Class 8 Ganita Prakash
Fractions in Disguise
Achievement Test - Part 4
Welcome to Part 4 of the Chapter Achievement Test! We continue the Foundational Skills section with questions 27 through 32.
These questions focus on mental math strategies (such as scaling from 10% to 40%) and logically comparing percentages to fractions. Every question includes our 4-pillar interactive breakdown with concept notes, hints, common pitfalls, handwritten solutions, and the official 2-mark scoring scheme.
These questions focus on mental math strategies (such as scaling from 10% to 40%) and logically comparing percentages to fractions. Every question includes our 4-pillar interactive breakdown with concept notes, hints, common pitfalls, handwritten solutions, and the official 2-mark scoring scheme.
27 Mental Percentage Scaling I
If 10% of a certain number \(N\) is 18, mentally calculate what 40% of the same number \(N\) would be. Briefly write your reasoning.[cite: 1]
Concept:
1
Percentages scale proportionally. If you multiply the percentage value by a constant factor, the resulting quantity scales by that exact same factor.[cite: 1]
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Hint:
1
How many times larger is 40% compared to 10%?
2
Multiply 18 by that scale factor.
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Mistakes to Avoid:
!
Finding the full number \(N\) unnecessarily: You do not need to calculate \(N = 180\) first. Scaling 10% directly to 40% is much faster and less prone to calculation errors.
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Solution:
Given: \(10\%\text{ of } N = 18\)
We need \(40\%\text{ of } N\).
Since \(40\%\) is \(4 \times 10\%\), we scale the value accordingly:
\(40\%\text{ of } N = 4 \times 18 = 72\)
Ans: 72 (Reasoning: 40% is four times 10%)
🎯 Marking Scheme:
• 1 Mark for identifying the scale factor (4 times).
• 1 Mark for the correct final answer (72) with brief reasoning.[cite: 1]
• 1 Mark for identifying the scale factor (4 times).
• 1 Mark for the correct final answer (72) with brief reasoning.[cite: 1]
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28 Mental Percentage Scaling II
If 10% of a certain number \(N\) is 24, mentally calculate what 30% of the same number \(N\) would be. Briefly write your reasoning.[cite: 3]
Concept:
1
Percentages scale linearly with their values. Scaling 10% to 30% requires multiplying by 3.[cite: 3]
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Hint:
1
Calculate \(3 \times 24\).
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Mistakes to Avoid:
!
Calculating \(N\) first: While finding \(N = 240\) works, it introduces unnecessary multiplication steps.
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Solution:
Given: \(10\%\text{ of } N = 24\)
We need \(30\%\text{ of } N\).
Since \(30\%\) is \(3 \times 10\%\), we scale the value:
\(30\%\text{ of } N = 3 \times 24 = 72\)
Ans: 72 (Reasoning: 30% is three times 10%)
🎯 Marking Scheme:
• 1 Mark for the scale factor (3 times).
• 1 Mark for the final correct answer (72) with brief reasoning.[cite: 3]
• 1 Mark for the scale factor (3 times).
• 1 Mark for the final correct answer (72) with brief reasoning.[cite: 3]
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29 Mental Percentage Scaling III
If 10% of a certain number \(N\) is 32, mentally calculate what 20% of the same number \(N\) would be. Briefly write your reasoning.[cite: 5]
Concept:
1
Proportional scaling allows quick calculations when percentages are direct multiples.[cite: 5]
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Hint:
1
Calculate \(2 \times 32\).
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Mistakes to Avoid:
!
Division errors: Do not divide 32 by 2. You are scaling up from 10% to 20%, so you must multiply!
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Solution:
Given: \(10\%\text{ of } N = 32\)
We need \(20\%\text{ of } N\).
Since \(20\%\) is \(2 \times 10\%\):
\(20\%\text{ of } N = 2 \times 32 = 64\)
Ans: 64 (Reasoning: 20% is double 10%)
🎯 Marking Scheme:
• 1 Mark for identifying double scaling.
• 1 Mark for the correct final answer (64) with reasoning.[cite: 5]
• 1 Mark for identifying double scaling.
• 1 Mark for the correct final answer (64) with reasoning.[cite: 5]
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30 Mental Percentage Scaling IV
If 10% of a certain number \(N\) is 45, mentally calculate what 40% of the same number \(N\) would be. Briefly write your reasoning.[cite: 7]
Concept:
1
Scaling 10% to 40% requires multiplying the partial value by 4.[cite: 7]
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Hint:
1
Calculate \(4 \times 45\).
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Mistakes to Avoid:
!
Arithmetic errors: \(4 \times 45\) is \(180\), not \(160\). Double-check your multiplication!
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Solution:
Given: \(10\%\text{ of } N = 45\)
We need \(40\%\text{ of } N\).
Since \(40\%\) is \(4 \times 10\%\):
\(40\%\text{ of } N = 4 \times 45 = 180\)
Ans: 180 (Reasoning: 40% is four times 10%)
🎯 Marking Scheme:
• 1 Mark for the scale factor (4 times).
• 1 Mark for the correct final answer (180) with reasoning.[cite: 7]
• 1 Mark for the scale factor (4 times).
• 1 Mark for the correct final answer (180) with reasoning.[cite: 7]
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31 Proportional Comparison I
Insert '<', '>', or '=' to make the following mathematical statements true without performing detailed calculations:
(i) \(80\%\) ____ \(\frac{4}{5}\)
(ii) \(60\%\) ____ \(\frac{2}{3}\)[cite: 5]
(i) \(80\%\) ____ \(\frac{4}{5}\)
(ii) \(60\%\) ____ \(\frac{2}{3}\)[cite: 5]
Concept:
1
Convert either the percentage to a fraction or the fraction to a percentage to compare their magnitudes directly.[cite: 5]
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Hint:
1
\(\frac{4}{5} = 80\%\). Compare with \(80\%\).
2
\(\frac{2}{3} \approx 66.66\%\). Compare with \(60\%\).
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Mistakes to Avoid:
!
Guessing symbols: Always convert both sides to the same unit (percentages) before inserting the inequality sign.
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Solution:
For (i): \(\frac{4}{5} = 4 \times 20\% = 80\%\).
Therefore, \(80\% \textbf{ = } \frac{4}{5}\).
For (ii): \(\frac{2}{3} \approx 66.66\%\).
Since \(60\% < 66.66\%\), we have \(60\% \textbf{ < } \frac{2}{3}\).
🎯 Marking Scheme:
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[cite: 5]
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[cite: 5]
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32 Proportional Comparison II
Insert '<', '>', or '=' to make the following mathematical statements true without performing detailed calculations:
(i) \(75\%\) ____ \(\frac{3}{4}\)
(ii) \(50\%\) ____ \(\frac{3}{5}\)[cite: 7]
(i) \(75\%\) ____ \(\frac{3}{4}\)
(ii) \(50\%\) ____ \(\frac{3}{5}\)[cite: 7]
Concept:
1
Convert fractions to percentages (\(\frac{3}{4} = 75\%\), \(\frac{3}{5} = 60\%\)) to compare.[cite: 7]
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Hint:
1
\(75\%\) vs \(75\%\).
2
\(50\%\) vs \(60\%\).
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Mistakes to Avoid:
!
Confusing numbers: Carefully check the inequality direction for part (ii). 50% is less than 60%.
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Solution:
For (i): \(\frac{3}{4} = 3 \times 25\% = 75\%\).
Therefore, \(75\% \textbf{ = } \frac{3}{4}\).
For (ii): \(\frac{3}{5} = 3 \times 20\% = 60\%\).
Since \(50\% < 60\%\), we have \(50\% \textbf{ < } \frac{3}{5}\).
🎯 Marking Scheme:
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[cite: 7]
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[cite: 7]
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×
Chapter Achievement Test 1.1
Part 1: Objective Mastery I
Q1 - Q10 (1 Mark Each)
Part 2: Objective Mastery II
Q11 - Q20 (1 Mark Each)
Part 3: Foundational Skills I
Q21 - Q26 (2 Marks Each)
Part 4: Foundational Skills II
Q27 - Q32 (2 Marks Each)
Part 5: Applied Problem Solving I
Q33 - Q38 (3 Marks Each)
Part 6: Applied Problem Solving II
Q39 - Q44 (3 Marks Each)
Part 7: Advanced Analysis
Q45 - Q48 (4 Marks Each)
Part 8: Real-World Applications I
Q49 - Q50 (6 Marks Each)
Part 9: Real-World Applications II
Q51 - Q52 (6 Marks Each)
