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class 8 ganita prakash chapter 1
📖 Textbook Theory: Percentage Increase
When a price goes up over time, we calculate the percentage increase to understand the rate of inflation or growth.
The Golden Rule: The formula is always: $\frac{\text{Amount of Increase}}{\text{Original Base Value}} \times 100$. The "Original Base" is always the starting price chronologically (e.g., the price in 2015).
5. The petrol price in 2015 was ₹60 and ₹100 in 2025. What is the percentage increase in the price of petrol?
1
Calculate the raw difference (the increase in price) from 2015 to 2025.
2
To find the percentage increase, divide the raw increase by the Original Starting Price (the 2015 price) and multiply by 100.
1
Increase = $100 - 60 = 40$.
2
Set up the fraction: $\frac{40}{60}$. Simplify this fraction to its lowest terms before multiplying by 100.
!
The Wrong Base: If you divide 40 by 100, you will get 40% (Option ii). This is a trick! You cannot divide by the *new* price. The growth happened *on top of* the old price (60), so 60 is the denominator.
Petrol Price Calculation
Original Price (2015) = ₹60.
New Price (2025) = ₹100.
Increase Amount = $100 - 60 =$ ₹40.
Percentage Increase = $\left( \frac{\text{Increase}}{\text{Original Base}} \right) \times 100$
$= \frac{40}{60} \times 100$
Simplify fraction: $= \frac{2}{3} \times 100 = \frac{200}{3}$
$= 66.66...\%$
Ans: Option (iv) 66.66%
class 8 ganita prakash chapter 1
📖 Textbook Theory: Finding the Original Price
A discount is calculated as a percentage of the Original Marked Price (MRP).
The Algebra: If a discount is 15%, it means the customer only pays $100\% - 15\% = 85\%$ of the original price. So, $0.85 \times \text{Original Price} = \text{Final Price}$. We use this equation to work backwards.
3. Samson bought a car for ₹4,40,000 after getting a 15% discount from the car dealer. What was the original price of the car?
1
A 15% discount means the customer paid $100\% - 15\% = 85\%$ of the Original Price.
2
Use algebra: Let $X$ be the Original Price. Set up the equation $85\% \text{ of } X = \text{Final Paid Price}$.
1
Equation: $0.85 \times X = 4,40,000$.
2
To isolate $X$, divide $4,40,000$ by $0.85$.
!
The "Reverse" Fallacy: Do NOT calculate 15% of ₹4,40,000 (which is 66,000) and add it back to get ₹5,06,000. The 15% discount was calculated on the unknown, larger Original Price, not the discounted price! The only correct way to reverse it is division ($440000 \div 0.85$).
Samson's Car Calculation
Let Original Price = $P$.
Discount = 15%.
Paid Price = $(100\% - 15\%) \text{ of } P = 85\% \text{ of } P$.
$0.85 \times P = 4,40,000$
$P = \frac{4,40,000}{0.85}$
Multiply by 100 to remove decimal: $P = \frac{44000000}{85}$
$P = 5,17,647.05...$
Ans: The original price was approximately ₹5,17,647.
(Proof: 15% of 517647 is roughly 77647. 517647 - 77647 = 440000. Correct!)
A smartphone's price dropped from ₹20,000 to ₹15,000 over a year.
(A) What is the percentage decrease in the price of the smartphone?
(B) If the current ₹15,000 price is the result of a fresh 25% discount, what was the price just before the discount was applied?
(A) What is the percentage decrease in the price of the smartphone?
(B) If the current ₹15,000 price is the result of a fresh 25% discount, what was the price just before the discount was applied?
1
This tests both concepts from the section above. Part (A) asks you to find a percentage. Part (B) asks you to work backward from a discounted price.
1
Part (A): Difference is 5000. Base is 20000. Calculate $\frac{5000}{20000} \times 100$.
2
Part (B): A 25% discount means 15000 is exactly 75% of the old price. Solve: $0.75 \times P = 15000$.
!
In part (A), do not divide by 15000. The original base price was 20000. In part (B), do not calculate 25% of 15000 and add it.
Step-by-Step Calculation
Part (A):
Decrease = $20000 - 15000 =$ ₹5000.
Percentage Decrease = $\frac{5000}{20000} \times 100$
$= \frac{1}{4} \times 100 = 25\%$.
Part (B):
Let Pre-Discount Price = $P$.
Discount = 25%, so Paid Price = 75% of $P$.
$0.75 \times P = 15000$.
$P = \frac{15000}{0.75} =$ ₹20,000.
Ans: (A) 25% decrease (B) ₹20,000
In Question 5, why is the percentage increase calculated by dividing the ₹40 difference by ₹60 instead of ₹100?
A. Because 60 is a smaller number, resulting in a larger percentage.
B. Because you always divide by the current year's price.
C. Because percentage change measures how much a value shifted relative to where it started. The growth happened on top of the 2015 price (₹60), making it the 100% base.
D. Because dividing by 100 makes the calculation too simple.
1
The Golden Rule of percentage change: The denominator MUST be the chronological starting value.
1
The ₹40 increase was physically added to the ₹60 price tag.
!
Dividing by the new number (100) is a logical error that tells you "how much of the new price is made up of the increase," not "how much the old price grew."
Logical Explanation
The formula is $\frac{\text{Change}}{\text{Original Base}} \times 100$.
Because the price grew *from* ₹60 in 2015, that ₹60 serves as the anchor point (the 100% foundation).
Ans: Option (C)
In Question 3, Samson receives a 15% discount on his car. To find the original price '$P$', which of the following equations is mathematically correct?
A. $P = 440000 + (0.15 \times 440000)$
B. $0.85 \times P = 440000$
C. $0.15 \times P = 440000$
D. $P = 440000 \times 1.15$
1
A discount is removed from the unknown original price ($P$).
2
If 15% is removed, exactly 85% of $P$ remains. This remaining 85% equals the paid price.
Algebraic Modeling
The formula is $\text{Paid Price} = \text{Original} - (\text{Discount \%} \times \text{Original})$.
This simplifies to $\text{Paid Price} = (100\% - 15\%) \text{ of } P$.
$440000 = 85\% \text{ of } P$, which is $0.85 \times P$.
Option A incorrectly calculates the 15% on the discounted price instead of the original price. Option D incorrectly applies a 15% markup multiplier.
Ans: Option (B)
