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class 8 ganita prakash chapter 1
📖 Textbook Theory: Reverse Percentage
When a number is increased by a percentage, you are adding that percentage to the original 100% base.
The Algebra: An increase of 20% means the new number is $100\% + 20\% = 120\%$ of the original number. Mathematically, $1.20 \times \text{Original} = \text{New Number}$.
6. A number increased by 20% becomes 90. What is the number?
1
A percentage increase means the new number is greater than 100% of the original number.
2
If the increase is 20%, the new number represents exactly $120\%$ of the unknown original base.
1
Let the unknown number be $x$.
2
Set up the equation: $120\% \text{ of } x = 90$.
3
Convert 120% to a decimal ($1.2$) and solve for $x$ by dividing $90$ by $1.2$.
!
The Backward Fallacy: A common mistake is calculating 20% of the *new* number (90), which is 18, and subtracting it to get 72. But if you increase 72 by 20% ($72 \times 1.2$), you get 86.4, not 90! You must use algebra to reverse the multiplier properly.
Algebraic Solution
Let the original number be $x$.
The number increased by 20% becomes $100\% + 20\% = 120\%$ of $x$.
$1.20 \times x = 90$
$x = \frac{90}{1.20}$
Multiply numerator and denominator by 10 to remove the decimal:
$x = \frac{900}{12}$
$x = 75$
Ans: The original number is 75.
(Proof: 20% of 75 is 15. 75 + 15 = 90. Correct!)
class 8 ganita prakash chapter 1
📖 Textbook Theory: Multiple Transactions
This is a classic "trap" question. When a person sells two different items, you cannot simply average the percentage profit and loss.
The Golden Rule: Percentages are tied to their specific Cost Price (CP) base. Since the buffaloes were sold for the same price, they must have had very different original cost prices. You must find the individual CP for each buffalo first, then compare Total CP to Total SP.
7. A milkman sold two buffaloes for ₹80,000 each. On one of them, he made a profit of 5% and on the other a loss of 10%. Find his overall profit or loss.
1
You must find the Cost Price (CP) for each buffalo individually.
2
For Buffalo 1 (5% profit), the SP is 105% of its CP.
3
For Buffalo 2 (10% loss), the SP is 90% of its CP.
4
Calculate $\text{Total CP}$ and compare it to $\text{Total SP}$ to find the final monetary loss.
1
Buffalo 1: $1.05 \times \text{CP}_1 = 80,000$. (Use fractions: $\frac{105}{100} = \frac{21}{20}$).
2
Buffalo 2: $0.90 \times \text{CP}_2 = 80,000$. (Use fractions: $\frac{90}{100} = \frac{9}{10}$).
3
Add $\text{CP}_1$ and $\text{CP}_2$. The math will be messy! Use a common denominator.
!
The Averaging Trap: Do NOT say "+5% profit and -10% loss = -5% overall loss". Percentages cannot be added or subtracted unless they are calculated on the exact same 100% base. Since Buffalo 2 cost much more than Buffalo 1, its 10% slice is much heavier!
Buffalo 1: Finding Cost Price
SP = ₹80,000. Profit = 5%.
$\text{SP} = 105\% \text{ of CP}_1 = 1.05 \times \text{CP}_1$
$\text{CP}_1 = \frac{80000}{1.05} = 80000 \times \frac{100}{105} = 80000 \times \frac{20}{21}$
$\text{CP}_1 = \frac{1600000}{21} \approx 76,190.48$
Buffalo 2: Finding Cost Price
SP = ₹80,000. Loss = 10%.
$\text{SP} = 90\% \text{ of CP}_2 = 0.90 \times \text{CP}_2$
$\text{CP}_2 = \frac{80000}{0.90} = 80000 \times \frac{100}{90} = 80000 \times \frac{10}{9}$
$\text{CP}_2 = \frac{800000}{9} \approx 88,888.89$
Overall Result
Total SP = $80,000 + 80,000 = 1,60,000$.
Total CP = $76,190.48 + 88,888.89 = 1,65,079.37$.
Since Total CP > Total SP, it is a loss.
Loss = $1,65,079.37 - 1,60,000 = 5,079.37$.
Ans: His overall loss is approx ₹5,079.37.
A man sells two vintage watches for ₹6,000 each.
On one watch he makes a 20% profit. On the other watch, he makes a 20% loss.
Calculate his exact overall monetary profit or loss for the combined transaction.
On one watch he makes a 20% profit. On the other watch, he makes a 20% loss.
Calculate his exact overall monetary profit or loss for the combined transaction.
1
This mirrors the buffalo question but uses much cleaner numbers.
1
Watch 1 (20% Profit): $1.20 \times \text{CP}_1 = 6000$. Find $\text{CP}_1$.
2
Watch 2 (20% Loss): $0.80 \times \text{CP}_2 = 6000$. Find $\text{CP}_2$.
!
"Since it's +20% and -20%, he broke even!" No! The 20% loss occurred on a highly expensive watch, while the 20% profit occurred on a cheaper watch. He definitively lost money.
Step-by-Step Calculation
Watch 1 (Profit):
$1.20 \times \text{CP}_1 = 6000 \implies \text{CP}_1 = \frac{6000}{1.2} = 5000$.
He bought it for ₹5000 and sold it for ₹6000 (+₹1000 profit).
Watch 2 (Loss):
$0.80 \times \text{CP}_2 = 6000 \implies \text{CP}_2 = \frac{6000}{0.8} = 7500$.
He bought it for ₹7500 and sold it for ₹6000 (-₹1500 loss).
Overall:
Total CP = $5000 + 7500 = 12500$.
Total SP = $6000 + 6000 = 12000$.
Loss = $12500 - 12000 = 500$.
Ans: He took an overall loss of ₹500.
In Question 6, a number is increased by 20% to become 90. If the question instead asked "A number decreased by 20% becomes 90", what equation would you use to find the unknown number $x$?
A. $1.20 \times x = 90$
B. $0.80 \times x = 90$
C. $x - 0.20 = 90$
D. $90 \times 0.20 = x$
1
A decrease means you subtract the percentage from the 100% base.
1
$100\% - 20\% = 80\%$.
!
Option C subtracts the decimal $0.20$ as a raw number, not as a multiplier of $x$.
Logical Explanation
The formula is $\text{Final Value} = (100\% - \text{Decrease \%}) \text{ of Original}$.
Therefore, $90 = (100\% - 20\%) \text{ of } x$.
This translates algebraically to $0.80x = 90$.
Ans: Option (B)
In Question 7, why does the 10% loss on the second buffalo result in a larger monetary value than the 5% profit on the first buffalo?
A. Because 10 is simply a larger number than 5.
B. Because losses are always heavier than profits.
C. Because the Cost Price (the base) of the second buffalo was much higher than the first. Taking 10% of a massive number yields a larger chunk than taking 5% of a smaller number.
D. It doesn't; the monetary amounts were exactly the same.
1
A percentage is meaningless without knowing its base.
The Base Effect
For both buffaloes to reach the exact same selling price (₹80,000), their starting costs had to be very different.
Buffalo 1 gained value to reach 80k, so it started lower (approx ₹76k). Buffalo 2 lost value to reach 80k, so it started much higher (approx ₹88k).
Taking a 10% cut out of ₹88,000 creates a much larger absolute loss than taking a 5% gain out of ₹76,000.
Ans: Option (C)
