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Class 8 - Ganita Prakash Chapter 1 - Part 9

Class 8 Ganita Prakash Fractions in Disguise

Achievement Test - Part 9

Welcome to Part 9, the Grand Finale of the Class 8 Fractions in Disguise Module!

In this ultimate challenge, we conquer the deepest logical puzzles of the chapter. We will demystify the famous "Left-handed Paradox", calculate the true mathematical value behind "Buy X Get Y" store deals, and analyze statistical bar graphs. Master these textbook solutions (Pages 29-30) and complete the final MCQ gauntlet![cite: 33, 34]

1 Successive Fare Increases Pg 29

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?[cite: 33]

Concept:

This is a successive percentage increase problem. Assume a base of 100. Multiply by $1.03$ (Year 1) and then by $1.04$ (Year 2).[cite: 33]

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Hint:

Final Price = $100 \times 1.03 \times 1.04$.

$1.03 \times 1.04 = 1.0712$. Multiply by 100 to get the new percentage.

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Mistakes to Avoid:

Adding the percentages: $3\% + 4\% = 7\%$ is incorrect. The 4% increase is applied to the newly inflated 103% base, generating an extra $0.12\%$.

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Solution:

Let original fare = 100.

After Year 1: $100 \times 1.03 = 103$

After Year 2: $103 \times 1.04 = 107.12$

The new fare is 107.12% of the original.

Overall Increase = $107.12 - 100 = 7.12\%$

Ans: The overall increase is 7.12%.

🎯 Marking Scheme:
• 1 Mark for using successive multiplication.
• 1 Mark for $107.12$.
• 1 Mark for final answer (7.12%).[cite: 33]

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2 Rectangle Dimensions Pg 29

If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease by?[cite: 33]

Concept:

Area = Length $\times$ Breadth. If $L_{new} = 1.1L$, then to keep the Area equal to $L \times B$, the new breadth must be $B_{new} = \frac{1}{1.1}B$.[cite: 33]

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Hint:

$B_{new} = \frac{1}{1.1}B = \frac{10}{11}B$.

The decrease is $1 - \frac{10}{11} = \frac{1}{11}$. What is $\frac{1}{11}$ as a percentage?

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Mistakes to Avoid:

Assuming symmetric 10% decrease: If Length goes up by 10% and Breadth drops by 10%, Area = $1.1 \times 0.9 = 0.99$. The area would drop by 1%. The breadth must drop by slightly *less* than 10%.

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Solution:

Area $A = L \times B$.

New Length $L' = L + 10\%L = 1.1L$.

Since Area is unchanged:

$1.1L \times B' = L \times B$

$B' = \frac{B}{1.1} = \frac{10}{11}B$

Decrease in Breadth = $B - \frac{10}{11}B = \frac{1}{11}B$

Percentage = $\frac{1}{11} \times 100 = \frac{100}{11}\%$

$= 9 \frac{1}{11}\%$ or approx $9.09\%$

Ans: Breadth decreases by $9 \frac{1}{11}\%$.

🎯 Marking Scheme:
• 1 Mark for stating $1.1L \times B' = LB$.
• 1 Mark for $B' = \frac{10}{11}B$.
• 1 Mark for final percentage ($9.09\%$ or $\frac{100}{11}\%$).[cite: 33]

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3 Shopping Deals Pg 29

Three shops sell the same items at ₹100.
Shop A: "Buy 1 get 1 free"
Shop B: "Buy 2 get 1 free"
Shop C: "Buy 3 get 1 free"
Find the effective price per item, the percentage discount, and determine the best shop if you need exactly 4 items.[cite: 33]

Concept:

Discount % = $\frac{\text{Free Items}}{\text{Total Items Received}} \times 100$. Effective price = $\frac{\text{Money Paid}}{\text{Total Items Received}}$.[cite: 33]

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Hint:

Shop A: Pay for 1, get 2. Discount = $\frac{1}{2} = 50\%$.

Shop B: Pay for 2, get 3. Discount = $\frac{1}{3} = 33.33\%$.

If you need 4 items at Shop B, the 4th item is full price!

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Mistakes to Avoid:

Wrong denominator: "Buy 2 get 1 free" is not a 50% discount ($\frac{1}{2}$). It is a $33.33\%$ discount because you receive 3 items in total ($\frac{1}{3}$).

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Solution:

(i) Price per item & Order:

Shop A: Pay ₹100 for 2. Price = ₹50.

Shop B: Pay ₹200 for 3. Price = ₹66.67.

Shop C: Pay ₹300 for 4. Price = ₹75.

Cheapest to Costliest: A, B, C.

(ii) Percentage Discount:

Shop A: $\frac{1}{2} \times 100 = 50\%$

Shop B: $\frac{1}{3} \times 100 = 33.33\%$

Shop C: $\frac{1}{4} \times 100 = 25\%$

(iii) Buying exactly 4 items:

Shop A: Buy 2, get 2 free. Pay ₹200.

Shop B: Buy 2, get 1 free. Buy 1 full price. Pay ₹300.

Shop C: Buy 3, get 1 free. Pay ₹300.

Ans: Shop A is the best choice (Pay ₹200).

🎯 Marking Scheme:
• 1 Mark for Effective Prices.
• 1 Mark for Discount Percentages.
• 1 Mark for concluding Shop A is best for 4 items.[cite: 33]

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4 The Left-Handed Paradox Pg 30

In a room of 100 people, 99% are left-handed. How many left-handed people have to leave the room to bring that percentage down to 98%?[cite: 34]

Concept:

This is a variation of the famous "Potato Paradox". Focus on the group that is NOT leaving. There is 1 right-handed person (1% of 100). If the left-handers become 98% of the new group, the 1 right-hander must now represent 2% of the new group's total.[cite: 34]

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Hint:

If 1 person = 2% of the New Total, what is the New Total?

$1 = 0.02 \times \text{New Total}$. Solve for New Total.

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Mistakes to Avoid:

Answering "1 person": If 1 left-hander leaves, you have 98 left-handers and 1 right-hander. Total = 99. The percentage is $\frac{98}{99} = 98.98\%$, not 98%.

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Solution:

Initial State: 100 Total (99 Left, 1 Right)

Goal State: Left is 98%, which means Right is 2%.

The Right-handed person does NOT leave.

$2\%$ of New Total = 1 Right-handed person

$0.02 \times N = 1$

$N = \frac{1}{0.02} = 50$

The new total population must be 50.

Therefore, $100 - 50 = 50$ people must leave.

Ans: 50 left-handed people must leave.

🎯 Marking Scheme:
• 1 Mark for identifying Right-handers become 2%.
• 1 Mark for calculating New Total = 50.
• 1 Mark for concluding 50 must leave.[cite: 34]

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5 Computer Usage Graph Pg 30

Based on the Computer Usage Graph:
(i) Twenties are most computer-literate.
(ii) Women lag behind across all ages.
(iii) More people in twenties than teenagers.
(iv) >25% in thirties can use computers.
(v) <10% aged 60+ can use computers.
(vi) Half of twenties use computers.
Which statements are valid?[cite: 34]

Concept:

A percentage graph only shows proportions of a specific group, NOT the absolute population numbers. You cannot determine total population counts from this graph.[cite: 34]

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Hint:

For (iii), does the graph tell you how many people live in the country?

For (iv), the thirties average is between 14% (F) and 25% (M). Is that >25%?

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Mistakes to Avoid:

Validating statement (vi): The male twenties bar is 37%, female is 26%. Neither is 50%, and the average is ~31%. It is false.

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Solution:

(i) True. Bars are longest (26% F, 37% M).

(ii) True. Blue (Female) is shorter than Yellow (Male) everywhere.

(iii) False. Graph shows %, not raw numbers.

(iv) False. Max for thirties is 25%, average is lower.

(v) True. Seniors are 2% F, 4% M. Both < 10%.

(vi) False. Bars are 26% and 37%, not 50%.

Ans: Valid statements: (i), (ii), (v).

🎯 Marking Scheme:
• 1 Mark for identifying (i) & (ii).
• 1 Mark for rejecting population/averaging fallacies.
• 1 Mark for accepting (v).[cite: 34]

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1 Offer Calculation

A store offers a "Buy 4 Get 1 Free" deal. What is the effective percentage discount the buyer receives?

A. 20%

B. 25%

C. 80%

D. 10%

Concept:

Discount % = $\frac{\text{Free Items}}{\text{Total Items Received}} \times 100$.

Hint:

You pay for 4, you get 1 free. Total items received = 5.

Discount = $\frac{1}{5}$.

Mistakes to Avoid:

Dividing by Paid Items: Doing $\frac{1}{4} = 25\%$ (Option B) is a classic mistake. You must divide by the total volume of goods the customer walks out with.

Solution:

Total items = $4 + 1 = 5$.

Discount = $\frac{1}{5} \times 100 = 20\%$

Ans: Option A (20%)

2 Area Percentage Shift

If the radius of a circle increases by 10%, by what percentage does its area increase?

A. 10%

B. 20%

C. 21%

D. 100%

Concept:

Area = $\pi r^2$. Since $r$ is squared, a multiplier on $r$ gets squared as well. New Area = $\pi (1.1r)^2$.

Hint:

Calculate $1.1 \times 1.1$.

$1.21$ is $121\%$, which means a $21\%$ increase.

Mistakes to Avoid:

Assuming linear scaling: The area does not increase by 10% (Option A). Area scales quadratically, not linearly.

Solution:

New Area = $\pi(1.1r)^2 = 1.21 \pi r^2$.

Multiplier is $1.21$, which equals $121\%$.

Increase = $121\% - 100\% = 21\%$.

Ans: Option C (21%)

3 Data Interpretation Logic

If 40% of Group A is equal in number to 50% of Group B, which statement MUST be true?

A. Group B is larger

B. Group A is larger

C. They are equal

D. Cannot be determined

Concept:

$0.40 A = 0.50 B$. Solve for the ratio of A to B. $A/B = 0.50 / 0.40 = 5/4$.

Hint:

Since it takes a *smaller* percentage of A to equal a *larger* percentage of B, group A's total pool must be larger.

Mistakes to Avoid:

Judging by percentage size: Seeing "50%" next to B and assuming B is larger. 50% is half. 40% is less than half. If less than half of A equals half of B, A is massive.

Solution:

$0.4A = 0.5B \implies A = 1.25B$.

Therefore $A > B$.

Ans: Option B (A is larger)

4 The Evaporation Trap

Fresh fruit contains 80% water. Dry fruit contains 20% water. How many kg of dry fruit can be obtained from 100 kg of fresh fruit?

A. 20 kg

B. 25 kg

C. 80 kg

D. 60 kg

Concept:

During evaporation, water leaves, but the solid "pulp" mass remains perfectly constant. Equate the pulp weight of fresh fruit to the pulp weight of dry fruit.

Hint:

Fresh fruit: $80\%$ water, so $20\%$ pulp. $20\%$ of $100\text{kg} = 20\text{kg}$ solid pulp.

Dry fruit: $20\%$ water, so $80\%$ pulp. Therefore, $80\%$ of (Dry Total) = $20\text{kg}$.

Mistakes to Avoid:

Assuming 20kg directly: Just because it's 20% water doesn't mean it weighs 20kg. The 20kg of solid pulp must account for 80% of the new mass.

Solution:

Pulp in 100kg fresh = $20\%$ of $100 = 20\text{kg}$.

Dry fruit is $80\%$ pulp.

$0.80 \times \text{Total Dry} = 20$

Total Dry = $\frac{20}{0.8} = \frac{200}{8} = 25\text{kg}$

Ans: Option B (25kg)

5 Successive Increment Rule

If a price increases by 5% and then by another 5%, the total percentage increase is:

A. Exactly 10%

B. Less than 10%

C. More than 10%

D. Cannot be determined

Concept:

Because the second 5% increase acts on an already inflated base (105%), it will yield a larger absolute increase than the first 5%.

Hint:

Formula for successive increments $A$ and $B$: Total % = $A + B + \frac{AB}{100}$.

$5 + 5 + \frac{25}{100} = 10.25\%$.

Mistakes to Avoid:

Assuming 10% flat: $5+5=10$ ignores compound growth.

Solution:

Total = $1.05 \times 1.05 = 1.1025$.

$1.1025 = 110.25\%$, which is a $10.25\%$ increase.

Ans: Option C (>10%)

1 Deal Conversion

A clothing store offers a "Buy 3 Get 2 Free" promotion. What is the effective percentage discount on the entire purchase?

Concept:

Calculate the ratio of free items to the TOTAL items obtained. Total items = Paid Items + Free Items.

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Hint:

Total items = $3 + 2 = 5$.

You got 2 free out of 5. Calculate $\frac{2}{5} \times 100$.

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Mistakes to Avoid:

Calculating 2/3: $\frac{2}{3} = 66.66\%$ discount is wrong because it ignores the 2 free items in the total denominator.

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Solution:

Total items received = 5

Free items = 2

Discount \% = $\frac{2}{5} \times 100 = 40\%$

Ans: The effective discount is 40%.

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2 Dimension Balancing

If the length of a rectangle decreases by 20%, by what percentage must the breadth increase to keep the exact same area?

Concept:

$L_{new} \times B_{new} = L \times B$. If $L$ drops by 20%, it becomes $0.8L$. Set $0.8L \times B_{new} = L \times B$ and solve for $B_{new}$.

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Hint:

$B_{new} = \frac{1}{0.8} B = \frac{10}{8} B = 1.25 B$.

A multiplier of $1.25$ means an increase of how much?

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Mistakes to Avoid:

Symmetric 20% increase: A 20% increase does NOT balance a 20% decrease. $0.8 \times 1.2 = 0.96$ (4% loss of area). It must increase by more than 20%.

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Solution:

$0.8L \times B' = LB \implies B' = \frac{1}{0.8}B = 1.25B$.

Multiplier 1.25 equals $125\%$.

Increase = $125\% - 100\% = 25\%$.

Ans: Breadth must increase by 25%.