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class 8 ganita prakash chapter 1
📖 Textbook Theory: Decimal Multipliers
When a quantity increases by a certain percentage, you can find the final value in one step using a decimal multiplier.
The Golden Rule: An increase of $X\%$ means the new total is $(100 + X)\%$ of the original base. To turn a percentage into a decimal multiplier, simply divide by 100.
8. The population of elephants in a national park increased by 5% in the last decade. If the population of the elephants last decade is $p$, the population now is:
1
An "increase" means we are adding to the original 100% amount.
2
To find the final value using a multiplier, add the percentage increase to 100%, and convert the total percentage into a decimal.
1
Original population = $100\%$.
2
New population = $100\% + 5\% = 105\%$.
3
How do you write 105% as a decimal multiplier? (Divide by 100).
!
The Decimal Placement Trap:
Option (ii) $p \times 0.05$ represents ONLY the newly born elephants (the 5% increase), not the total population.
Option (i) $p \times 0.5$ means 50%, not 5%.
Option (ii) $p \times 0.05$ represents ONLY the newly born elephants (the 5% increase), not the total population.
Option (i) $p \times 0.5$ means 50%, not 5%.
Algebraic Translation
Let original population = $p$.
New Population = Original + Increase
$= 100\% \text{ of } p + 5\% \text{ of } p$
$= 105\% \text{ of } p$
Convert 105% to a decimal: $\frac{105}{100} = 1.05$.
Therefore, New Population = $1.05 \times p$.
Ans: Option (iv) $p \times 1.05$
class 8 ganita prakash chapter 1
📖 Textbook Theory: Translating Decreases
The English language uses specific phrasing for percentages that must be translated carefully into math.
"Fallen By" vs "Fallen To": If demand has "fallen by" 85%, it means 85% of the original demand was wiped out. What remains is the difference: $100\% - 85\%$.
9. Which of the following statement(s) mean the same as "The demand for cameras has fallen by 85% in the last decade"?
1
"Fallen by" describes the portion of the 100% base that was removed or destroyed.
2
To find what the demand is *now*, you must subtract the fallen portion from the original 100% baseline.
1
Original demand a decade ago = 100%.
2
It fell by 85%. Calculate: $100\% - 85\%$.
3
Read the options carefully to find the one that says the current demand is that remaining percentage of the old demand.
!
The "By vs To" Trap: Option (i) says "The demand now is 85%...". This would only be true if the text said demand "fallen TO 85%". Because it fell BY 85%, that huge chunk is gone forever.
Logical Translation
Let original demand a decade ago = 100%.
The demand fell by 85%.
This means the amount of decrease is 85% of the original.
Current Demand = $100\% - 85\% = 15\%$.
Therefore, the demand now is exactly 15% of the demand a decade ago.
Ans: Option (iii)
A smartphone battery degrades over time. If the battery capacity has "fallen by 8%", which mathematical equation accurately represents its current capacity ($C$) relative to its original factory capacity ($F$)?
1
This mirrors the Camera Demand logic perfectly. Transform the "fallen by" percentage into the "remaining" percentage.
1
Original capacity $F$ is 100%.
2
Subtract 8% from 100%. Convert that remaining percentage into a decimal multiplier for $F$.
!
Do not write $C = 0.08 \times F$. That would mean the battery only holds 8% of its original charge!
Algebraic Modeling
Original Capacity $F$ = 100%.
Amount lost = 8%.
Remaining Capacity $C$ = $100\% - 8\% = 92\%$.
To write 92% of $F$ algebraically, convert 92% to a decimal ($0.92$).
Ans: $C = 0.92 \times F$
In Question 8, why is Option (i) "$p \times 0.5$" an incorrect multiplier for a 5% increase?
A. Because 0.5 calculates a 5% decrease, not an increase.
B. Because 0.5 represents 50%, not 5%. Furthermore, it fails to include the original 100% base population, effectively cutting the population in half instead of increasing it.
C. Because 5% cannot be written as a decimal.
D. It is not incorrect; $0.5$ is the standard way to write 5%.
1
A decimal multiplier has two parts: the whole number (which retains the original base) and the decimal (which represents the change).
1
$5\% = \frac{5}{100} = 0.05$.
!
Confusing 0.5 (which is 50%) with 0.05 (which is 5%) is one of the most common math errors in the world.
Logical Explanation
To increase by 5%, the final multiplier must represent 105%.
$\frac{105}{100} = 1.05$.
Option (i) uses 0.5, which is $\frac{50}{100}$ (50%). Because it lacks the "1" in front, it destroys the original population and only leaves half of it behind!
Ans: Option (B)
In Question 9, why is Option (iv) "The demand a decade ago was 15% of the demand now" mathematically false, even though it contains the correct number "15"?
A. Because the number should be 85.
B. Because the statement is grammatically incorrect.
C. Because it reverses the relationship. If the old demand was 15% of the new demand, it would mean demand violently increased over the decade, rather than falling.
D. Because 15% cannot be calculated on current demand.
1
The order of subjects in a percentage statement matters completely.
Relational Logic
The correct truth is: $\text{Current Demand} = 0.15 \times \text{Old Demand}$.
Option (iv) states: $\text{Old Demand} = 0.15 \times \text{Current Demand}$.
If Old = 0.15 of Current, that means the Current demand is nearly 7 times larger than the old demand! That describes massive growth, not an 85% fall.
Ans: Option (C)
