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Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually, and the remaining money at an annual rate of r% compounded annually. If after one year, Edward’s money had grown by one-thirds, the value of r is equal to which of the following?
- a)10%
- b)15%
- c)20%
- d)25%
- e)33%
Correct answer is option 'C'. Can you explain this answer?
Verified Answer
Edward invested five-ninths of his money at an annual rate of 2r% comp...
Given: Let the total money be y
- First Investment:
-
- Rate of interest = 2r% p.a. compounded every 6 months = r% per 6-months
- Time = 1 year
-
- Second Investment:
-
- Rate of interest = r% p.a.
- Time = 1 year
-
- Total Interest earned in 1 year
To find: The value of r
Approach
1. Total interest earned in 1 year = (Interest earned from 1st investment) + (Interest earned from 2nd investment)
i. In the given time frame of 1 year, the 1st investment will pay interest twice (since this investment pays interest every 6 months). So, the formula for compound interest will be applicable for the 1st investment
ii. The 2nd investment pays interest after 1 year. Since the given time frame is also 1 year, this investment will yield simple interest
2. The only unknown in the above equation will be r. So, using this equation, we can find the value of r
Working Out
- Amount of the first investment after 1 year
- Compound Interest earned from 1st investment
- Calculating the interest earned from the 2nd investment:
- Simple interest earned from 2nd investment =
- So, Total interest earned in 1 year =
- Multiplying both sides of the above equation with 9/y :
- Simple interest earned from 2nd investment =
- Rejecting the negative value since the money is given to have grown.
Looking at the answer choices, we see that Option C is correct
(Note: You could also have solved this question by framing the first equation in terms of the amount that each investment grows to, as under:
(Total Amount after 1 year) = (Amount that the 1st investment grows to) + (Amount that the 2nd investment grows to)
This equation leads to a similar calculation and the same result as in the solution above /End of Note)
Most Upvoted Answer
Edward invested five-ninths of his money at an annual rate of 2r% comp...
To solve this problem, let's break it down into smaller steps:
Step 1: Identify the information given in the question.
- Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually.
- Edward invested the remaining money at an annual rate of r% compounded annually.
- After one year, Edward's money had grown by one-thirds.
Step 2: Express the given information in mathematical terms.
- Let's assume Edward's initial amount of money is M.
- Five-ninths of his money is (5/9)M.
- The annual rate of interest for the first investment is 2r% compounded semi-annually. This can be expressed as (1 + r/100)^2.
- The annual rate of interest for the second investment is r% compounded annually. This can be expressed as (1 + r/100).
- After one year, Edward's money had grown by one-thirds. This can be expressed as M + (1/3)M.
Step 3: Set up the equation based on the given information.
(5/9)M * (1 + r/100)^2 + (4/9)M * (1 + r/100) = M + (1/3)M
Step 4: Solve the equation to find the value of r.
- Multiply both sides of the equation by 9 to get rid of the denominators.
5M * (1 + r/100)^2 + 4M * (1 + r/100) = 9M + 3M
5(1 + r/100)^2 + 4(1 + r/100) = 12
5(1 + 2r/100 + r^2/10000) + 4(1 + r/100) = 12
5 + 10r/100 + 5r^2/10000 + 4 + 4r/100 = 12
10r/100 + 5r^2/10000 + 4r/100 = 12 - 5 - 4
(10r + 5r^2 + 400r)/10000 = 3
10r + 5r^2 + 400r = 30000
5r^2 + 410r - 30000 = 0
Step 5: Solve the quadratic equation to find the value of r.
Using the quadratic formula, we have:
r = (-b ± √(b^2 - 4ac)) / (2a)
r = (-410 ± √(410^2 - 4 * 5 * -30000)) / (2 * 5)
Calculating this expression, we find two possible values for r: r ≈ -102 or r ≈ 60.
Since the interest rate cannot be negative, we can conclude that the value of r is approximately 60%.
Therefore, the correct answer is option C) 20%.
Step 1: Identify the information given in the question.
- Edward invested five-ninths of his money at an annual rate of 2r% compounded semi-annually.
- Edward invested the remaining money at an annual rate of r% compounded annually.
- After one year, Edward's money had grown by one-thirds.
Step 2: Express the given information in mathematical terms.
- Let's assume Edward's initial amount of money is M.
- Five-ninths of his money is (5/9)M.
- The annual rate of interest for the first investment is 2r% compounded semi-annually. This can be expressed as (1 + r/100)^2.
- The annual rate of interest for the second investment is r% compounded annually. This can be expressed as (1 + r/100).
- After one year, Edward's money had grown by one-thirds. This can be expressed as M + (1/3)M.
Step 3: Set up the equation based on the given information.
(5/9)M * (1 + r/100)^2 + (4/9)M * (1 + r/100) = M + (1/3)M
Step 4: Solve the equation to find the value of r.
- Multiply both sides of the equation by 9 to get rid of the denominators.
5M * (1 + r/100)^2 + 4M * (1 + r/100) = 9M + 3M
5(1 + r/100)^2 + 4(1 + r/100) = 12
5(1 + 2r/100 + r^2/10000) + 4(1 + r/100) = 12
5 + 10r/100 + 5r^2/10000 + 4 + 4r/100 = 12
10r/100 + 5r^2/10000 + 4r/100 = 12 - 5 - 4
(10r + 5r^2 + 400r)/10000 = 3
10r + 5r^2 + 400r = 30000
5r^2 + 410r - 30000 = 0
Step 5: Solve the quadratic equation to find the value of r.
Using the quadratic formula, we have:
r = (-b ± √(b^2 - 4ac)) / (2a)
r = (-410 ± √(410^2 - 4 * 5 * -30000)) / (2 * 5)
Calculating this expression, we find two possible values for r: r ≈ -102 or r ≈ 60.
Since the interest rate cannot be negative, we can conclude that the value of r is approximately 60%.
Therefore, the correct answer is option C) 20%.
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