0
⇒ (x – 36)(x – 9) = 0
Thus, we can say,
x – 36 = 0 or x – 9 = 0
⇒ x = 36 or x = 9
Therefore,
If John’s marbles = 36
Then, Jivanti’s marbles = 45 – 36 = 9
And if John’s marbles = 9
Then, Jivanti’s marbles = 45 – 9 = 36
(ii) Let us say the number of toys produced in a day is x.
Therefore, cost of production of each toy = Rs(55 – x)
Given the total cost of production of the toys = Rs 750
∴ x(55 – x) = 750
⇒ x2 – 55x + 750 = 0
⇒ x2 – 25x – 30x + 750 = 0
⇒ x(x – 25) -30(x – 25) = 0
⇒ (x – 25)(x – 30) = 0
Thus, either x -25 = 0 or x – 30 = 0
⇒ x = 25 or x = 30
Hence, the number of toys produced in a day will be either 25 or 30.