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.

Subject

Maths

Standard

Class 10

Views

887

Asked By

Georgia

Vedhika
Answer / Solution

Solution: Let

The cost of 1st prize be Rs.P and cost of 2nd prize = Rs.P − 20, and cost of 3rd prize = Rs.P − 40.
We can see that the cost of these prizes is in the form of A.P. Therefore, first term is P and common difference (P − 40) − (P − 20) = −40 + 20 = −20. Thus, a = P and d = −20

Also given that the som of 7 ters are 700, that is S7 = 700. By the formula of sum of nth term, we know the formula Sn = n/2 [2a + (n – 1)d]. Replace the values weget, 

7/2 [2a + (7 – 1)d] = 700

7/2 [2a + (6) (–20)] = 700

2a + 6(−20) = 200
2a − 120 = 200
2a = 320 ⇒ a = 160

Therefore, the value of each of the prizes was Rs 160, Rs 140, Rs 120, Rs 100, Rs 80, Rs 60, and Rs 40.

Seema
Answer / Solution

Let the least value price be X.

Then the next value is X + 20, and the next value is X + 40 and on and on up to 7 values.

Using an arithmetic Progression with sum 700. You can do it either manually or using the formula of AP.

Method 1: (Manually)

X + X+20 + X + 40 + …  X + 120 = 700

⇒ 7X + (20 + 40 + … + 120) = 700

⇒ 7X + 20(1 + 2 + 3 + … + 6) = 700

⇒ 7X + 20(21) = 700

⇒ 7X = 280

⇒ X = 40

Method 2: (Using AP Formula)

Number of prizes = 7 (N). So sum till nth term will be:

n/2 (2X+ (n - 1) x 20) = 700.

Put the value of N as 7.

⇒ 7/2 (2X + 120) = 700

⇒ (2X + 120) = 700 * 2/7

⇒ 2X + 120 = 200

⇒ 2X = 200 - 120

⇒ 2X = 80

⇒ X = 40

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