Arithmetic

Simple & Compound Interest

Medium

Bank $A$ offers $6 \%$ interest rate per annum compounded half yearly. Bank $B$ and Bank $C$ offer simple interest but the annual interest rate offered by Bank $C$ is twice that of Bank $B$ . Raju invests a certain amount in Bank B for a certain period and Rupa invests $₹ 10,000$ in Bank $C$ for twice that period. The interest that would accrue to Raju during that period is equal to the interest that would have accrued had he invested the same amount in Bank $A$ for one year. The interest accrued, in INR, to Rupa is

$1436$

$3436$

$2346$

$2436$

✅ Correct Option: 4

Given Information:

Bank A: 6% per annum compounded half yearly

Bank B: Simple interest at rate R%

Bank C: Simple interest at rate 2R% (twice Bank B's rate)

Investment Details:

Raju: Invests amount P in Bank B for time T

Rupa: Invests ₹10000 in Bank C for time 2T

Key Condition: Raju's interest from Bank B equals the interest he would earn from Bank A in 1 year.

Since Bank A compounds half yearly, we apply the compound interest formula:

Amount = Principal × $(1 + \frac{\text{Rate}}{100})^n$

Here:

Principal = P

Rate per half year = $\frac{6}{2}$ = 3%

Number of half years in 1 year = 2

Amount after 1 year = $P × (1 + \frac{3}{100})^2$

= $P × (1.03)^2$

= $P × 1.0609$

Interest from Bank A = $P × 1.0609 - P$

= $P × 0.0609$

Raju's interest from Bank B = $\frac{P × R × T}{100}$

We know that Raju's interest from Bank B equals his potential interest from Bank A:

$\frac{P × R × T}{100} = P × 0.0609$

Dividing both sides by P:

$\frac{R × T}{100} = 0.0609$

Therefore: $R × T = 6.09$

Key Insight: This relationship $(R × T = 6.09)$ is crucial for solving the rest of the problem.

Rupa's Investment Details:

Principal = ₹10000

Rate = 2R% (twice Bank B's rate)

Time = 2T (twice Raju's period)

Rupa's Interest = $\frac{10000 × 2R × 2T}{100}$

= $\frac{10000 × 4RT}{100}$

= $\frac{10000 × 4 × 6.09}{100}$ [Since $RT = 6.09$ ]

= $\frac{10000 × 24.36}{100}$

= ₹2436

Rupa's interest = ₹2436

Method Summary: We recognized that the compound interest calculation from Bank A gives us the relationship $RT = 6.09$ , which we then used to find Rupa's simple interest with the doubled rate and doubled time period.

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