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Difficulty: Very hardAlgebraic Word Problems and Modeling

A total initial capital of $100,000\$100,000 is split between Fund X and Fund Y. Fund X yields an annual simple interest rate of r%r\%, while Fund Y yields an annual simple interest rate of (r+2)%(r + 2)\%. Under the original capital allocation, the ratio of the annual interest earned from Fund X to the annual interest earned from Fund Y is 15:1415 : 14. If the initial allocation to Fund X had been increased by 25%25\% by transferring funds directly from Fund Y, the total annual interest earned from both funds combined would have been $5,500\$5,500. What was the original amount allocated to Fund X?

  1. A
    $40,000\$40,000
  2. B
    $50,000\$50,000
  3. $60,000\$60,000Answer
  4. D
    $64,000\$64,000
  5. E
    $75,000\$75,000

Answer

The original amount allocated to Fund X was $60,000\$60,000.
The correct option is $60,000\$60,000. Substituting X=60,000X = 60,000 gives Fund Y allocation Y=40,000Y = 40,000. From the modified interest condition, 100,000r−2.5(60,000)=350,000100,000r - 2.5(60,000) = 350,000, giving r=5%r = 5\%. Fund X yields 60,000×0.05=$3,00060,000 \times 0.05 = \$3,000 and Fund Y yields 40,000×0.07=$2,80040,000 \times 0.07 = \$2,800, which satisfies the ratio 3,000:2,800=15:143,000 : 2,800 = 15 : 14. Under the modified allocation (75,00075,000 in X and 25,00025,000 in Y), interest is 75,000(0.05)+25,000(0.07)=3,750+1,750=$5,50075,000(0.05) + 25,000(0.07) = 3,750 + 1,750 = \$5,500.

Step-by-Step Solution

1
Formulate variables and express initial conditions.
Let XX be the initial capital in Fund X and Y=100,000−XY = 100,000 - X be the initial capital in Fund Y. The interest rates are r100\frac{r}{100} and r+2100\frac{r+2}{100} respectively.
Establish a single-variable representation for the fund allocations.
2
Set up the interest ratio equation.
\frac{X \cdot r}{(100,000 - X)(r+2)} = \frac{15}{14} \implies 14 X r = 15(100,000 - X)(r+2).$
Relate the original interest outputs according to the 15:1415:14 ratio.
3
Model the modified allocation scenario.
Fund X becomes 1.25X1.25X and Fund Y becomes 100,000−1.25X100,000 - 1.25X. Total interest equation: (1.25X)(r100)+(100,000−1.25X)(r+2100)=5,500.(1.25X)\left(\frac{r}{100}\right) + (100,000 - 1.25X)\left(\frac{r+2}{100}\right) = 5,500.
Express total combined interest under the hypothetical 25%25\% transfer.
4
Simplify the total interest equation to express rr in terms of XX.
1.25Xr + 100,000r + 200,000 - 1.25Xr - 2.5X = 550,000 \implies 100,000r - 2.5X = 350,000 \implies r = 3.5 + 0.000025X.$
Eliminate the XrXr product term to solve for rr linearly.
5
Substitute rr into the ratio equation and solve for XX.
Expanding 29Xr+30X=1,500,000r+3,000,00029Xr + 30X = 1,500,000r + 3,000,000 with r=3.5+0.000025Xr = 3.5 + 0.000025X yields 0.000725X2+94X−8,250,000=00.000725X^2 + 94X - 8,250,000 = 0, which factors to give X=60,000X = 60,000.
Determine the exact value for the initial allocation to Fund X.

Key Concept

Algebraic modeling of multi-variable financial rate and allocation systems.
Estimated Time:3m 0s
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Topics

Algebraic Word Problems and Modeling

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