Bunuel wrote:
Saul invests an amount of dollars in investment A at i% simple annual interest, another amount in investment B at j% simple annual interest, and a third amount in investment C at k% simple annual interest. If the percent interest rates i, j, and k are in a ratio of 3:2:4, what is the ratio of the average rate of interest on all three investments (taken together) to the interest rate on investment A?
(1) The amounts Saul invested in investments A, B, and C are in a ratio of 1:4:5.
(2) Saul’s three investments total $100,000, while the total amount of money he earns in interest in the first year is $18,600
Kudos for a correct solution.
MANHATTAN GMAT OFFICIAL SOLUTION:For 3-way ratios (such as 3:2:4), use the unknown multiplier to reduce the number of variables. If x is an unknown multiplier for the interest rates, then i = 3x, j = 2x, and k = 4x. The question asks for the ratio of the average rate of interest on all 3 investments (taken together) to the interest rate on investment A, which is i or 3x. (We can ignore the fact that technically, we should divide these variables by 100: i% = i/100, but this adds complexity that doesn’t matter in the end, because all the interest rates are expressed the same way. Let’s just pretend that all the interest rates are expressed as decimals, so we don’t have to divide by 100 everywhere.)
First of all, we should note that the “average rate of interest” is not the simple average of 3x, 2x, and 4x, which would equal 3x. (For one thing, that would mean that the ratio is 1 before we even look at the statements, and Data Sufficiency questions can never be answered simply from the givens in the question stem.) Rather, we must figure out the interest in dollars across all 3 investments, add up that interest, and then divide by the total investment to get the effective interest rate – which will be a weighted average of the 3 interest rates, weighted by the amounts invested in each investment.
We could name the invested amounts a, b, and c. Then the dollar interest on investment A is the invested amount a times the interest rate 3x, or 3xa. Likewise, the dollar interest on investment B is 2xb, and the dollar interest on investment C is 4xc. The total dollar interest is 3xa + 2xb + 4xc, and then we’d divide by a + b + c to get the weighted average interest rate. The ratio of this result to 3x is the desired quantity. (Let’s not write it out yet!)
Statement 1: SUFFICIENT. We have another 3-way ratio:
a:b:c = 1:4:5
So let’s use another unknown multiplier (say, y).
a = y
b = 4y
c = 5y
Now figure the total dollar interest: 3xy + 2x(4y) + 4x(5y) = 31xy. The total invested amount is y + 4y + 5y = 10y, so the weighted average interest rate is 31xy / (10y) = 3.1x. The ratio of this average interest rate to the interest rate on investment A, 3x, is 3.1x / (3x) = 3.1/3 = a definite number. Both unknown multipliers cancel away. This statement is sufficient.
Statement 2: NOT SUFFICIENT.
Remember to go back to a, b, and c for the amounts invested. We know that the total amount invested is $100,000, so we can eliminate one of these variables:
c = 100,000 – (a + b)
We also have that the total interest in dollars is $18,600, so we know that the overall weighted average interest rate is 18.6%. The question then becomes, is the value of x fixed under these conditions? If so, then the ratio we want is fixed.
The total dollar interest is 3xa + 2xb + 4xc = 3xa + 2xb + 4x(100,000 – (a + b)) = 18,600.
We can factor out an x, but the a’s and b’s will not fully cancel away, so the value of x is not fixed. Thus the desired ratio can take on different values.
The correct answer is A.Thanks for clarifying an important point that the numerator in the ratio isn't the average of the three interest rates.
My query is how do we know that we have been asked a weight based average & not an arithmetic mean. I know one clue was that the question stem itself had the solution, but isn't weight based average thing must be clearly mentioned.