Example 1
Represent the following situations mathematically.
(i) John and Jivanti together have 45 marbles. Both of them lose 5 marbles each, and the product of the number of marbles they now have is 124. Represent the situation mathematically.
Solution
Let the number of marbles John initially had be x.
Then, the number of marbles Jivanti initially had is 45 − x.
After losing 5 marbles each:
- John's marbles = x − 5
- Jivanti's marbles = (45 − x) − 5 = 40 − x
According to the given condition,
(x − 5)(40 − x) = 124
Expanding the equation,
40x − x² − 200 + 5x = 124
−x² + 45x − 200 = 124
−x² + 45x − 324 = 0
Multiplying both sides by −1,
x² − 45x + 324 = 0
(ii) A cottage industry produces a certain number of toys in a day. The cost of production of each toy (in ₹) is 55 minus the number of toys produced in a day. On a particular day, the total cost of production was ₹750. Represent the situation mathematically.
Solution
Let the number of toys produced in a day be x.
Therefore, the cost of producing each toy is (55 − x) rupees.
Total Cost = Number of Toys × Cost per Toy
750 = x(55 − x)
Expanding the equation,
750 = 55x − x²
x² − 55x + 750 = 0
Required quadratic equation: x² − 55x + 750 = 0