Class 8 Ganita Prakash
Fractions in Disguise
Achievement Test - Part 3
Welcome to Part 3 of the Class 8 Fractions in Disguise Test Module!
In this section, we tackle backward calculations (finding the total 'base' when given a percentage) and situations where percentages smash through the 100% limit (like target sales and harvest growth). Review your detailed textbook solutions (Pages 10 to 14), test your logic with the fully interactive 4-Pillar MCQs, and finally try the extra practice questions.[cite: 13, 14, 15, 16, 17]
In this section, we tackle backward calculations (finding the total 'base' when given a percentage) and situations where percentages smash through the 100% limit (like target sales and harvest growth). Review your detailed textbook solutions (Pages 10 to 14), test your logic with the fully interactive 4-Pillar MCQs, and finally try the extra practice questions.[cite: 13, 14, 15, 16, 17]
1 Cyclist's Journey Pg 10
A cyclist cycles from Delhi to Agra and completes 40% of the journey. If he has covered 92 km, how many more kilometres does he have to travel to reach Agra?[cite: 13, 14]
Concept:
1
If a certain percentage represents a value, you can find the remaining percentage directly using proportionality. Here, 40% represents 92 km. The remaining journey is $100\% - 40\% = 60\%$.[cite: 13, 14]
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Hint:
1
If 40% is 92 km, find what 20% is by dividing by 2.
2
Then multiply that 20% value by 3 to find 60%.
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Mistakes to Avoid:
!
Finding the total and stopping: If you calculate the total distance (100% = 230 km), you must remember to subtract the 92 km already covered. Finding 60% directly avoids this extra step.
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Solution:
Distance covered = $40\% = 92\text{ km}$
Remaining distance = $100\% - 40\% = 60\%$
If $40\% = 92$
Then $20\% = \frac{92}{2} = 46$
Therefore, $60\% = 46 \times 3 = 138\text{ km}$
Ans: He has to travel 138 km more.
🎯 Marking Scheme:
• 1 Mark for setting up the proportion or finding 100%.
• 1 Mark for the correct remaining distance (138 km).[cite: 13, 14]
• 1 Mark for setting up the proportion or finding 100%.
• 1 Mark for the correct remaining distance (138 km).[cite: 13, 14]
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2 Target Sales Pg 10
Kishanlal aims to achieve daily sales of at least ₹5000. On Day 4, he made ₹6000. What percentage of the target was achieved on Day 4?[cite: 14, 15]
Concept:
1
Percentage achieved = $\frac{\text{Actual Sales}}{\text{Target Sales}} \times 100$. If the actual sales exceed the target, the percentage will be greater than 100%.[cite: 14, 15]
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Hint:
1
Setup the fraction: $\frac{6000}{5000} \times 100$.
2
Simplify the fraction to $\frac{6}{5}$ and solve.
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Mistakes to Avoid:
!
Reporting only the excess: While his sales were 20% *more* than the target, the question asks "What percentage of the target was achieved". The correct answer is 120%, not 20%.
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Solution:
Target Sales = ₹5000
Actual Sales = ₹6000
Percentage = $\frac{6000}{5000} \times 100$
$= \frac{6}{5} \times 100$
$= 6 \times 20 = 120\%$
Ans: He achieved 120% of his target.
🎯 Marking Scheme:
• 1 Mark for setting up the equation.
• 1 Mark for the final answer (120%).[cite: 15]
• 1 Mark for setting up the equation.
• 1 Mark for the final answer (120%).[cite: 15]
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3 Wheat Harvest Pg 12
A farmer harvested 260 kg of wheat last year. This year, they harvested 650 kg of wheat. What percentage of last year's harvest is this year's harvest?[cite: 16]
Concept:
1
We are comparing this year to last year. So last year's harvest is the "base" (denominator). Percentage = $\frac{\text{This Year}}{\text{Last Year}} \times 100$.[cite: 16]
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Hint:
1
Calculate $\frac{650}{260} \times 100$.
2
Cancel the zeros: $\frac{65}{26}$. Both are divisible by 13!
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Mistakes to Avoid:
!
Wrong Denominator: If you calculate $\frac{650 - 260}{260}$, you are finding the *percentage increase*, not what percentage the current harvest is of the previous one.
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Solution:
Last year = 260 kg
This year = 650 kg
Percentage = $\frac{650}{260} \times 100$
$= \frac{65}{26} \times 100$
Divide by 13: $= \frac{5}{2} \times 100$
$= 5 \times 50 = 250\%$
Ans: This year's harvest is 250% of last year's.
🎯 Marking Scheme:
• 1 Mark for setting up $\frac{650}{260} \times 100$.
• 1 Mark for the final calculation (250%).[cite: 16]
• 1 Mark for setting up $\frac{650}{260} \times 100$.
• 1 Mark for the final calculation (250%).[cite: 16]
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1 Backward Calculation
If 150% of a number is 45, what is 100% of the number?
A. 20
B. 30
C. 40
D. 60
Concept:
1
Let the number be $x$. $\frac{150}{100} \times x = 45$. Solve for $x$.
Hint:
1
$150\% = \frac{3}{2}$. So $\frac{3}{2}x = 45$. Multiply 45 by $\frac{2}{3}$.
Mistakes to Avoid:
!
Finding 150% of 45: Do not calculate $1.5 \times 45 = 67.5$. You are solving backwards for the base.
Solution:
$\frac{3}{2}x = 45$
$x = 45 \times \frac{2}{3} = 15 \times 2 = 30$
Ans: Option B (30)
2 200% Growth
A company's revenue grew to 200% of last year's revenue. If last year's revenue was ₹5 Lakhs, what is this year's revenue?
A. 5L
B. 10L
C. 15L
D. 200L
Concept:
1
$200\% = \frac{200}{100} = 2$. So 200% of a value is exactly double that value.
Hint:
1
Multiply 5 Lakhs by 2.
Mistakes to Avoid:
!
Adding 200% to the base: The phrase is "grew *to* 200%", not "grew *by* 200%".
Solution:
Revenue = $200\%$ of 5L
$= 2 \times 5L = 10L$
Ans: Option B (10L)
3 Distance Proportions
If 60% of a journey is 120 km, what is the total distance?
A. 72
B. 180
C. 200
D. 240
Concept:
1
Total distance represents $100\%$. If $60\% = 120$, find $10\%$ and then multiply by 10.
Hint:
1
If 60 parts = 120, then 1 part (1%) = 2.
2
100 parts (100%) = $100 \times 2$.
Mistakes to Avoid:
!
Finding 60% of 120: You will get 72, which is wrong. 120 is the *result* of the percentage, not the base.
Solution:
$\frac{60}{100} \times D = 120$
$D = 120 \times \frac{100}{60} = 2 \times 100 = 200$
Ans: Option C (200)
4 Improper Percentages
What is 125% represented as a fraction in lowest terms?
A. $\frac{5}{4}$
B. $\frac{4}{5}$
C. $\frac{1}{4}$
D. $\frac{12}{5}$
Concept:
1
$125\% = \frac{125}{100}$. Simplify by dividing numerator and denominator by 25.
Hint:
1
How many times does 25 go into 125? How many times into 100?
Mistakes to Avoid:
!
Inverting: Option B is $\frac{4}{5}$, which equals 80%, not 125%.
Solution:
$\frac{125}{100} = \frac{125 \div 25}{100 \div 25} = \frac{5}{4}$
Ans: Option A ($\frac{5}{4}$)
5 Proportional Linkage
If 25% of $x$ is 10, then what is 75% of $x$?
A. 20
B. 30
C. 40
D. 50
Concept:
1
You don't need to solve for $x$. 75% is exactly three times 25%. So multiply the value by 3.
Hint:
1
$10 \times 3 = ?$
Mistakes to Avoid:
!
Wasting time finding x: Finding $x = 40$ and then recalculating 75% of 40 is prone to math errors and wastes exam time. Direct scaling is faster.
Solution:
$75\% = 3 \times 25\%$
Therefore, value = $3 \times 10 = 30$.
Ans: Option B (30)
1 Target Baker Sales
A baker sets a daily target of selling 40 cakes. Today, they sold 50 cakes. What percentage of their target did they achieve?
Concept:
1
Just like Kishanlal's sales, percentage achieved = $\frac{\text{Actual Sales}}{\text{Target Sales}} \times 100$. Because actual > target, the answer will exceed 100%.
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Hint:
1
Setup: $\frac{50}{40} \times 100$.
2
Simplify to $\frac{5}{4} \times 100$. (What is 100 divided by 4?)
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Mistakes to Avoid:
!
Finding the percentage increase: Do not calculate $\frac{10}{40} \times 100 = 25\%$. That is the *increase*, but the question asks for the *total target achieved* (125%).
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Solution:
Percentage = $\frac{50}{40} \times 100$
$= \frac{5}{4} \times 100 = 5 \times 25$
$= 125\%$
Ans: They achieved 125% of their target.
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2 Tank Capacity
30% of a water tank's capacity is 600 liters. What is the total capacity (100%) of the tank?
Concept:
1
This is a backward calculation. If $30\% = 600$, find $10\%$ first, and then multiply by 10 to reach $100\%$.
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Hint:
1
Divide 600 by 3 to find 10%.
2
$600 \div 3 = 200$. Now what is $200 \times 10$?
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Mistakes to Avoid:
!
Finding 30% of 600: Do not calculate $0.30 \times 600 = 180$. The 600 liters is already the 30% portion, not the base total.
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Solution:
If $30\% = 600$
Then $10\% = \frac{600}{3} = 200$
Total capacity ($100\%$) = $200 \times 10 = 2000$ liters.
Ans: The total capacity is 2000 liters.
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Fractions in Disguise
Part 1: Fraction & % Equivalence
Pages 1 - 4
Part 2: % of a Quantity
Pages 6 - 9
Part 3: Thresholds & >100%
Pages 10 - 14
Part 4: Proportions, Profit & Loss
Pages 14 - 17
Part 5: Discounts, Taxes & Margins
Pages 18 - 20
Part 6: Simple & Compound Growth
Pages 20 - 24
Part 7: Depreciation & Successive %
Pages 24 - 27
Part 8: Advanced Applications I
Pages 28 - 29
Part 9: Advanced Apps II & Data
Pages 29 - 30
