Question

In a Mathematics quiz, 30 prizes consisting of 1st and 2nd prizes only are to be given. 1st and 2nd prizes are worth ₹ 2000 and ₹ 1000, respectively. If the total prize money is ₹ 52,000 then show that:

1. If 1st prizes are x in number the number of 2nd prizes are ______.
2. The total value of prizes in terms of x are ______.
3. The equation formed is ______.
4. The solution of the equation is ______.
5. The number of 1st prizes are ______ and the number of 2nd prizes are ______.

Answer 1

1. If 1st prizes are x in number the number of 2nd prizes are 30.
2. The total value of prizes in terms of x are 2000x + 1000(30 – x).
3. The equation formed is 1000x + 30000 = 52000.
4. The solution of the equation is x = 22.
5. The number of 1st prizes are 22 and the number of 2nd prizes are 8.

Explanation:

Given, the number of prizes = 30

Total prize money = ₹ 52000, 1st and 2nd prizes are worth ₹ 2000 and ₹ 1000, respectively.

a. 1st prizes are x in number, the number of 2nd prizes are (30 – x), because total number of prizes are 30.

b. Total value of prizes in terms of x are 2000x + 1000(30 – x).

c. The equation formed is 1000x + 30000 = 52000 

From (b), 2000x + 1000(30 – x) = 52000

⇒ 2000x + 30000 – 1000x = 54000

⇒ 1000x + 30000 = 52000

d. The solution of the equation is 22.

From (c), 1000x + 30000 = 52000

⇒ 1000x = 52000 – 30000 = 22000

⇒ x = `22000/1000` = 22

e. From (b), 2000x + 1000(30 – x) = 52000

2x + (30 – x) = 52   ...[Dividing both sides by 1000]

x + 30 = 52 

⇒ x = 52 – 30 = 22

∴ Number of 2nd prizes = 30 – 22 = 8