Question

In a mathematic quiz, 30 prizes consisting of 1st and 2nd prizes only are to be given 1st and 2nd prizes are worth Rs. 2000 and Rs. 1000, respectively. If the total prize money is Rs. 52,000 then show that
(a) If 1st prizes are x in number, the number of 2nd prizes are ____
(b) The total value of prizes in term of x are ____
(c) The equation formed is ____
(d) The solution of the equation is ____.
(e) The number of 1st prizes are ____ and the number of 2nd prizes are ____.

Answer 1

Step 1: Find the number of 2nd prizes​
Given: The total number of prizes = 30
The number of first prizes = x
So, x + Number of second prizes = 30
Number of second prizes $=30-x$
Hence, the number of 2nd prizes are $30-x$

Step 2: Find the total value of prizes​
Value of first prize = Rs. 2000
Value of x first prizes $=2000x$
Value of second prize = Rs. 1000
Value of $(30-x)$ second prizes $=1000(30-x)$
So, total value of prizes $=2000x+1000(30-x)$
Hence, the total value of prizes in term of x is $2000x+1000(30-x)$.
Step 3: Find the equation​
Total prize money $=52000$
$2000x+1000(30-x)=52000$
$2000x+30000-1000x=52000$
$\therefore 1000x+30,000=52,000$
Hence, the equation formed is $1000x+30,000=52,000$

Step 4: Find the solution of the equation​
$1000x+30,000=52,000$
$1000x=52,000-30,000$
$1000x=22,000$
$x=\frac{22,000}{1000}$
$\therefore x=22$
Hence, the solution of the equation is 22.

Step 5: Find the number of 1st and 2nd prizes​
The number of 1st prizes $=x=22$
The number of 2nd prizes $=30-x$
$=30-22=8$
Hence, the number of 1st prizes are 22 and the number of 2nd prizes are 8.