Question

A sum of Rs. 700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is Rs. 20 less than its preceding prize. Find the value of each of the prizes.

Answer 1

Hint: To solve this question, we will, first of all, assume a variable for the least value prize. Then as the difference between the prize value is 20, consecutive prize values are obtained as $x, x + 20$, etc. Finally, we will sum them up to Rs. 700 to get the value of x and all others.

Complete step-by-step solution:
The sum of the amount is given to be Rs. 700. It is to be given as 7 cash prize. Let the least one of the last students getting the prize to be of amount x. Then the student above the last student would get a prize of amount $x + 20$ as each prize is 20 less than the preceding prize. The next student after the above one will get a prize of amount $x + 20 + 20 = x + 40.$ So, these were the prizes of the last three students.
Following the same strategy as the above consecutive students, we will get,
\[x,x+20,x+40,x+60,x+80,x+100,x+120......\left( i \right)\]
Now because the total sum is given to be 700. Adding all the values of (i) will give Rs. 700.
\[\Rightarrow x+x+20+x+40+x+60+x+80+x+100+x+120=700\]
\[\Rightarrow 7x+420=700\]
Substituting 420 to both the sides, we have,
\[\Rightarrow 7x=700-420\]
\[\Rightarrow x=\dfrac{280}{7}\]
\[\Rightarrow x=40\]
Therefore the last student will get the price of Rs. 40. Putting the value of x in the equation (i) to get all the prize values, we have them as
\[\Rightarrow 40\]
\[\Rightarrow 40+20=60\]
\[\Rightarrow 40+40=80\]
\[\Rightarrow 40+60=100\]
\[\Rightarrow 40+80=120\]
\[\Rightarrow 40+100=140\]
\[\Rightarrow 40+120=160\]
Therefore, the value of the prizes are 40, 60, 80, 100, 120, 140 and 160.

Note: Another method to solve this question can be assuming a variable ‘y’ for the largest or the biggest prize value. Then the consecutive prize values will be $y, y – 20, y – 40, y – 60, y – 80, y – 100, y – 120.$ Then adding all the values, we will get y as 160 and the sum is Rs. 700. So, anyways the answer would be the same.