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class 8 ganita prakash chapter 1
📖 Textbook Theory: Compounding Mechanics
When interest is not compounded (Simple Interest), the interest for every year is identical because it is only calculated on the original starting deposit.
When interest is compounded annually, the interest earned each year is added to the account balance. The next year's interest is calculated on this new, larger balance. This creates a "snowball effect."
1. Bank of Yahapur offers an interest of 10% p.a. Compare how much one gets if they deposit ₹20,000 for a period of 2 years with compounding and without compounding annually.
1
Without Compounding: Calculate 10% of ₹20,000 once. Add that exact same amount for Year 1 and Year 2.
2
With Compounding: Calculate 10% of ₹20,000 for Year 1. Add it to the total. Then calculate 10% of the new total for Year 2.
1
$10\%$ of $20,000 = 2,000$.
2
Without: $20,000 + 2,000 + 2,000$.
3
With: Year 1 ends at $22,000$. What is $10\%$ of $22,000$? Add that to $22,000$.
!
The First Year Trap: A common mistake is thinking compound interest is magically higher from day 1. It isn't! Notice that at the end of Year 1, both accounts have exactly ₹22,000. Compounding only takes effect in Year 2 and beyond, when "interest earns interest".
Without Compounding (Simple)
| Year | Amount in FD | Interest Returned (10%) |
|---|---|---|
| Year 1 | ₹20,000 | $0.10 \times 20000 =$ ₹2,000 |
| Year 2 | ₹20,000 | $0.10 \times 20000 =$ ₹2,000 |
| Total | Principal (20,000) + Total Interest (4,000) = ₹24,000 | |
With Compounding
| Year | Amount in FD (New Base) | Interest Added (10%) |
|---|---|---|
| Year 1 | ₹20,000 | $0.10 \times 20000 =$ ₹2,000 |
| Year 2 | ₹22,000 (20k+2k) | $0.10 \times 22000 =$ ₹2,200 |
| Total | Final Amount in FD = ₹24,200 | |
Ans: Without: ₹24,000. With: ₹24,200. (₹200 more with compounding)
class 8 ganita prakash chapter 1
📖 Textbook Theory: The Power of Time
When you increase the number of years (the duration), compound interest becomes significantly more powerful because the "snowball" rolls down the hill longer, picking up more and more interest on top of the previously accumulated interest.
2. Bank of Wahapur offers an interest of 5% p.a. Compare how much one gets if one deposits ₹20,000 for a period of 4 years with compounding and without compounding annually.
1
Calculate simple interest by finding 5% of ₹20,000 just once, and adding it 4 times.
2
Calculate compound interest by finding 5% of the new total at the end of every single year, sequentially for 4 years.
1
$5\%$ of $20,000 = 1,000$.
2
Without: $20,000 + (1000 \times 4)$.
3
With: Yr 1 = 21,000. Yr 2 = $21,000 + (5\% \text{ of } 21,000)$. Keep going until Year 4!
!
Decimal Fatigue: In Year 3 and Year 4 of compounding, the numbers will get messy and have decimals. Do not round off the decimals during the middle of your calculation! Only round off at the very end to get an accurate final amount.
Without Compounding (Simple)
| Year | Amount in FD | Interest Returned (5%) |
|---|---|---|
| Year 1-4 | ₹20,000 | $0.05 \times 20000 =$ ₹1,000 per year |
| Total | Principal (20,000) + Total Interest ($1000 \times 4$) = ₹24,000 | |
With Compounding
| Year | Amount in FD (New Base) | Interest Added (5%) |
|---|---|---|
| Year 1 | ₹20,000 | $0.05 \times 20000 =$ ₹1,000 |
| Year 2 | ₹21,000 (20k+1k) | $0.05 \times 21000 =$ ₹1,050 |
| Year 3 | ₹22,050 (21000+1050) | $0.05 \times 22050 =$ ₹1,102.50 |
| Year 4 | ₹23,152.50 (22050+1102.50) | $0.05 \times 23152.50 =$ ₹1,157.625 |
| Total | Final Amount in FD = ₹24,310.125 (≈ ₹24,310.13) | |
Ans: Without: ₹24,000. With: ₹24,310.13. (₹310.13 more with compounding)
Based on Textbook Question 3: Notice that the Simple Interest for Bank of Yahapur (10% for 2 yrs) and Bank of Wahapur (5% for 4 yrs) was exactly the same (₹24,000). But why was the Compound Interest higher for Wahapur (₹24,310) than Yahapur (₹24,200)?
A. Because 5% is an easier number for banks to compound than 10%.
B. Because compounding happens more times (4 cycles vs 2 cycles). More cycles mean more chances to earn "interest on interest", creating a larger snowball effect over time.
C. Because the base deposit for Wahapur must have been larger.
D. It was purely a mathematical coincidence.
1
This explores the secret power of Compound Interest: Frequency and Time.
1
Simple interest is just multiplication ($10 \times 2 = 20$ vs $5 \times 4 = 20$).
2
Compound interest is exponential. It matters how many times the interest gets dumped back into the main bucket to grow again.
!
Option C is false because both problems explicitly stated the deposit was exactly ₹20,000.
Logical Explanation
In simple interest, $10\% \times 2$ years and $5\% \times 4$ years yield the exact same flat 20% total return on the original base.
But compounding is a snowball. Bank of Wahapur updates the base 4 times. Bank of Yahapur only updates it 2 times.
Even with a lower rate, giving the snowball more time/cycles to roll makes it grow much larger in the end.
Ans: Option (B)
Look at the Year 1 rows for both options in Bank of Yahapur. If a person withdraws their money exactly at the end of Year 1, which option gives them more money?
A. Without Compounding (Simple)
B. With Compounding
C. They give the exact same amount.
D. It depends on the month they withdraw.
1
Compounding requires the interest from a previous period to be added to the base for the *next* period.
The First Year Rule
In Year 1, both accounts start with the exact same base of ₹20,000.
Both accounts apply the exact same 10% rate to that base.
Therefore, at the end of Year 1, both accounts have exactly ₹22,000.
The divergence only happens in Year 2, when Simple Interest sticks to the original ₹20,000 base, while Compound Interest upgrades to the new ₹22,000 base.
Ans: Option (C)
