Question 2: Solve the problems given in Example 1.

(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.

**Solution:** Given, John and Jivanti together have number of marbles = 45

After losing of 5 marbles by each of them, number of marble `= 45 – 5 – 5 = 45 – 10 = 35`

Let us assume, John has x marbles

Hence; marbles with Jivanti `= 35 – x`

As per question; product of marbles after loss = 124

Therefore; `x(35 – x) = 124`

Or, `35x – x^2 = 124`

Or, `- x^2 + 35x – 124 = 0`

Or, `x^2 – 35x + 124 = 0`

Or, `x^2 – 4x – 31x + 124 = 0`

Or, `x(x – 4) – 31 (x – 4) = 0`

Or, `(x – 31)(x – 4) = 0`

Hence, `x = 31` and `x = 4`

One person has 31 marbles and another has 4 marbles

At the beginning; one person had 36 marbles and another had 9 marbles.

(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 Rs 750. We would like to find out the number of toys produced on that day.

**Solution:** Let us assume, number of toys `= x`

Then, cost of production of each toy `= x – 55`

Hence, total cost of production `= x(55 – x) = 750`

Or, `55x – x^2 = 750`

Or, `x^2 - 55x + 750 = 0`

Or, `x^2 - 30x - 25x + 750 = 0`

Or, `x(x - 30) – 25(x – 30) = 0`

Or, `(x – 25)(x – 30) = 0`

Hence, `x = 25` and `x = 30`

Thus, number of toys is 25 or 30

Question 3: Find two numbers whose sum is 27 and product is 182.

**Solution:** Let us assume, one of the numbers `= x`

Hence, second number `= 27 – x`

As per question; `x(27 – x) = 182`

Or, `27x – x^2 = 182`

Or, `27x – x^2 – 182 = 0`

Or, `x^2 – 27x + 182 = 0`

Or, `x^2 – 14x – 13x + 182 = 0`

Or, `x(x – 14) – 13(x – 14) = 0`

Or, `(x – 13)(x – 14) = 0`

Hence, `x = 13` and `x = 14`

Hence, the numbers are 13 and 14

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