A sum of Rs.1600 is to be used, to give 10 cash prizes to students of school, for their overall academic performances. If each prize is Rs. 20 less than its preceding prize, find the value of each of the prizes.
Answers
Given,
10 cash prizes are given of a total of Rs. 1600.
Each prize is less than Rs. 20 less than its preceding prize.
To find,
The value of each of the prizes.
Solution,
The value of the 10 prizes will be Rs. 250, Rs. 230, Rs. 210, Rs. 190, Rs. 170, Rs. 150, Rs. 130, Rs. 110, Rs. 90, and Rs. 70.
We can easily solve this problem by following the given steps.
Let's take the first cash prize to be Rs.x.
According to the question,
10 prizes = x, (x-20), (x-40), (x-60), (x-80), ---
Sum of all the prizes = Rs. 1600
If we carefully observe the sequence of the cash prizes, we will observe that the difference between each prize and its preceding prize is the same. So, the sequence formed is an A.P.
Common difference(d) = Second term - first term
d = x-20-x
d = -20
Total terms(n) = 10
The first term (a) = x
Sum of all the terms (Sn) = 1600
Using the formula of sum of all the terms in an AP,
Sn = n/2 [2a+(n-1)d]
1600 = 10/2 [2 × x + (10-1)(-20)]
5 [2x + 9(-20)] = 1600
5[2x - 180] = 1600
10x - 900 = 1600
10x = 1600+900 ( Moving 900 from the left-hand side to the right-hand side will result in the change of the sign from minus to plus.)
10x = 2500
x = 2500/10
x = 250
So, the first prize is Rs. 250.
And the 10 prizes are Rs. 250, Rs. 230, Rs. 210, Rs. 190, Rs. 170, Rs. 150, Rs. 130, Rs. 110, Rs. 90, and Rs. 70 (because each prize is Rs. 20 less than its preceding prize.)
Hence, the value of the 10 prizes are Rs. 250, Rs. 230, Rs. 210, Rs. 190, Rs. 170, Rs. 150, Rs. 130, Rs. 110, Rs. 90, and Rs. 70.
Answer:
250,230,210.......
Step-by-step
let the first prize be x
therefore x, x-20, x-40,.....
a=x, d = -20, n=10, Sn= 1600
Sn=n/2(2a+(n-1)d)
1600=10/2(2x+9(-20))
160= 1/2(2(x-90))
160=x-90
160+90=x
250=x
So the ap is 250, 230, 210.... 70
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