Class 8 Ganita Prakash
Fractions in Disguise
Achievement Test - Part 2
Welcome to Part 2 of the Class 8 Fractions in Disguise Test Module!
In this section, we move beyond simple conversions and start calculating the percentage of specific quantities. We will explore the powerful "FDP Trio" (Fractions, Decimals, Percentages) to compute values mentally without pen and paper. Dive into your detailed textbook solutions (Pages 6 to 9), test your reflexes with the 4-Pillar MCQs, and finally try the extra practice questions.[cite: 10, 11, 12, 13]
In this section, we move beyond simple conversions and start calculating the percentage of specific quantities. We will explore the powerful "FDP Trio" (Fractions, Decimals, Percentages) to compute values mentally without pen and paper. Dive into your detailed textbook solutions (Pages 6 to 9), test your reflexes with the 4-Pillar MCQs, and finally try the extra practice questions.[cite: 10, 11, 12, 13]
1 Sugar Consumption Pg 6
Madhu ate 120 g of biscuits having 25% sugar and Madhav ate 95 g of biscuits having 35% sugar. Who consumed more sugar?[cite: 10]
Concept:
1
Percentages represent proportions, not absolute amounts. To find the exact amount of sugar, you must calculate the specific percentage of the total weight of biscuits eaten by each person.[cite: 10]
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Hint:
1
For Madhu: Calculate 25% of 120g. Since 25% is $\frac{1}{4}$, just divide 120 by 4.
2
For Madhav: Calculate 35% of 95g. $\frac{35}{100} \times 95$.
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Mistakes to Avoid:
!
Comparing percentages directly: Assuming Madhav ate more simply because 35% > 25% is dangerous. You must calculate the actual grams because their base weights (120g vs 95g) are different.
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Solution:
1. Sugar consumed by Madhu:
Amount = $25\%$ of $120\text{g}$
$= \frac{25}{100} \times 120 = \frac{1}{4} \times 120 = 30\text{g}$
2. Sugar consumed by Madhav:
Amount = $35\%$ of $95\text{g}$
$= \frac{35}{100} \times 95 = 0.35 \times 95 = 33.25\text{g}$
Since $33.25\text{g} > 30\text{g}$, Madhav consumed more.
Ans: Madhav consumed more sugar.
🎯 Marking Scheme:
• 1 Mark for calculating Madhu's sugar (30g).
• 1 Mark for calculating Madhav's sugar (33.25g).
• 1 Mark for the correct final conclusion.[cite: 10]
• 1 Mark for calculating Madhu's sugar (30g).
• 1 Mark for calculating Madhav's sugar (33.25g).
• 1 Mark for the correct final conclusion.[cite: 10]
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2 The FDP Trio Logic Pg 8
We can find 50% of a value by multiplying with $\frac{1}{2}$. Will multiplying the value by $0.5$ also give the answer for 50% of the value?[cite: 12]
Concept:
1
Fractions, Decimals, and Percentages (The FDP Trio) are just three different ways to write the exact same proportion. $50\% = \frac{50}{100} = \frac{1}{2} = 0.5$.[cite: 12]
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Hint:
1
What is the decimal equivalent of $\frac{1}{2}$?
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Mistakes to Avoid:
!
Assuming they act differently: Multiplying by $\frac{1}{2}$ or multiplying by $0.5$ will mathematically yield the exact same product every time.
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Solution:
Yes, multiplying a value by 0.5 will also give 50% of the value.
Because: $50\% = \frac{50}{100}$
$\frac{50}{100} = \frac{1}{2}$ (as a fraction)
$\frac{1}{2} = 0.5$ (as a decimal)
Ans: Yes, because 50% = 1/2 = 0.5.
🎯 Marking Scheme:
• 1 Mark for stating 'Yes' with the correct equivalency justification.[cite: 12]
• 1 Mark for stating 'Yes' with the correct equivalency justification.[cite: 12]
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3 Grading Threshold Pg 8
The maximum marks in a test are 75. If students score 80% or above in the test, they get an A grade. How much should Zubin score at least to get an A grade?[cite: 12]
Concept:
1
Calculate a percentage of a total quantity by multiplying the percentage fraction by the total amount. Here, you need $80\%$ of 75.[cite: 12]
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Hint:
1
Set up the expression: $\frac{80}{100} \times 75$.
2
Simplify $\frac{80}{100}$ to $\frac{4}{5}$. Then calculate $4 \times (75 \div 5)$.
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Mistakes to Avoid:
!
Finding the remaining percentage: The question asks for the target score, not the marks lost. Don't calculate 20% of 75 unless you plan to subtract it from 75 afterwards.
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Solution:
Total Marks = 75
Target for A grade = $80\%$
Using Fraction Multiplication:
Target Score = $\frac{80}{100} \times 75$
$= \frac{4}{5} \times 75$
$= 4 \times 15 = 60$
Ans: Zubin must score at least 60 marks.
🎯 Marking Scheme:
• 1 Mark for setting up the equation.
• 1 Mark for final calculation (60).[cite: 12]
• 1 Mark for setting up the equation.
• 1 Mark for final calculation (60).[cite: 12]
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4 Ratio to Percentage Pg 9
To prepare a particular millet kanji (porridge), suppose the ratio of millet to water to be mixed for boiling is 2:7. What percentage does the millet constitute in this mixture? If 500 ml of the mixture is to be made, how much millet should be used?[cite: 13]
Concept:
1
When given a ratio like $A:B$, the total number of parts in the whole mixture is $A + B$. The fraction of $A$ is $\frac{A}{A+B}$. Convert this fraction to a percentage.[cite: 13]
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Hint:
1
Total parts = $2 + 7 = 9$.
2
Fraction of millet = $\frac{2}{9}$. Multiply by 100 for the percentage.
3
For 500ml mixture, calculate $\frac{2}{9}$ of 500.
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Mistakes to Avoid:
!
Using the wrong denominator: Writing the fraction as $\frac{2}{7}$ is a critical error. 7 is just the water parts, not the *total* mixture.
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Solution:
Ratio of millet to water = $2 : 7$
Total parts = $2 + 7 = 9$ units.
1. Percentage of millet:
Fraction = $\frac{2}{9}$
Percentage = $\frac{2}{9} \times 100 = \frac{200}{9} = 22.22\%$
2. Quantity in 500ml mixture:
Amount = $22.22\%$ of $500\text{ml}$
Amount = $\frac{2}{9} \times 500 = \frac{1000}{9} \approx 111.11\text{ml}$
Ans: Millet is 22.22%. Required amount is ~111.11ml.
🎯 Marking Scheme:
• 1 Mark for calculating total parts (9).
• 1 Mark for millet percentage ($22.22\%$).
• 1 Mark for quantity in 500ml ($111.11\text{ml}$).[cite: 13]
• 1 Mark for calculating total parts (9).
• 1 Mark for millet percentage ($22.22\%$).
• 1 Mark for quantity in 500ml ($111.11\text{ml}$).[cite: 13]
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1 Mental Calculation
What is 25% of 160?
A. 20
B. 40
C. 60
D. 80
Concept:
1
$25\%$ is equivalent to $\frac{1}{4}$. Dividing the number by 4 is the fastest mental math method.
Hint:
1
What is $160 \div 4$?
Mistakes to Avoid:
!
Dividing by 25: $160 \div 25$ is incorrect. $25\%$ means $\frac{25}{100}$, which simplifies to $\frac{1}{4}$.
Solution:
$25\% = \frac{1}{4}$
$\frac{1}{4} \times 160 = 40$
Ans: Option B (40)
2 The FDP Trio
What decimal value should be multiplied to find 10% of a quantity?
A. 1.0
B. 0.01
C. 0.1
D. 10.0
Concept:
1
To convert a percentage to a decimal, divide by 100.
Hint:
1
$10 \div 100 = \frac{10}{100} = \frac{1}{10}$. What is this as a decimal?
Mistakes to Avoid:
!
Choosing 0.01: $0.01$ is $1\%$, not $10\%$.
Solution:
$10\% = \frac{10}{100} = 0.1$
Ans: Option C (0.1)
3 Quick Percentage
What is 20% of 45?
A. 4.5
B. 8
C. 9
D. 10
Concept:
1
$20\%$ is equivalent to $\frac{1}{5}$. Divide the number by 5.
Hint:
1
What is $45 \div 5$?
Mistakes to Avoid:
!
Finding 10% instead: 4.5 is $10\%$ of 45. You need $20\%$, so double it!
Solution:
$20\% = \frac{1}{5}$
$\frac{1}{5} \times 45 = 9$
Ans: Option C (9)
4 Ratio Conversion
If a mixture has a 1:4 ratio of syrup to water, what percentage of the total mixture is syrup?
A. 25%
B. 20%
C. 40%
D. 80%
Concept:
1
Total parts = Sum of all ratio parts. The fraction of syrup is $\frac{\text{Syrup Parts}}{\text{Total Parts}}$.
Hint:
1
Total parts = $1 + 4 = 5$.
2
Syrup is 1 part. Calculate $\frac{1}{5} \times 100$.
Mistakes to Avoid:
!
Using the wrong denominator: Doing $\frac{1}{4} = 25\%$ is wrong because 4 is just the water, not the *total* mixture.
Solution:
Total = $1 + 4 = 5$
Fraction = $\frac{1}{5}$
Percentage = $\frac{1}{5} \times 100 = 20\%$
Ans: Option B (20%)
5 FDP Conversion
Which of the following represents 75% in the FDP Trio format?
A. 7/5 & 0.75
B. 3/4 & 0.75
C. 1/4 & 0.25
D. 4/3 & 1.33
Concept:
1
Convert 75% to a fraction by putting it over 100 and simplifying. Convert to decimal by shifting the point two places left.
Hint:
1
$\frac{75}{100}$ simplifies to what? Divide by 25.
2
$75 \div 100 = ?$
Mistakes to Avoid:
!
Choosing 7/5: $\frac{7}{5}$ equals $140\%$, not $75\%$.
Solution:
Fraction = $\frac{75}{100} = \frac{3}{4}$
Decimal = $75 \div 100 = 0.75$
Ans: Option B
1 Baking Recipe Ratios
A cake recipe needs flour and sugar in a 3:5 ratio by weight. What percentage of the total weight is flour?
Concept:
1
Calculate the total number of parts in the ratio. The percentage of one component is its parts divided by the total parts, multiplied by 100.
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Hint:
1
Total parts = $3 + 5 = 8$. Flour is 3 parts.
2
Calculate $\frac{3}{8} \times 100$. (Note: $\frac{1}{8} = 12.5\%$).
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Mistakes to Avoid:
!
Calculating the wrong ingredient: Make sure you put 3 in the numerator for flour, not 5 (which is sugar).
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Solution:
Total parts = $3 + 5 = 8$
Fraction of flour = $\frac{3}{8}$
Percentage = $\frac{3}{8} \times 100 = 3 \times 12.5 = 37.5\%$
Ans: Flour is 37.5% of the total weight.
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2 Test Score Percentage
A student scores 45 marks out of a total of 60 marks. What percentage did the student score?
Concept:
1
Create a fraction: $\frac{\text{Marks Obtained}}{\text{Total Marks}}$. Then multiply by 100.
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Hint:
1
Calculate $\frac{45}{60} \times 100$. Simplify the fraction to $\frac{3}{4}$ first!
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Mistakes to Avoid:
!
Inverting the fraction: Doing $\frac{60}{45} \times 100$ will result in > 100%, which is impossible for a standard test score.
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Solution:
Percentage = $\frac{45}{60} \times 100$
$= \frac{3}{4} \times 100$
$= 3 \times 25 = 75\%$
Ans: 75%
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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
