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A person invested a certain sum for five years at the rate of r% per annum simple interest. If he had invested the same sum at the same rate compounded half yearly for one year, he would have received the same amount. Find the value of r.?
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A person invested a certain sum for five years at the rate of r% per a...
**Problem Analysis:**
Let's assume the principal amount invested is P and the rate of interest is r%. The interest for the simple interest (SI) investment will be P * r * 5/100, as it is invested for 5 years.
For the compound interest (CI) investment, the interest will be calculated half-yearly. So, the rate of interest for each half-year will be r/2%. The time period will be 1 year, so there will be 2 compounding periods.
**Solution:**
Let's calculate the amount obtained from the simple interest investment after 5 years.
Simple Interest (SI) formula: SI = P * r * t/100
Amount (A) = P + SI = P + (P * r * 5/100) = P(1 + 5r/100)
Now, let's calculate the amount obtained from the compound interest investment after 1 year.
Compound Interest (CI) formula: A = P(1 + r/100)^n
where n is the number of compounding periods.
In this case, n = 2 (as it is compounded half-yearly for 1 year)
A = P(1 + r/200)^2
According to the problem, the amount obtained from both investments is the same.
P(1 + 5r/100) = P(1 + r/200)^2
Let's simplify the equation:
1 + 5r/100 = (1 + r/200)^2
Expanding the brackets:
1 + 5r/100 = 1 + 2r/200 + (r/200)^2
Simplifying further:
5r/100 = 2r/200 + (r/200)^2
Multiplying both sides by 200:
10r = 4r + r^2/200
2000r = 800r + r^2
Rearranging the equation:
r^2 + 1200r = 0
r(r + 1200) = 0
So, r = 0 or r = -1200
Since the rate of interest cannot be negative, the value of r is 0.
Therefore, the value of r is 0%.
Let's assume the principal amount invested is P and the rate of interest is r%. The interest for the simple interest (SI) investment will be P * r * 5/100, as it is invested for 5 years.
For the compound interest (CI) investment, the interest will be calculated half-yearly. So, the rate of interest for each half-year will be r/2%. The time period will be 1 year, so there will be 2 compounding periods.
**Solution:**
Let's calculate the amount obtained from the simple interest investment after 5 years.
Simple Interest (SI) formula: SI = P * r * t/100
Amount (A) = P + SI = P + (P * r * 5/100) = P(1 + 5r/100)
Now, let's calculate the amount obtained from the compound interest investment after 1 year.
Compound Interest (CI) formula: A = P(1 + r/100)^n
where n is the number of compounding periods.
In this case, n = 2 (as it is compounded half-yearly for 1 year)
A = P(1 + r/200)^2
According to the problem, the amount obtained from both investments is the same.
P(1 + 5r/100) = P(1 + r/200)^2
Let's simplify the equation:
1 + 5r/100 = (1 + r/200)^2
Expanding the brackets:
1 + 5r/100 = 1 + 2r/200 + (r/200)^2
Simplifying further:
5r/100 = 2r/200 + (r/200)^2
Multiplying both sides by 200:
10r = 4r + r^2/200
2000r = 800r + r^2
Rearranging the equation:
r^2 + 1200r = 0
r(r + 1200) = 0
So, r = 0 or r = -1200
Since the rate of interest cannot be negative, the value of r is 0.
Therefore, the value of r is 0%.
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A person invested a certain sum for five years at the rate of r% per annum simple interest. If he had invested the same sum at the same rate compounded half yearly for one year, he would have received the same amount. Find the value of r.?
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