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class 8 ganita prakash chapter 1
📖 Textbook Theory: Shrinking Base
When a population decreases by a percentage over multiple cycles, you cannot simply multiply the percentage by the time. The 100% base shrinks after every cycle, so the amount lost in the second cycle will be smaller than the amount lost in the first cycle.
Example 11: The population of a village was observed to be reducing by about 10% every decade. If the current population is 1250, what is the expected population after 3 decades?
1
Because the value is decreasing successively over multiple cycles, we use the negative compounding formula: $A = P \left( 1 - \frac{R}{100} \right)^n$.
1
$P = 1250$, Rate ($R$) = $10$, Time ($n$) = $3$.
2
The multiplier inside the bracket is $\left(1 - 0.10\right) = 0.90$.
3
Calculate $1250 \times (0.90)^3$.
!
The Flat 30% Mistake: Do NOT say "10% per decade for 3 decades is a flat 30% drop. 30% of 1250 is 375. So the answer is $1250 - 375 = 875$." That is Simple Interest math, and it ignores the fact that the 10% slice gets smaller as the population shrinks! The true answer will be slightly higher than 875.
Depreciation Formula Calculation
Given: $P = 1250$, Decline Rate = $10\%$ per cycle, Time ($n$) = 3 cycles.
Formula: $A = P \left( 1 - \frac{R}{100} \right)^n$
$A = 1250 \left( 1 - \frac{10}{100} \right)^3$
$A = 1250 \times (0.90)^3$
$A = 1250 \times (0.9 \times 0.9 \times 0.9)$
$A = 1250 \times 0.729$
$A = 911.25$
Since we can't have a fraction of a person, we round to the nearest whole number.
Ans: The expected population is around 911.
class 8 ganita prakash chapter 1
📖 Textbook Theory: The Illusion of High Percentages
Percentages are relative. A massive percentage increase means nothing if the starting base is tiny. Conversely, a small percentage increase on a huge base can yield a massive absolute gain.
The Golden Rule: Never compare two percentages without checking their 100% base. If you want absolute wealth, you must calculate the actual monetary gain, not just the percentage rate.
Tricky Percentages: You have won a contest. The organisers offer you two options to choose from:
Option A: You deposit ₹100 and you get back ₹300.
Option B: You deposit ₹1000 and you get back ₹1500.
What is the percentage gain each option gives? You can choose any option only once. Which option would you choose? Why?
Option A: You deposit ₹100 and you get back ₹300.
Option B: You deposit ₹1000 and you get back ₹1500.
What is the percentage gain each option gives? You can choose any option only once. Which option would you choose? Why?
1
Calculate the absolute Gain: $\text{Gain} = \text{Amount Returned} - \text{Deposit}$.
2
Calculate Percentage Gain: $\frac{\text{Gain}}{\text{Deposit}} \times 100$.
3
Look closely at the actual monetary gain. Which one gives you more physical money?
1
Option A Gain = $300 - 100 = 200$. $\frac{200}{100} \times 100$.
2
Option B Gain = $1500 - 1000 = 500$. $\frac{500}{1000} \times 100$.
!
Being blinded by percentages: Many people will blindly choose Option A because "200% sounds incredible!" But if you can only choose once, you are walking away with ₹200 profit in your pocket, while the person who chose the "boring" 50% option walks away with ₹500 profit.
Evaluating Option A
Deposit = ₹100. Returned = ₹300.
Absolute Gain = $300 - 100 =$ ₹200.
Percentage Gain = $\left( \frac{200}{100} \right) \times 100 = 200\%$.
Evaluating Option B
Deposit = ₹1000. Returned = ₹1500.
Absolute Gain = $1500 - 1000 =$ ₹500.
Percentage Gain = $\left( \frac{500}{1000} \right) \times 100 = 50\%$.
The Decision
Ans: I would choose Option B.
Even though Option A has a drastically higher percentage gain (200%), the actual amount of profit in Option B (₹500) is much larger than Option A (₹200). Since you can only do the transaction once, you want to maximize absolute profit, not percentage.
A smartphone was bought for ₹50,000. It loses 20% of its value in the first year. In the second year, it loses an additional 20% of its current value.
Calculate the exact value of the smartphone after 2 years.
Calculate the exact value of the smartphone after 2 years.
1
This mirrors the Village Population question. It is successive depreciation.
1
$P = 50000$, Rate = 20, $n = 2$.
2
The multiplier inside the bracket is $\left(1 - 0.20\right) = 0.80$.
3
Calculate $50000 \times (0.80)^2$.
!
Do not say "20% plus 20% is a 40% loss. $50,000 - 20,000 = 30,000$." The 20% in the second year is taken off a much lower base!
Step-by-Step Calculation
$A = P \left( 1 - \frac{R}{100} \right)^n$
$A = 50000 \left( 1 - \frac{20}{100} \right)^2$
$A = 50000 \times (0.80)^2$
$A = 50000 \times 0.64$
$A = 32,000$
Ans: The value of the phone is ₹32,000.
In Example 11, the population of 1250 declined by 10% for 3 successive decades. Why does this NOT result in a 30% total decline from the original 1250?
A. Because 10% is a very small number.
B. Because the base population shrinks every decade. The 10% drop in Decade 2 is calculated on 1125 people, not the original 1250. This results in smaller and smaller drops.
C. Because 10 times 3 is not 30.
D. It does result in exactly a 30% drop.
1
A flat 30% drop would mean multiplying 1250 by 0.70.
2
Successive 10% drops mean multiplying 1250 by 0.9, then 0.9, then 0.9.
1
Look at the animated staircase. Is the physical amount lost the same on every step?
!
Option D is the most common mathematical error in calculating depreciation.
Logical Explanation
A flat 30% drop is Simple Interest math. It assumes the 100% base never changes.
But in reality, a 10% decline is "negative compounding". After one decade, the village only has 1125 people.
In the second decade, you lose 10% of 1125 (which is 112.5 people). You do NOT lose 10% of 1250 again.
Because the base shrinks, the amount lost shrinks, meaning the total loss will be less than a flat 30% hit.
Ans: Option (B)
What is the core mathematical lesson from the "Would You Rather" question (Option A vs Option B)?
A. You should always pick the option with the highest percentage.
B. Percentage is the only true measure of wealth.
C. A percentage is just a ratio. Without knowing the size of the base (the Principal), a percentage cannot tell you the absolute monetary value of a transaction.
D. Depositing ₹100 is always safer than ₹1000.
1
A 1000% gain on ₹1 is just ₹10. A 1% gain on ₹1,000,000 is ₹10,000.
The Illusion of Ratios
A percentage only tells you the efficiency of the growth, not the magnitude of the growth.
Option A was highly efficient (200% growth), but the base was so tiny that it only produced ₹200.
Option B was much less efficient (50% growth), but it acted on a base 10 times larger, producing a much larger absolute wealth (₹500).
Ans: Option (C)
