UP Board Class 10 Mathematics: Compound Interest Problems Step by Step

Ever wondered why a small amount in a bank can grow faster than you expect? That’s the magic of compound interest, and it’s a favourite topic in UP Board exams.

💡 In Simple Words: Compound interest means you earn interest not only on the original money you saved, but also on the interest that money has already earned. It’s like a snowball rolling down a hill – it gets bigger and bigger as it rolls.

What is Compound Interest?

Compound interest (often shortened to CI) is the interest calculated on the principal (the original sum of money) plus any interest that has already been added. Think of it as a garden: the first seed (principal) grows a plant, and each new leaf (interest) helps the next leaf grow bigger.

Formula explained

The standard formula is:

A = P \(1 + \frac{r}{n}\)^{n t}

A – the amount you’ll have after the time period.
P – the principal, the money you start with.
r – the annual interest rate written as a decimal (so 7% becomes 0.07).
n – how many times a year the interest is added (compounded). If it’s yearly, n = 1; half‑yearly, n = 2; quarterly, n = 4; monthly, n = 12.
t – the number of years the money stays in the account.

Notice how the exponent n t tells you the total number of times interest gets added.

Step‑by‑step method to solve a CI problem

Follow these five easy steps, and you’ll never get stuck.

Read the question carefully and write down P, r, n, and t.
Convert the percentage rate to a decimal.
Plug the numbers into the formula.
Do the calculation inside the brackets first, then raise to the power, and finally multiply by P.
Round the answer as the question asks (usually to the nearest rupee).

graph TD A[Read the question] --> B[Identify P, r, n, t] B --> C[Convert r to decimal] C --> D["Substitute into A = P(1+r/n)^(n*t)"] D --> E[Calculate step by step] E --> F[Write the final amount]

Worked Example 1 – Yearly compounding

Question: Riya deposits ₹5,000 in a bank that pays 8% interest per year, compounded annually. How much will she have after 3 years?

Solution:

P = 5,000, r = 8% = 0.08, n = 1 (yearly), t = 3.
Put them in the formula: A = 5,000 \(1 + \frac{0.08}{1}\)^{1×3} = 5,000 \(1.08\)^3.
Calculate 1.08³ = 1.259712.
Multiply: 5,000 × 1.259712 = 6,298.56.
Rounded to the nearest rupee, Riya will have ₹6,299.

Worked Example 2 – Quarterly compounding

Question: A sum of ₹10,000 is invested at 6% per annum, compounded quarterly. Find the amount after 2 years.

Solution:

P = 10,000, r = 0.06, n = 4 (quarterly), t = 2.
Formula: A = 10,000 \(1 + \frac{0.06}{4}\)^{4×2} = 10,000 \(1 + 0.015\)^8 = 10,000 \(1.015\)^8.
1.015⁸ ≈ 1.126162.
Amount = 10,000 × 1.126162 = 11,261.62.
Rounded, the amount is ₹11,262.

Quick Comparison: Simple vs Compound Interest

Feature	Simple Interest (SI)	Compound Interest (CI)
What earns interest?	Only the principal.	Principal + previously earned interest.
Formula	SI = P r t	CI = P\(1+\frac{r}{n}\)^{n t}
Growth pattern	Linear (straight line).	Exponential (curve that gets steeper).
Typical use in exams	Basic word problems.	Problems involving yearly, half‑yearly, quarterly, or monthly compounding.

Common Mistakes to Avoid

Forgetting to change the percentage into a decimal.
Using the wrong value for n (e.g., treating quarterly as yearly).
Skipping the step of calculating the bracket before raising to the power.
Not rounding as per the question – sometimes they ask for the nearest rupee, sometimes for two decimal places.
Mixing up “time in years” with “number of periods”. If the interest is half‑yearly for 3 years, t = 3 but n = 2, so n t = 6.

📝 Likely Exam Questions

Question: A sum of ₹12,000 is invested at 9% per annum, compounded half‑yearly. Find the amount after 5 years.
Answer: n = 2, r = 0.09, t = 5 → A = 12,000(1+0.09/2)^{2×5}=12,000(1.045)^{10}=12,000×1.5513≈₹18,616.
Question: How much interest is earned on ₹8,000 at 7.5% per annum, compounded monthly for 2 years?
Answer: n=12, r=0.075, t=2 → A=8,000(1+0.075/12)^{24}=8,000(1.00625)^{24}=8,000×1.1616≈₹9,293. Total interest = 9,293‑8,000=₹1,293.
Question: If the amount after 3 years of quarterly compounding is ₹15,000 and the rate is 5% per annum, what was the original principal?
Answer: n=4, r=0.05, t=3 → A = P(1+0.05/4)^{12} → 15,000 = P(1.0125)^{12}=P×1.16075 → P≈15,000/1.16075≈₹12,925.
Question: A bank offers 10% per annum compounded annually. How long will it take for ₹2,000 to become ₹4,000?
Answer: Use A = P(1+r)^t → 4,000 = 2,000(1.10)^t → 2 = (1.10)^t → t = log 2 / log 1.10 ≈ 7.27 years (≈7 years 3 months).
Question: Compare the amount after 3 years for ₹5,000 at 6% p.a. simple interest versus compound interest (yearly).
Answer: SI = 5,000×0.06×3 = ₹900 → total = ₹5,900.
CI = 5,000(1.06)^3 = 5,000×1.191016 = ₹5,955.08. CI gives about ₹55 more.

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