Math worksheet with two interest calculation problems.

Let's solve both problems step by step.

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Problem 5:

Given:
- Simple Interest (SI) = $1200
- Rate (R) = 5% per annum
- Time (T) = 3 years

We are to find the Compound Interest (CI) on the same sum for the same period at the same rate.

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#### Step 1: Find the Principal (P)

Use the simple interest formula:

$$
SI = \frac{P \times R \times T}{100}
$$

Substitute the known values:

$$
1200 = \frac{P \times 5 \times 3}{100}
$$

$$
1200 = \frac{15P}{100}
$$

$$
1200 = 0.15P
$$

$$
P = \frac{1200}{0.15} = 8000
$$

So, the principal is $8000.

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#### Step 2: Calculate Compound Interest

Formula for compound interest (compounded annually):

$$
A = P \left(1 + \frac{R}{100}\right)^T
$$

$$
A = 8000 \left(1 + \frac{5}{100}\right)^3 = 8000 \left(1.05\right)^3
$$

Now calculate $ (1.05)^3 $:

$$
1.05^2 = 1.1025
$$
$$
1.05^3 = 1.1025 \times 1.05 = 1.157625
$$

$$
A = 8000 \times 1.157625 = 9261
$$

So, amount after 3 years = $9261

Now, compound interest:

$$
CI = A - P = 9261 - 8000 = 1261
$$

✔ So, Compound Interest = $1261

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Answer: (d) $1261

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Problem 6:

Given:
- Compound Interest (CI) = $510
- Rate (R) = $12\frac{1}{2}\% = 12.5\% = \frac{25}{2}\% $
- Time (T) = 2 years

We are to find the Simple Interest (SI) on the same sum at the same rate and time.

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#### Step 1: Use CI formula to find Principal

For compound interest compounded annually:

$$
CI = P \left(1 + \frac{R}{100}\right)^T - P
$$

$$
510 = P \left(1 + \frac{12.5}{100}\right)^2 - P
$$

$$
510 = P \left(1.125\right)^2 - P
$$

Calculate $ (1.125)^2 $:

$$
1.125^2 = 1.265625
$$

$$
510 = P(1.265625 - 1) = P(0.265625)
$$

$$
P = \frac{510}{0.265625} = 1918.367...
$$

Wait — let’s do it more precisely using fractions.

Convert 12.5% = $ \frac{25}{2} \% $

So,

$$
1 + \frac{R}{100} = 1 + \frac{25}{200} = 1 + \frac{1}{8} = \frac{9}{8}
$$

So,

$$
A = P \left(\frac{9}{8}\right)^2 = P \times \frac{81}{64}
$$

$$
CI = A - P = P \left(\frac{81}{64} - 1\right) = P \left(\frac{17}{64}\right)
$$

Given CI = 510:

$$
\frac{17}{64} P = 510
$$

$$
P = 510 \times \frac{64}{17} = 30 \times 64 = 1920
$$

Because $ 510 \div 17 = 30 $, then $ 30 \times 64 = 1920 $

So, Principal = $1920

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#### Step 2: Find Simple Interest

Use SI formula:

$$
SI = \frac{P \times R \times T}{100} = \frac{1920 \times 12.5 \times 2}{100}
$$

First, $ 12.5 \times 2 = 25 $

$$
SI = \frac{1920 \times 25}{100} = \frac{48000}{100} = 480
$$

✔ So, Simple Interest = $480

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Answer: (d) $480

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✔ Final Answers:

- Problem 5: (d) $1261
- Problem 6: (d) $480

Let me know if you'd like a visual explanation or shortcut methods!