Question

A profile of the investments owned by an agent's clients follows:
i) 228 own annuities.
ii) 220 own mutual funds.
iii) 98 own life insurance and mutual funds.
iv) 93 own annuities and mutual funds.
v) 16 own annuities, mutual funds, and life insurance.
vi) 45 more clients own only life insurance than own only annuities.
vii) 290 own only one type of investment (i.e., annuity, mutual fund, or life
insurance).
Calculate the agent's total number of clients.

The answer is 500. Could you please explain why? Thank you so much!

Answer 1

Let :

A denote the clients having annuities ;

M denote the clients having mutual funds;

L denote the clients having Life Insurance ;

n() denote the no. of clients having a particular combination of annuities, mututal funds and life insurance.

Now, we are given the following information :

n(A) = 228 ; n(M) = 220 ;n(LM) = 98 ; n(AM) = 93 ; n(AML) = 16 ; n(L only) - n(A only) = 45

n(only one type of investment) = n(L only) + n(A only) + n(M only) = 290

We need to find the total no. of clients the agent has which is equal to :

n(AML) = n(A) + n(M) + n(L) - n(AM) - n(ML) - n(LA) + n(AML)

Substituting the values that are known, we get :

n(AML) = 228 + 220 + n(L) - 93 - 98 - n(LA) + 16

=> n(AML) = 273 + n(L) - n(LA)

=> n(AML) = 273 + n(LA^c)   ................(1)

Now,

n(L only) - n(A only) = 45 => n(A only) = n(L only) - 45

Substituting the above equation in : n(L only) + n(A only) + n(M only) = 290 , we get :

2*n(L only) + n(M only) - 45 = 290

=> 2*n(L only) + n(M only) = 335

Now,

n(M only) = n(M(AL)^c)

= n(M) - n(M(AL))

= n(M) - n((MA)(ML))

= n(M) - n(MA) - n(ML) + n(MAL) [Using n(AB) = n(A) + n(B) - n(AB)]

= 220 - 93 - 98 + 16

= 45

Thus,

2*n(L only) + 45 = 335

=> n(L only) = 145

Now, n(L only) = n(L) - n(LA) - n(ML) + n(MAL) [We derived the formula while calculating n(M only)]

=> 145 = n(LA^c) - 98 + 16

=> n(LA^c) = 227

Substituting the above value in equation (1), we get :

n(AML) = 273 + 227 = 500

Thus, the agent has total 500 clients.

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