If a sum of money at a certain rate of simple interest per year doubles in 5 years and at a different rate of simple interest per year becomes three times in 12 years, then the difference in the two rates of simple interest per year is
10/3%
Analyzing Simple Interest Rates
This problem involves two scenarios where a sum of money grows under simple interest, but at different rates and over different periods. We need to find the difference between these two simple interest rates.
Understanding Simple Interest
Simple interest is calculated only on the initial principal amount. The formula for Simple Interest (SI) is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
Where:
- P is the Principal amount
- R is the Rate of Interest per year
- T is the Time period in years
The total Amount (A) after T years is the Principal plus the Simple Interest:
\[ A = P + \text{SI} \]
Scenario 1: Money Doubles in 5 Years
In the first scenario, let the principal be P. The amount becomes double the principal, which is 2P, in 5 years. The simple interest earned in this case is the difference between the amount and the principal.
- Principal = P
- Amount = 2P
- Simple Interest (SI1) = Amount - Principal = 2P - P = P
- Time (T1) = 5 years
- Let the rate of interest be R1% per year.
Using the simple interest formula:
\[ \text{SI}_1 = \frac{P \times R_1 \times T_1}{100} \]
Substituting the values:
\[ P = \frac{P \times R_1 \times 5}{100} \]
We can cancel P from both sides (assuming P > 0):
\[ 1 = \frac{R_1 \times 5}{100} \]
Now, solve for R1:
\[ R_1 \times 5 = 100 \]
\[ R_1 = \frac{100}{5} \]
\[ R_1 = 20\% \]
So, the first simple interest rate is 20% per year.
Scenario 2: Money Becomes Three Times in 12 Years
In the second scenario, the principal is P. The amount becomes three times the principal, which is 3P, in 12 years. The simple interest earned here is the difference between this amount and the principal.
- Principal = P
- Amount = 3P
- Simple Interest (SI2) = Amount - Principal = 3P - P = 2P
- Time (T2) = 12 years
- Let the rate of interest be R2% per year.
Using the simple interest formula:
\[ \text{SI}_2 = \frac{P \times R_2 \times T_2}{100} \]
Substituting the values:
\[ 2P = \frac{P \times R_2 \times 12}{100} \]
Cancel P from both sides (assuming P > 0):
\[ 2 = \frac{R_2 \times 12}{100} \]
Now, solve for R2:
\[ R_2 \times 12 = 2 \times 100 \]
\[ R_2 \times 12 = 200 \]
\[ R_2 = \frac{200}{12} \]
Simplify the fraction:
\[ R_2 = \frac{100}{6} \]
\[ R_2 = \frac{50}{3}\% \]
So, the second simple interest rate is 50/3% per year.
Calculating the Difference in Rates
We need to find the difference between the two rates, R1 and R2.
Difference = R1 - R2
\[ \text{Difference} = 20\% - \frac{50}{3}\% \]
To subtract these fractions, we need a common denominator, which is 3. Convert 20% to a fraction with a denominator of 3:
\[ 20 = \frac{20 \times 3}{3} = \frac{60}{3} \]
Now subtract:
\[ \text{Difference} = \frac{60}{3}\% - \frac{50}{3}\% \]
\[ \text{Difference} = \frac{60 - 50}{3}\% \]
\[ \text{Difference} = \frac{10}{3}\% \]
The difference in the two simple interest rates per year is 10/3%.
| Scenario | Principal (P) | Amount (A) | Simple Interest (SI) | Time (T) | Rate (R) |
|---|---|---|---|---|---|
| 1 | P | 2P | P | 5 years | 20% |
| 2 | P | 3P | 2P | 12 years | 50/3 % |
Revision Table: Simple Interest Rates Difference
| Concept | Description | Formula/Calculation |
|---|---|---|
| Simple Interest (SI) | Interest calculated only on the principal amount. | \[ \frac{P \times R \times T}{100} \] |
| Amount (A) | Total money after adding interest to principal. | \[ P + \text{SI} \] |
| Rate R1 Calculation | Determined when principal doubles in 5 years. | \[ P = \frac{P \times R_1 \times 5}{100} \implies R_1 = 20\% \] |
| Rate R2 Calculation | Determined when principal triples in 12 years. | \[ 2P = \frac{P \times R_2 \times 12}{100} \implies R_2 = \frac{50}{3}\% \] |
| Difference in Rates | Subtracting R2 from R1. | \[ 20\% - \frac{50}{3}\% = \frac{60}{3}\% - \frac{50}{3}\% = \frac{10}{3}\% \] |
Additional Information on Simple Interest
Simple interest is one of the most basic concepts in finance. Here are a few points to note:
- The interest earned each year is constant because it is always calculated on the original principal.
- Unlike compound interest, simple interest does not earn interest on previously accumulated interest.
- Simple interest is often used for short-term loans or basic financial calculations.
- The rate of interest is typically given as an annual percentage (per year). It's important to ensure that the time period is also in years when using the formula. If time is given in months, convert it to years by dividing by 12.
- Understanding how money grows at simple interest rates over time is fundamental for various financial calculations and problems.
Similar Questions
A sum of money was invested at simple interest at a certain rate for 5 years. Had it been invested at a 5% higher rate, it would have fetched Rs.500 more. What was the principal amount?
A sum was put at simple interest at certain rate for 2 years. Had it been put at 1% higher rate of interest, it would have fetched Rs. 24 more. What is the sum?
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Important Questions from Simple Interest
How much time will it take for an amount of Rs. 450 to yield Rs. 81 as interest at 4.5% per annum of simple interest ?
Nirav and Mehul borrowed Rs.4000 and Rs.5000 respectively for 2.5 years at the rate of x% per annum. Mehul paid Rs 125 more interest than Nirav. Find x.
If the interest on a sum of Rs.1200 is more than the interest on Rs.1000 by Rs.120 in three years, then what is the rate of interest per annum?.
The difference between the simple interest received from two banks on Rs. 500 for two years is Rs. 2.50. What is the difference between their rates?
A sum of Rs.1200 becomes Rs.1560 at a rate of simple interest in 3 years. In how many years will the sum of Rs.800 amount to Rs.1120 at the same rate of simple interest?
