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class 8 ganita prakash chapter 1
📖 Textbook Theory: Assuming a Base Value
When a problem gives you abstract ratios (like "the price of 3 equals the cost of 5") without any actual currency values, the easiest mathematical strategy is to invent a convenient Cost Price.
The Method: Let the Cost Price of 1 item be exactly ₹10. Calculate the total Cost Price for the transaction, and then use the ratio to find the total Selling Price.
9. A shopkeeper sells pencils at a price such that the selling price of 3 pencils is equal to the cost of 5 pencils. Does he make a profit or a loss? What is his profit or loss percentage?
1
Profit occurs when the Selling Price (SP) is strictly greater than the Cost Price (CP) for the exact same number of items. You must calculate the SP and CP for a batch of 3 pencils to compare them fairly.
1
Let the Cost Price of 1 pencil = ₹10.
2
What is the Cost Price of 3 pencils? ($3 \times 10 = 30$).
3
The problem says the Selling Price of 3 pencils equals the Cost of 5 pencils. What is the Cost of 5 pencils? ($5 \times 10 = 50$).
4
Therefore, he bought 3 pencils for ₹30, and sold those 3 pencils for ₹50. Calculate the profit percentage based on his ₹30 investment.
!
The Backward Ratio Mistake: If you say $\text{SP} = 3$ and $\text{CP} = 5$, then $3 < 5$, meaning it's a loss. That is backwards! The problem says the money gained from selling 3 equals the money spent on 5. Since he got his money back by selling fewer items, he is making a massive profit.
Concrete Number Method
Let Cost Price (CP) of 1 pencil = ₹10.
Step 1: Find the Transaction Values
He invested in 3 pencils: $\text{CP of } 3 = 3 \times 10 =$ ₹30.
He sold those 3 pencils: $\text{SP of } 3 = \text{CP of } 5$.
$\text{SP of } 3 = 5 \times 10 =$ ₹50.
Step 2: Calculate Profit
Since $\text{SP } (50) > \text{CP } (30)$, he made a profit.
Profit Amount = $50 - 30 =$ ₹20.
Profit \% = $\left( \frac{\text{Profit Amount}}{\text{Investment Base}} \right) \times 100$.
Profit \% = $\left( \frac{20}{30} \right) \times 100 = \frac{2}{3} \times 100 \approx 66.66\%$.
Ans: He makes a profit of 66.66%.
class 8 ganita prakash chapter 1
📖 Textbook Theory: The Additive Fallacy
When prices are increased sequentially over time, it is mathematically identical to Compound Interest. The new percentage must be calculated on the newly inflated price, not the original starting price.
10. The bus fares were increased by 3% last year and by 4% this year. What is the overall percentage price increase in the last 2 years?
1
Assume the original bus fare was exactly ₹100.
2
Apply the 3% increase to 100. Then apply the 4% increase to the newly inflated price. Compare the final price back to 100 to find the exact percentage difference.
1
Original = 100. Year 1 = $100 + 3 = 103$.
2
Year 2 = 103 + (4% of 103). Calculate $0.04 \times 103$.
!
The Additive Trap: The most common mistake is to say $3\% + 4\% = 7\%$. This is mathematically incorrect because the 4% increase in the second year is applied to a fare that is already more expensive than the original! The true increase must be slightly larger than 7%.
Compounding Inflation
Let Original Fare = 100.
Year 1 (+3%)
New Fare = $100 \times 1.03 = 103$.
Year 2 (+4%)
Final Fare = $103 \times 1.04$.
$103 \times (1 + 0.04) = 103 + 4.12$.
Final Fare = $107.12$.
Overall Increase
Total Increase = $107.12 - 100 = 7.12$.
Ans: The overall percentage price increase is 7.12%.
(Notice it is higher than a flat 7%!).
class 8 ganita prakash chapter 1
📖 Textbook Theory: Maintaining the Product
If you increase one factor in a multiplication equation, you must decrease the other factor to keep the final result exactly the same. However, the percentages will not be symmetrical!
11. If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease by?
1
Let the original Length be 10 and Breadth be 10. Area = 100.
2
Increase the length by 10%. $L_{new} = 11$.
3
Since Area must remain 100, $11 \times B_{new} = 100$. Solve for $B_{new}$, then find the percentage decrease from the original 10.
1
$B_{new} = \frac{100}{11} = 9.0909...$
2
Original Breadth was 10. Decrease Amount = $10 - 9.0909... = 0.909...$
3
Percentage Decrease = $\left( \frac{0.909}{10} \right) \times 100$.
!
The Symmetrical Trap: The most common intuitive answer is "10%". But $11 \times 9 = 99$, not 100! Because the Length (11) is now larger than the Breadth (10), decreasing the Breadth by 1 unit takes a smaller percentage chunk out of the total than increasing the Length by 1 unit did.
Algebraic Solution
Let Area ($A$) = $L \times B$.
New Length ($L_{new}$) = $1.1L$ (a 10% increase).
New Area = Old Area $\implies 1.1L \times B_{new} = L \times B$.
Divide both sides by $1.1L$:
$B_{new} = \frac{B}{1.1} = \frac{10}{11}B$.
Calculate the Decrease
Decrease Amount = Original $B$ - New $B$
$= B - \frac{10}{11}B = \frac{1}{11}B$.
Percentage Decrease = $\left( \frac{1/11}{1} \right) \times 100 = \frac{100}{11}\%$.
Ans: Exactly $9 \frac{1}{11}\%$ (or approx 9.09%).
The price of a streaming subscription was increased by 20% in January. Due to customer complaints, the company decreased the new price by 20% in February.
What is the overall percentage change compared to the original price?
What is the overall percentage change compared to the original price?
1
This tests the "Successive Multipliers" rule again.
1
Increase of 20% = Multiplier of $1.20$.
2
Decrease of 20% = Multiplier of $0.80$.
3
Calculate $1.20 \times 0.80$.
!
The Cancellation Trap: $+20\% - 20\% = 0\%$ is incorrect! The 20% decrease is taking a bite out of a much larger, already inflated price. The final price will drop below the starting point.
Step-by-Step Translation
Assume original price = 100.
January Price = $100 \times 1.20 = 120$.
February Price = $120 \times 0.80 = 96$.
Since the final price is 96, it has dropped by exactly 4 units from the original 100 base.
Ans: An overall decrease of 4%.
In Question 9 (the pencil problem), the Profit Percentage was calculated out of the Cost Price ($20 \div 30 = 66.6\%$). Why do we use the Cost Price as the base (the denominator) instead of the Selling Price ($20 \div 50 = 40\%$)?
A. Because Profit Percentage is meant to measure "Return on Investment". The money you spent (Cost Price) is your investment, so it must be the 100% baseline.
B. Because the Cost Price is always a smaller number, resulting in a higher percentage to make the shopkeeper look better.
C. Because the Selling Price changes depending on the customer.
D. It doesn't matter; both 66.6% and 40% are mathematically valid answers for "Profit Percentage".
1
A percentage is a ratio of a Part to a Whole (Base). Defining the correct Base is critical.
1
When you ask "How much profit did I make?", you are usually asking "How much money did my initial investment grow by?"
!
Option D is false. In accounting and mathematics, standard "Profit Margin" or "Profit Percentage" is strictly defined as relative to the Cost (Investment), unless specifically labeled as "Gross Margin" (which uses Revenue).
Logical Explanation
When evaluating a business transaction, the money you spend to buy the goods is your starting point—your 100% Base.
If you invest ₹30 and get ₹50 back, your ₹30 has generated an extra ₹20.
To measure the efficiency of your investment, you must divide the profit by the investment ($20 \div 30$), proving your money grew by 66.6%.
Ans: Option (A)
In Question 11, the breadth had to decrease by roughly 9.09% to compensate for a 10% increase in length. What is the mathematical reason why the required decrease percentage is smaller than the increase percentage?
A. Because area calculations use subtraction.
B. Because the rectangle changed its shape.
C. Because the length is now longer (11 units). Since the base is larger, a smaller percentage drop on 11 is mathematically equal in absolute value to a larger percentage jump on 10.
D. It is an error; it should be exactly 10%.
1
This is the "Cancellation Trap" acting geometrically.
The Shifting Base
Original Area = $10 \times 10 = 100$.
If Length goes up by 1 unit (+10%), it becomes 11.
To get the area back down to 100, we must remove 10 total area units. Since the length is now 11, we have to divide that removal across all 11 units.
$\frac{10}{11} \approx 0.909$. Because we are dividing the burden across a larger physical number (11), the relative percentage required (9.09%) is smaller than the original 10%.
Ans: Option (C)
