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class 8 ganita prakash chapter 1
📖 Textbook Theory: Reverse Profit Calculation
In standard mathematical problems, Profit Margin is calculated as a percentage of the Cost Price (Expenditure). Cost is the initial investment, making it the 100% base.
The Golden Equation:
Revenue (Selling Price) = Cost + Profit
Revenue = 100% of Cost + 25% of Cost
Revenue = 125% of Cost.
3. The total sales of a company (also called revenue) was ₹2.5 crore last year. They had a healthy profit margin of 25%. What was the total expenditure (costs) of the company last year?
1
In standard business math, a profit margin is applied to the Cost (Expenditure). This means Cost is the 100% baseline.
2
Total Sales (Revenue) is the final amount after profit is added. Therefore, $\text{Revenue} = \text{Cost} + \text{Profit}$.
1
Let the Cost be $C$. The Profit is 25% of $C$ (or $0.25C$).
2
Set up the equation: $C + 0.25C = 2.5$. This simplifies to $1.25 \times C = 2.5$. Divide to find $C$.
!
The "Reverse Profit" Trap: The most common mistake is calculating 25% of ₹2.5 crore (which is 0.625) and subtracting it to get 1.875 crore. This is incorrect! The 25% profit was calculated on the unknown Cost, not on the Revenue. You cannot apply a multiplier to the wrong base.
Company Expenditure Calculation
Let $C$ = Total Expenditure (Costs).
Profit = 25% of $C$ = $0.25 \times C$.
Revenue = Cost + Profit
Revenue = $1.00C + 0.25C = 1.25C$.
We know Revenue = ₹2.5 crore.
$1.25 \times C = 2.5$
$C = \frac{2.5}{1.25}$
$C = 2.0$
Ans: The total expenditure was ₹2.0 crore.
(Proof: 25% of 2.0 cr is 0.5 cr. 2.0 + 0.5 = 2.5 cr Revenue. Perfect!)
class 8 ganita prakash chapter 1
📖 Textbook Theory: Discounts
A discount is a reduction in the price of an item. It is a form of Percentage Decrease.
The Formula:
Discount Amount = (Discount %) × (Original Marked Price).
Final Selling Price = Original Marked Price - Discount Amount.
4. A clothing shop offers a 25% discount on all shirts. If the original price of a shirt is ₹300, how much will Anwar have to pay to buy this shirt?
1
A discount is always calculated based on the Original Price (the Marked Price).
2
You can find the final price in two ways: (Method 1) Find the discount amount and subtract it. (Method 2) Subtract the percentage from 100% first, and directly multiply.
1
Method 1: 25% is exactly $\frac{1}{4}$. What is $\frac{1}{4}$ of 300? Subtract that from 300.
2
Method 2: If the discount is 25%, Anwar must pay $100\% - 25\% = 75\%$ of the price. Calculate 75% of 300.
!
Subtracting raw numbers: Do not simply subtract the number 25 from 300 to get ₹275. 25% is a proportion of the whole, not a raw currency amount!
Discount Calculation
Original Price = ₹300. Discount = 25%.
Method 1:
Discount Amount = 25% of 300
Since 25% is $\frac{1}{4}$, $\text{Discount} = \frac{300}{4} =$ ₹75.
Final Price = $300 - 75 =$ ₹225.
Method 2:
Final Price = $(100\% - 25\%) \text{ of } 300$
$= 75\% \text{ of } 300$
$= \frac{75}{100} \times 300 = 75 \times 3 =$ ₹225.
Ans: Anwar has to pay ₹225.
A bakery had a total revenue of ₹10,000 yesterday. They made a profit margin of 20% on the cost of their ingredients.
(A) What was the total cost of the ingredients?
(B) If they offer a 10% discount on all their cakes today, what percentage of the original price will customers pay?
(A) What was the total cost of the ingredients?
(B) If they offer a 10% discount on all their cakes today, what percentage of the original price will customers pay?
1
This mirrors the Company Revenue and Shirt problems.
1
Part (A): Remember, $\text{Revenue} = 120\% \text{ of Cost}$. So, $1.20 \times C = 10000$.
2
Part (B): The original price is always 100%. If you take 10% off, what is left?
!
In part (A), do not calculate 20% of 10,000 (which is 2000) and subtract it. The profit is calculated on the unknown cost, not the revenue!
Step-by-Step Calculation
Part (A): Find the Cost
Revenue = Cost + 20% of Cost.
$1.20 \times \text{Cost} = 10000$
$\text{Cost} = \frac{10000}{1.20} =$ ₹8333.33.
Part (B): The Discount
Original Price = 100%.
Discount = 10%.
Customer pays = $100\% - 10\% = 90\%$.
Ans: (A) ₹8333.33 (B) 90%
In Question 3, why is it incorrect to calculate the total expenditure by finding 25% of ₹2.5 crore and subtracting it?
A. Because 25% is too small a margin for a company.
B. Because the 25% profit margin was originally applied to the Cost (Expenditure) as the 100% base, not the Revenue. Taking 25% of the Revenue applies the percentage to the wrong base.
C. Because 2.5 crore cannot be easily divided by 4.
D. It is not incorrect; that is the standard way to find profit.
1
A percentage is a multiplier attached to a specific base variable. You cannot freely swap bases.
1
Profit is a percentage of the cost. Revenue is the final total.
!
Option D is the exact trap many students fall into. $0.25 \times 2.5 = 0.625$. $2.5 - 0.625 = 1.875$. If the cost was 1.875, a 25% profit on it would be $1.875 \times 0.25 = 0.46875$. $1.875 + 0.46875 = 2.34375$ Revenue (NOT 2.5). The math breaks if you use the wrong base!
Logical Explanation
The formula is $\text{Revenue} = \text{Cost} + (25\% \text{ of Cost})$.
The 25% multiplier is firmly attached to the Cost variable, which is unknown.
If you calculate 25% of Revenue, you are finding a completely different number. To mathematically undo an increase of 25%, you must divide by 1.25.
Ans: Option (B)
In Question 4, what is the fastest mental-math pathway to find the final price of the ₹300 shirt after a 25% discount?
A. Subtract 25 from 300 to get ₹275.
B. Multiply 300 by 0.25 to get the final price directly.
C. Recognize that a 25% discount leaves 75% to pay. 75% is $\frac{3}{4}$. Find $\frac{3}{4}$ of 300.
D. Divide 300 by 1.25.
1
Mental math relies on turning percentages into simple fractions.
Efficiency Breakdown
Option A subtracts a raw number, which is mathematically invalid.
Option B calculates the discount amount, not the final price.
Option C uses the complement percentage. If 25% is gone, 75% remains. $75\% = \frac{3}{4}$.
$\frac{3}{4} \times 300 \rightarrow 300 \div 4 = 75 \rightarrow 75 \times 3 = 225$. This is the fastest mental pathway.
Ans: Option (C)
