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class 8 ganita prakash chapter 1
📖 Textbook Theory: The +50% and -50% Illusion
Does increasing a price by 50% and then decreasing it by 50% bring you back to the exact same starting price? No!
The Golden Rule: When you increase the price, the 50% is calculated on the small original base. But when you decrease it, the 50% is calculated on the new, much larger base. Taking 50% away from a massive number will always hurt more than adding 50% to a small number!
Example 13: After Surbhi bought cookware from the wholesaler, she kept a profit margin of 50% on all the products. To clear off the remaining stock, she thought she would offer a 50% discount and come out without any loss.
1
Let the original Cost Price be $x$. She marks it up by 50%. The Marked Price becomes $1.5x$.
2
She gives a 50% discount on the Marked Price. Half of $1.5x$ is taken away, leaving her with $0.75x$. Since $0.75x$ is less than her original $1.0x$, she lost money.
1
For (ii): We know the final selling price is $0.75x$, and we are told she sold the goods for ₹12,000. Set up the equation: $0.75x = 12000$. Solve for $x$ (the Cost Price).
2
For (iii): She wants to sell at exactly $x$. Her marked price is $1.5x$. The discount amount needed is $1.5x - 1.0x = 0.5x$. What percentage is $0.5x$ out of $1.5x$?
!
Calculating Discount % from Cost: In part (iii), you might think the discount should just be 50% because $0.5x$ is 50% of the Cost Price. But discounts are ALWAYS calculated as a percentage of the Marked Price! You must calculate $0.5x \div 1.5x$.
Part (i): Did she make a loss?
Let Cost Price (CP) = $x$.
Marked Price (MP) = $x + 0.5x = 1.5x$.
Selling Price (SP) = $50\%$ of MP = $0.5 \times 1.5x = 0.75x$.
Since SP ($0.75x$) is less than CP ($x$), yes, she made a loss.
Part (ii): Loss Amount & Percentage
We are given that SP = ₹12,000.
$0.75x = 12000$
$x = \frac{12000}{0.75} = 16,000$. (This is her Cost Price).
Loss Amount = $\text{CP} - \text{SP} = 16000 - 12000 = 4,000$.
Loss \% = $\left( \frac{\text{Loss}}{\text{CP}} \right) \times 100 = \left( \frac{4000}{16000} \right) \times 100 = 25\%$.
Ans (ii): She lost ₹4,000, which is a 25% loss.
Part (iii): The Break-Even Discount
To have no profit/loss, her final SP must equal her CP ($x$).
Her Marked Price is $1.5x$.
Discount needed = $\text{MP} - \text{Desired SP} = 1.5x - x = 0.5x$.
Discount \% = $\left( \frac{\text{Discount Amount}}{\text{Marked Price}} \right) \times 100$
$= \left( \frac{0.5x}{1.5x} \right) \times 100 = \left( \frac{1}{3} \right) \times 100 = 33.33\%$.
Ans (iii): She should have offered a 33.33% discount.
class 8 ganita prakash chapter 1
📖 Textbook Theory: Shifting the 100% Anchor
A percentage is always a comparison between two numbers. The number that comes after the word "of" is the Base (the 100% anchor).
The Mirror Illusion: If A is 120% of B, you cannot simply say B is 80% of A! When you reverse the sentence, the 100% anchor shifts. You must divide to find the true reverse percentage.
Ariba and Arun have some marbles. Ariba says, "The number of marbles with me is 120% of the marbles Arun has". What would be an appropriate statement Arun could make comparing the number of marbles he has with Ariba's?
1
Translate the English to Algebra: $\text{Ariba} = 1.20 \times \text{Arun}$.
2
To find Arun's sentence, isolate 'Arun' on one side of the equation. $\text{Arun} = \frac{\text{Ariba}}{1.20}$.
1
Let's pick an easy number. Assume Arun has 100 marbles. How many does Ariba have? (120).
2
Now reverse it. Arun has 100. Ariba has 120. What percentage is 100 out of 120?
3
Calculate $\frac{100}{120} \times 100$.
!
The Subtraction Trap: Do NOT assume Arun's statement is "I have 80% of Ariba's marbles." You cannot just subtract the 20% overage from 100%. Percentages are not symmetrical because the base changes! 100 out of 120 is $83.33\%$, not $80\%$.
Algebraic Base Shifting
Ariba's statement: $\text{Ariba} = 120\% \text{ of Arun}$.
$\text{Ariba} = \frac{120}{100} \times \text{Arun}$.
$\text{Ariba} = \frac{6}{5} \times \text{Arun}$.
Now, solve for Arun:
Multiply both sides by $\frac{5}{6}$.
$\text{Arun} = \frac{5}{6} \times \text{Ariba}$.
Convert the fraction $\frac{5}{6}$ into a percentage:
$= \frac{5}{6} \times 100 \approx 83.33\%$.
Ans: Arun could say: "The number of marbles with me is 83.33% of the marbles Ariba has."
Company X's stock price fell by 50% on Monday.
By what exact percentage must the stock price rise on Tuesday just to get back to its original starting price?
By what exact percentage must the stock price rise on Tuesday just to get back to its original starting price?
1
This tests the "Cancellation Trap". Percentages do not cancel symmetrically.
1
Assume the stock started at ₹100.
2
It falls 50%. The new price is ₹50.
3
You need it to go from ₹50 back up to ₹100. The required gain is ₹50. What percentage is 50 out of the new base of 50?
!
"It needs to rise by 50%." False! If it is at ₹50, rising 50% only adds ₹25, leaving it at ₹75.
Step-by-Step Translation
Let original price = ₹100.
After 50% fall, new price = ₹50.
Target price = ₹100.
Required Increase Amount = $100 - 50 = 50$.
Required % Gain = $\frac{\text{Increase Amount}}{\text{Current Base}} \times 100$
$= \frac{50}{50} \times 100 = 100\%$.
Ans: The stock must rise by exactly 100% to break even.
In Example 13, Surbhi marked up her cost by 50% and then discounted it by 50%, resulting in a 25% final loss. If she had reversed the order—discounting the original cost by 50% first, and then marking it up by 50%—would her final loss be different?
A. Yes, she would make a profit.
B. Yes, the loss would be much larger.
C. No, the final loss would be exactly the same (25%). The order of multiplication does not change the final result.
D. Yes, they would cancel out to zero.
1
A 50% markup is a multiplier of $1.5$. A 50% discount is a multiplier of $0.5$.
1
Calculate $x \times 1.5 \times 0.5$.
2
Calculate $x \times 0.5 \times 1.5$. Are they different?
!
Many people think the order of discounts and markups changes the final price. Due to the commutative property of multiplication ($a \times b = b \times a$), the order never matters!
Logical Explanation
Let Base = 100.
Path 1: Markup then Discount
$100 \times 1.5 = 150$. Then $150 \times 0.5 = 75$.
Path 2: Discount then Markup
$100 \times 0.5 = 50$. Then $50 \times 1.5 = 75$.
Because multiplication is commutative, successive percentage changes will always yield the exact same final result regardless of the order they are applied.
Ans: Option (C)
In the Marbles question, Arun's marbles are approx 83.33% of Ariba's. If the question instead asked: "Ariba's marbles are 200% of Arun's," what percentage of Ariba's marbles would Arun have?
A. 100%
B. 50%. Since Ariba has double (200%), Arun must have exactly half (50%) of what she has.
C. -100%
D. 0%
1
Use the same algebraic reversal technique. $\text{Ariba} = 2.0 \times \text{Arun}$.
Base Reversal
Equation: $\text{Ariba} = 2 \times \text{Arun}$.
Divide both sides by 2:
$\text{Arun} = \frac{\text{Ariba}}{2} = 0.5 \times \text{Ariba}$.
Convert $0.5$ to a percentage: $50\%$.
Ans: Option (B)
