Problem Solving

Alphonsaj wrote:A father left a will of $71 million between his 2 daughters aged 8.5 years and 12 years such that they may get equal amounts when each of them reach the age of 18 years. The father instructed that the original amount of 71mn be invested at 10% pa till such time the daughters turned 18. How much did the elder daughter get at the time of the will?

A.31mn
B.49mn
C.39mn
D.35.5mn
E.40mn

We can PLUG IN THE ANSWERS, which represent the initial amount bequeathed to the elder daughter.
Since the elder daughter earns interest for only 6 years (from age 12 to age 18), while the younger daughter earns interest for 9.5 years (from age 8.5 to age 18), the elder daughter earns LESS TOTAL INTEREST than the younger daughter.
Yet both daughters must receive the SAME AMOUNT at age 18.
Implication:
To compensate for earning LESS INTEREST than the younger daughter, the elder daughter must receive a GREATER INITIAL AMOUNT than the younger daughter.
Thus, the elder daughter must receive MORE THAN HALF of the $71 million.
Eliminate A and D.

Remaining options:
C) 39 million
E) 40 million
B) 49 million

Test the middle value.

Answer choice E: 40 million for the elder daughter, implying 31 million for the younger daughter
Elder daughter:
Yearly interest = 10% of 40 million = 4 million.
Total interest after 6 years = 6*4 = 24 million.
Amount received at 18 years = initial amount + interest = 40+24 = 64 million.

Younger daughter:
Yearly interest = 10% of 31 million = 3.1 million.
Total interest after 9.5 years = (9.5)(3.1) ⩳ 29 million.
Amount received at 18 years = initial amount + interest ⩳ 31+29 = 60 million.

Here, the elder daughter receives too much money at age 18.
Eliminate E.

For the elder daughter to receive LESS MONEY at age 18, she must receive a SMALLER INITIAL AMOUNT.

The correct answer is C.

Answer choice C: 39 million for the elder daughter, implying 32 million for the younger daughter
Elder daughter:
Yearly interest = 10% of 39 million = 3.9 million.
Total interest after 6 years = (6)(3.9) = 23.4 million.
Amount received at 18 years = initial amount + interest = 39 + 23.4 = 62.4 million.

Younger daughter:
Yearly interest = 10% of 32 million = 3.2 million.
Total interest after 9.5 years = (9.5)(3.2) ⩳ 30.4 million.
Amount received at 18 years = initial amount + interest ⩳ 32 + 30.4 = 62.4 million.
Success!

Alphonsaj wrote:A father left a will of $71 million between his 2 daughters aged 8.5 years and 12 years such that they may get equal amounts when each of them reach the age of 18 years. The father instructed that the original amount of 71mn be invested at 10% pa till such time the daughters turned 18. How much did the elder daughter get at the time of the will?

A.31mn
B.49mn
C.39mn
D.35.5mn
E.40mn

Algebraic approach:

Let e = the amount bequeathed to the elder daughter and y = the amount bequeathed to the younger daughter.

Elder daughter:
Interest earned each year = (10/100)e.
Total interest earned over the 6 years from age 12 to 18 = (6)(10/100)e = (60/100)e.
Total received at age 18 = (initial amount) + interest = e + (60/100)e = (160/100)e.

Younger daughter:
Interest earned each year = (10/100)y.
Total interest earned over the 9.5 years from age 8.5 to 18 = (9.5)(10/100)y = (95/100)y.
Total received at age 18 = (initial amount) + interest = y + (95/100)y = (195/100)y.

Since the total received at 18 must be the same in each case, we get:
(160/100)e = (195/100)y
160e = 195y
32e = 39y
e/y = 39/32.

Thus, e = 39 million and y = 32 million, for a total of 71 million.

The correct answer is C.

I'd think of it this way.

The elder daughter will get her money in 6 years, so let's run the interest till then first, and call her initial amount x.

Elder daughter's net: x * (1 + .1*6)

The younger daughter's share (let's call that y) is similar:

Younger daughter's net: y * (1 + .1*9.5)

We know that the initial shares sum to 71, so we have (x + y) = 71.

We also know that the two daughters receive the same amounts, so x * (1 + .1*6) = y * (1 + .1*9.5)

We want to solve for x, so let's isolate y: x * (1.6)/(1.95) = y

Then plug this back into our first equation, x + y = 71:

x + x * (1.6)/(1.95) = 71

x * (1 + 160/195) = 71

x * (355 / 195) = 71

x = 195 * (71 / 355) => 195 * (1/5) => 39

Of course, I probably should've reduced the above to one equation from the start. We know that

Elder's Daughter Trust * Interest = Younger Daughter's Trust * Interest

or

x * (1 + .1*6) = (71 - x) * (1 + .1*9.5)

One variable, one equation, much easier to see and to solve ... go with this one.

BTW this is a really neat question: thinking about the setup is a bit baffling (I was stuck for a sec on how they could receive the same amount of money when they each turn 18!), but the equation ends up being so clean! Kudos to the author.