Class 8 Ganita Prakash
Fractions in Disguise
Achievement Test - Part 3
Welcome to the Subjective Section of the Chapter Achievement Test! We now move beyond multiple-choice into structured problem-solving with questions 21 through 26.
In this part, you will translate real-world scenarios into percentage values and perform rapid logical comparisons between fractions and percentages without doing heavy calculations. Let's begin!
In this part, you will translate real-world scenarios into percentage values and perform rapid logical comparisons between fractions and percentages without doing heavy calculations. Let's begin!
21 Percentage Word Problem
Aarav has a collection of 40 balloons, out of which 24 are blue. What percentage of his balloons are blue?[span_0](start_span)[span_0](end_span)
Concept:
1
To find the percentage of a part relative to a whole, construct the fraction \(\frac{\text{Part}}{\text{Whole}}\) and multiply it by 100.[span_1](start_span)[span_1](end_span)
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Hint:
1
Set up the fraction: \(\frac{24}{40}\).
2
Multiply by 100: \(\frac{24}{40} \times 100\). Simplify by canceling zeros!
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Mistakes to Avoid:
!
Inverting the fraction: Do not write \(\frac{40}{24} \times 100\). The "whole" total always goes in the denominator!
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Solution:
Total balloons = \(40\)
Blue balloons = \(24\)
\(\text{Percentage} = \frac{24}{40} \times 100\%\)
\(= \frac{24}{4} \times 10\%\)
\(= 6 \times 10\%\)
\(= 60\%\)
Ans: 60% of his balloons are blue.
🎯 Marking Scheme:
• 1 Mark for setting up the correct fraction (24/40 × 100).
• 1 Mark for the final correct percentage (60%).[span_2](start_span)[span_2](end_span)
• 1 Mark for setting up the correct fraction (24/40 × 100).
• 1 Mark for the final correct percentage (60%).[span_2](start_span)[span_2](end_span)
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22 Percentage Word Problem
Meera has a collection of 50 stickers, out of which 35 are shiny. What percentage of her stickers are shiny?[span_3](start_span)[span_3](end_span)
Concept:
1
Construct the fraction \(\frac{\text{Part}}{\text{Whole}}\) and multiply it by 100 to convert to a percentage.[span_4](start_span)[span_4](end_span)
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Hint:
1
Fraction: \(\frac{35}{50}\).
2
Multiply by 100. Note that 100 divided by 50 is exactly 2!
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Mistakes to Avoid:
!
Long division error: Do not try to evaluate \(35 \div 50\) as a decimal manually if you can just multiply by 100 first. The 50 perfectly cancels out the 100!
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Solution:
Total stickers = \(50\)
Shiny stickers = \(35\)
\(\text{Percentage} = \frac{35}{50} \times 100\%\)
\(= 35 \times 2\%\)
\(= 70\%\)
Ans: 70% of her stickers are shiny.
🎯 Marking Scheme:
• 1 Mark for setting up the correct fraction (35/50 × 100).
• 1 Mark for the final correct percentage (70%).[span_5](start_span)[span_5](end_span)
• 1 Mark for setting up the correct fraction (35/50 × 100).
• 1 Mark for the final correct percentage (70%).[span_5](start_span)[span_5](end_span)
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23 Percentage Word Problem
Rohan has a collection of 60 coins, out of which 27 are foreign currency. What percentage of his coins are foreign?[span_6](start_span)[span_6](end_span)
Concept:
1
Identify the 'whole' (60) and the 'part' (27). Calculate \(\frac{27}{60} \times 100\).[span_7](start_span)[span_7](end_span)
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Hint:
1
Simplify \(\frac{27}{60}\) first. Both divide by 3! It becomes \(\frac{9}{20}\).
2
Now multiply \(\frac{9}{20}\) by 100.
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Mistakes to Avoid:
!
Failing to simplify: Multiplying \(27 \times 100 = 2700\) and then dividing by \(60\) creates massive numbers that are easy to miscalculate. Always simplify the fraction first!
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Solution:
Total coins = \(60\)
Foreign coins = \(27\)
\(\text{Percentage} = \frac{27}{60} \times 100\%\)
Simplify fraction by 3: \(= \frac{9}{20} \times 100\%\)
\(= 9 \times 5\%\)
\(= 45\%\)
Ans: 45% of his coins are foreign.
🎯 Marking Scheme:
• 1 Mark for setting up the correct fraction (27/60 × 100).
• 1 Mark for the final correct percentage (45%).[span_8](start_span)[span_8](end_span)
• 1 Mark for setting up the correct fraction (27/60 × 100).
• 1 Mark for the final correct percentage (45%).[span_8](start_span)[span_8](end_span)
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24 Percentage Word Problem
Ananya has a collection of 80 books, out of which 56 are fiction. What percentage of her books are fiction?[span_9](start_span)[span_9](end_span)
Concept:
1
Set up the fraction \(\frac{56}{80}\) and multiply by 100.[span_10](start_span)[span_10](end_span)
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Hint:
1
Notice that 56 and 80 are both in the 8 times table!
2
Simplify \(\frac{56}{80}\) to \(\frac{7}{10}\).
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Mistakes to Avoid:
!
Finding the non-fiction percentage: The question asks for the percentage of fiction books (56). Do not subtract 56 from 80 and use 24!
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Solution:
Total books = \(80\)
Fiction books = \(56\)
\(\text{Percentage} = \frac{56}{80} \times 100\%\)
Simplify fraction by 8: \(= \frac{7}{10} \times 100\%\)
\(= 7 \times 10\%\)
\(= 70\%\)
Ans: 70% of her books are fiction.
🎯 Marking Scheme:
• 1 Mark for setting up the correct fraction (56/80 × 100).
• 1 Mark for the final correct percentage (70%).[span_11](start_span)[span_11](end_span)
• 1 Mark for setting up the correct fraction (56/80 × 100).
• 1 Mark for the final correct percentage (70%).[span_11](start_span)[span_11](end_span)
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25 Logical Comparisons I
Insert '<', '>', or '=' to make the following mathematical statements true without performing detailed calculations:
(i) \(40\%\) ____ \(\frac{2}{5}\)
(ii) \(60\%\) ____ \(\frac{3}{5}\)[span_12](start_span)[span_12](end_span)[span_13](start_span)[span_13](end_span)
(i) \(40\%\) ____ \(\frac{2}{5}\)
(ii) \(60\%\) ____ \(\frac{3}{5}\)[span_12](start_span)[span_12](end_span)[span_13](start_span)[span_13](end_span)
Concept:
1
Recognize standard fraction-to-percentage conversions. Knowing that \(\frac{1}{5} = 20\%\) allows you to instantly map these values without doing long math.[span_14](start_span)[span_14](end_span)[span_15](start_span)[span_15](end_span)
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Hint:
1
If \(\frac{1}{5}\) is \(20\%\), what is \(\frac{2}{5}\)?
2
What is \(\frac{3}{5}\)? Compare them directly!
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Mistakes to Avoid:
!
Assuming fractions are always smaller: A fraction like \(\frac{2}{5}\) might look "small", but it represents exactly 40 out of 100. Do not guess the symbol based on appearance.
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Solution:
We know that \(\frac{1}{5} = 20\%\).
For (i):
\(\frac{2}{5} = 2 \times 20\% = 40\%\).
Therefore, \(40\% \textbf{ = } \frac{2}{5}\).
For (ii):
\(\frac{3}{5} = 3 \times 20\% = 60\%\).
Therefore, \(60\% \textbf{ = } \frac{3}{5}\).
🎯 Marking Scheme:
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[span_16](start_span)[span_16](end_span)[span_17](start_span)[span_17](end_span)
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[span_16](start_span)[span_16](end_span)[span_17](start_span)[span_17](end_span)
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26 Logical Comparisons II
Insert '<', '>', or '=' to make the following mathematical statements true without performing detailed calculations:
(i) \(80\%\) ____ \(\frac{4}{5}\)
(ii) \(75\%\) ____ \(\frac{3}{4}\)[span_18](start_span)[span_18](end_span)[span_19](start_span)[span_19](end_span)
(i) \(80\%\) ____ \(\frac{4}{5}\)
(ii) \(75\%\) ____ \(\frac{3}{4}\)[span_18](start_span)[span_18](end_span)[span_19](start_span)[span_19](end_span)
Concept:
1
Utilize foundational percentage building blocks: \(\frac{1}{5} = 20\%\) and \(\frac{1}{4} = 25\%\).[span_20](start_span)[span_20](end_span)[span_21](start_span)[span_21](end_span)
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Hint:
1
Scale the blocks! If \(\frac{1}{5}\) is \(20\%\), multiply by 4.
2
If \(\frac{1}{4}\) is \(25\%\), multiply by 3.
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Mistakes to Avoid:
!
Performing long division: The prompt specifically asks you to avoid detailed calculations. Recognizing patterns is a required skill for this mark.
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Solution:
For (i): We know \(\frac{1}{5} = 20\%\).
\(\frac{4}{5} = 4 \times 20\% = 80\%\).
Therefore, \(80\% \textbf{ = } \frac{4}{5}\).
For (ii): We know \(\frac{1}{4} = 25\%\).
\(\frac{3}{4} = 3 \times 25\% = 75\%\).
Therefore, \(75\% \textbf{ = } \frac{3}{4}\).
🎯 Marking Scheme:
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[span_22](start_span)[span_22](end_span)[span_23](start_span)[span_23](end_span)
• 1 Mark for the correct symbol in (i).
• 1 Mark for the correct symbol in (ii).[span_22](start_span)[span_22](end_span)[span_23](start_span)[span_23](end_span)
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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)
