Question

Veeru gave Rupees 2400 on loan. Some amount he gave at 4% per annum simple interest and remaining at 5% per annum simple interest. After two years he got Rupees 220 as interest. Then the amount given at 4% and 5% per annum simple interest are, respectively?

• A. Rupees 1000, Rupees 1400
• B. Rupees 800, Rupees 1600
• C. Rupees 1800, Rupees 600
• D. Rupees 2200, Rupees 200

Hint:

For such problems, assign one variable and use SI = \( \frac{PRT}{100}\). Remember total principal = sum of parts, and total interest = sum of interests.

Answer 1

The Correct Option is A

Step 1: Let the amounts be variables.
Suppose \(\,x\) at 4% and \((2400-x)\) at 5%.

Step 2: Write simple interest equations.

Simple interest formula: \(SI = \dfrac{P \cdot R \cdot T}{100}\). For 2 years: Interest on \(x\): \( \dfrac{x \cdot 4 \cdot 2}{100} = \dfrac{8x}{100} = \dfrac{2x}{25}\). 

Interest on \((2400-x)\): \( \dfrac{(2400-x)\cdot 5\cdot 2}{100} = \dfrac{10(2400-x)}{100} = \dfrac{2400-x}{10}\). 

Step 3: Form the equation.
\[ \frac{2x}{25} + \frac{2400-x}{10} = 220 \] Multiply through by 50: \[ 4x + 5(2400-x) = 11000 \] \[ 4x + 12000 - 5x = 11000 \] \[ - x + 12000 = 11000 \quad \Rightarrow \quad x = 1000 \] 

So, at 4%: Rupees 1000; at 5%: \(2400-1000=1400\). \[ \boxed{1000,\ 1400} \]

Answer 2

The alligation (weighted average) method offers a quick alternative to setting up and solving an equation for the split amount.

1. Rupees 1000 at 4%, Rupees 1400 at 5%: Checking the total interest: \(1000\times4\%\times2+1400\times5\%\times2=80+140=220\), which exactly matches the given interest.
2. Rupees 800 at 4%, Rupees 1600 at 5%: Here the interest is \(800\times0.04\times2+1600\times0.05\times2=64+160=224\), which is more than the given 220, so this split is incorrect.
3. Rupees 1800 at 4%, Rupees 600 at 5%: Here the interest is \(1800\times0.04\times2+600\times0.05\times2=144+60=204\), less than 220, so this is also incorrect.
4. Rupees 2200 at 4%, Rupees 200 at 5%: Here the interest is \(2200\times0.04\times2+200\times0.05\times2=176+20=196\), well short of 220, so this fails too.

The required average annual rate of interest is \(\dfrac{220}{2400\times2}\times100\approx4.5833\%\). By alligation, the amounts at 4% and 5% must be in the ratio \((5-4.5833):(4.5833-4)=0.4167:0.5833=5:7\), so out of Rupees 2400 split into 12 equal parts of Rupees 200 each, the amount at 4% is \(5\times200=1000\) and at 5% is \(7\times200=1400\).

So the correct answer is Rupees 1000, Rupees 1400.