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class 8 ganita prakash chapter 1
📖 Textbook Reference: Calculating Loss
When the Selling Price (SP) is less than the Cost Price (CP), the transaction results in a Loss.
$\text{Loss} = \text{CP} - \text{SP}$.
Just like profit, the Loss Percentage is always calculated on the original Cost Price (CP).
$\text{Loss \%} = \frac{\text{Loss}}{\text{CP}} \times 100$.
The Golden Rule: If you are given the SP and the Loss %, the SP represents $(100\% - \text{Loss}\%)$ of the CP. You can set up an algebraic equation to work backwards and find the original CP.
Example 5: The rice stock in Raghu's provision store is getting old. He had purchased the rice at ₹35 per kg. To clear his stock, he sells 10 kg rice for ₹300. Find out the percentage loss.
(Could we have just calculated the loss percentage per kg instead? Would it be the same?)
(Could we have just calculated the loss percentage per kg instead? Would it be the same?)
- Example 6: Shyamala had procured decorative vases at ₹2650 per piece. She decides to sell a damaged one at a loss of 18%. How much will she get by selling this piece?
- Snehal's Strawberries: Due to heavy rains, Snehal sells some of his stock at ₹80 per kg with a 12% loss. What is the cost price?
1
When given CP and SP, calculate the difference to find the Loss amount. Divide that Loss by the CP and multiply by 100.
2
When given the SP and the Loss %, recognize that $\text{SP} = (100\% - \text{Loss}\%) \text{ of CP}$. Set up an equation to find the original CP.
1
Example 5: 10kg of rice at ₹35/kg means the total CP is ₹350. The SP is ₹300. Calculate $\frac{350 - 300}{350} \times 100$.
2
Example 6: The SP is $100\% - 18\% = 82\%$ of the CP. Calculate $0.82 \times 2650$.
3
Snehal: A 12% loss means the SP (₹80) is exactly 88% of the CP. Set up: $0.88 \times \text{CP} = 80$.
!
The "Reverse Loss" Fallacy: In Snehal's problem, do NOT calculate 12% of ₹80 (9.6) and add it back to get ₹89.6. The 12% loss was calculated on the unknown, larger Cost Price, not the Selling Price! You must use algebra: divide by 0.88.
Step-by-Step Calculation
Example 5: Raghu's Rice
Total CP = $10 \times 35 = 350$. Total SP = $300$.
Loss = $350 - 300 = 50$.
Loss % = $\frac{50}{350} \times 100 = \frac{1}{7} \times 100 \approx 14.28\%$.
(Yes, calculating per kg is the same: CP=35, SP=30. Loss=5. $\frac{5}{35} \times 100 = \frac{1}{7} \times 100$).
Example 6: Shyamala's Vase
SP is $100\% - 18\% = 82\%$ of CP.
$\text{SP} = 0.82 \times 2650 = ₹2173$.
Snehal's Strawberries (Finding CP)
SP is $100\% - 12\% = 88\%$ of CP.
$0.88 \times \text{CP} = 80 \implies \text{CP} = \frac{80}{0.88} = \frac{8000}{88} \approx ₹90.90$.
A car dealership sells a used car for ₹3,40,000, incurring a loss of 15% on their original investment. What was the exact Cost Price of the car?
1
A 15% loss means the dealership only recovered 85% of their original money.
2
Set up the equation: $85\% \text{ of } \text{CP} = \text{Selling Price}$.
1
Equation: $0.85 \times \text{CP} = 340000$.
2
Divide 3,40,000 by 0.85 to isolate CP.
!
If you add 15% of 3,40,000 back to the price, you get ₹3,91,000. Let's test that: 15% of 3,91,000 is 58,650. $391000 - 58650 = 332350$. This does NOT match the ₹3,40,000 selling price! This proves why you cannot calculate loss percentages on the Selling Price.
Step-by-Step Calculation
Selling Price is $100\% - 15\% = 85\%$ of Cost Price.
$0.85 \times \text{CP} = 3,40,000$
$\text{CP} = \frac{3,40,000}{0.85}$
$\text{CP} = 4,00,000$
Ans: The Cost Price was ₹4,00,000.
In Example 5, the text asks if calculating the loss percentage "per kg" yields the same result as calculating the loss percentage for the entire 10 kg batch. Why is the answer yes?
A. Because 10 kg is a multiple of 1 kg.
B. Because multiplying both the numerator (loss) and denominator (CP) by 10 creates an equivalent fraction, leaving the percentage unchanged.
C. Because the loss was exactly ₹5 per kg.
D. It is not yes; the total loss percentage is always higher.
1
A percentage is a ratio. Scaling the total quantity up or down does not change the ratio.
1
Per kg: Loss is ₹5, CP is ₹35. Fraction is $\frac{5}{35}$.
2
For 10 kg: Loss is ₹50, CP is ₹350. Fraction is $\frac{50}{350}$.
!
Do not confuse the amount of loss with the percentage of loss. The total amount lost (₹50) is indeed higher, but the percentage rate is identical.
Mathematical Proof
The per-kg loss percentage is $\frac{5}{35} \times 100$.
If you multiply the numerator and denominator by 10 (for 10 kg), you get $\frac{5 \times 10}{35 \times 10} = \frac{50}{350}$.
Since $\frac{5}{35}$ and $\frac{50}{350}$ both simplify to $\frac{1}{7}$, the resulting percentage ($14.28\%$) is perfectly identical.
Ans: Option (B)
In Snehal's strawberry problem, the SP is ₹80 and the loss is 12%. Which of the following equations correctly models this transaction to find the Cost Price (CP)?
A. $\text{CP} = 80 + (0.12 \times 80)$
B. $\text{CP} - 12 = 80$
C. $\text{CP} - (0.12 \times \text{CP}) = 80$
D. $0.12 \times \text{CP} = 80$
1
Translate the definition of Loss into algebra: $\text{SP} = \text{CP} - \text{Loss}$.
2
Loss is always a percentage of the CP. So, $\text{Loss} = 0.12 \times \text{CP}$.
Algebraic Translation
The formula is $\text{SP} = \text{CP} - \text{Loss Amount}$.
We know $\text{SP} = 80$, and $\text{Loss Amount} = 12\% \text{ of CP} = 0.12 \text{CP}$.
Substituting these values in gives: $80 = \text{CP} - 0.12\text{CP}$.
This simplifies to $80 = 0.88\text{CP}$.
Ans: Option (C)
