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Quantitative Aptitude Question on Arithmetic Progressions

A club consist of members whose age are in AP. The common difference between the ages is 3 months. If the youngest member is 7 years old and the sum of the ages of all the members is 250, then the number of members in the club is

A

15

B

25

C

20

D

30

Answer

25

Explanation

Solution

Let's denote the number of members in the club as 'n'.
We are given that the youngest member is 7 years old, and the common difference between the ages of the members is 3 months, which is equivalent to 14\frac{1}{4} years.
So, the ages of the members can be represented as an arithmetic progression (AP):
7, 7+$$\frac{1}{4}, 7++2(14)(\frac{1}{4}), 7++3(14)(\frac{1}{4}), …
To find the sum of the ages of all the members, we can use the formula for the sum of an arithmetic series:
Sum =(n2)×(\frac{n}{2})\times[2a + (n-1)d]
Where:
● n is the number of terms (number of members),
● a is the first term (age of the youngest member, which is 7 years),
● d is the common difference (14(\frac{1}{4} years)).
We're given that the sum of the ages of all the members is 250.
Substituting these values into the formula:
250 =(n2)×[2×7+(n−1)×(14)]=(\frac{n}{2})\times[2\times7+(n-1)\times(\frac{1}{4})]
Now, simplify:
250=(n2)×[14+(n−1)4]=(\frac{n}{2})\times[14+\frac{(n-1)}{4}]
Next, remove the fraction by multiplying both sides by 4:
1000=n×\times[56++(n-1)]
Now, distribute n on the right side:
1000 = 56n++ n(n-1)
Now, expand and simplify further:
1000 = 56n+$$n^2-n
Combine like terms: 0 =n2+n^2+55n- 1000
Now, we need to solve this quadratic equation for n.
You can factor it or use the quadratic formula:
n=−b+‾(b2−4ac)2an=-b\underline+\sqrt\frac{(b2-4ac)}{2a}
In this case, a = 1, b = 55, and c =-1000.
n=−55+‾(552−4(1)(−1000))2(1)n=-55\underline+\sqrt\frac{(552-4(1)(-1000))}{2(1)}
n=−55+‾(3025+4000)2n=-55\underline+\sqrt\frac{(3025+4000)}{2}
n=−55+‾70252n=-55\underline+\sqrt\frac{7025}{2}
n=−55+‾52812n=-55\underline+5\sqrt\frac{281}{2}
Now, we have two possible solutions:

  1. n=−55+52812n=-55+5\sqrt\frac{281}{2}
  2. n=−55−52812n=-55-5\sqrt\frac{281}{2}
    Since the number of members in the club should be a positive whole number, wecan ignore the negative solution.
    So, the number of members in the club is approximately:
    N≈−55+2812≈25.67N\approx-55+\sqrt\frac{281}{2}\approx25.67
    Since the number of members must be a whole number, we can round down to the nearest integer.
    Therefore, the number of members in the club is 25.
    So, the correct answer is (B): 25.