Class 8 Ganita Prakash
Fractions in Disguise
Achievement Test - Part 1
Welcome to Part 1 of the Class 8 Fractions in Disguise Test Module!
In this foundational section, we explore the fundamental meaning of the '%' symbol and master the art of converting fractions into percentages (and vice versa). Review your detailed subjective textbook solutions, test your speed with the fully interactive 4-Pillar MCQs, and finally try the extra practice questions.[cite: 4, 5, 6, 7]
In this foundational section, we explore the fundamental meaning of the '%' symbol and master the art of converting fractions into percentages (and vice versa). Review your detailed subjective textbook solutions, test your speed with the fully interactive 4-Pillar MCQs, and finally try the extra practice questions.[cite: 4, 5, 6, 7]
1 Meaning of Per cent Pg 1
Do you know what the symbol '%' means? Can any fraction be expressed as a percentage?[cite: 4]
Concept:
1
The word 'per cent' is derived from the Latin phrase 'per centum', meaning 'by the hundred' or 'out of hundred'. Therefore, percentages are simply fractions where the denominator is 100.[cite: 4]
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Hint:
1
Think of 25% as 25 out of every 100.
2
If you have a fraction like $\frac{1}{2}$, multiplying by 100 will convert it to a percentage.
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Mistakes to Avoid:
!
Assuming percentages must be whole numbers: Fractions like $\frac{1}{3}$ convert to 33.33%. Percentages do not have to be integers.
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Solution:
1. The symbol '%' is read as per cent.
It means "out of hundred". So, $50\%$ means $\frac{50}{100}$.
2. Yes, given any fraction, we can express it as a percentage by multiplying the fraction by 100.
Ans: '%' means out of hundred. Yes, any fraction can be a %.
🎯 Marking Scheme:
• 1 Mark for defining '%'.
• 1 Mark for confirming any fraction can be converted.[cite: 4]
• 1 Mark for defining '%'.
• 1 Mark for confirming any fraction can be converted.[cite: 4]
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2 Paint Mixture (Red) Pg 1
Surya wants to use a deep orange colour to capture the sunset. He mixes some red paint and yellow paint to make this colour. The red paint makes up $\frac{3}{4}$ of this mixture. What percentage of the colour is made with red?[cite: 4]
Concept:
1
To express $\frac{3}{4}$ as a percentage, we need to find its equivalent fraction with 100 as the denominator. We can do this by multiplying the numerator and denominator by the same number, or by simply multiplying the fraction by 100.[cite: 5]
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Hint:
1
Calculate $\frac{3}{4} \times 100$.
2
First, divide 100 by 4 to get 25. Then multiply by 3.
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Mistakes to Avoid:
!
Multiplying only the denominator: Do not write $\frac{3}{4 \times 25} = \frac{3}{100} = 3\%$. You must multiply both numerator and denominator!
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Solution:
Fraction of red paint = $\frac{3}{4}$
Method 1: Equivalent Fraction
$\frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 75\%$
Method 2: Multiply by 100
$\frac{3}{4} \times 100 = 3 \times 25 = 75\%$
Ans: 75% of the colour is red paint.
🎯 Marking Scheme:
• 1 Mark for identifying the correct conversion method.
• 1 Mark for the final answer (75%).[cite: 5]
• 1 Mark for identifying the correct conversion method.
• 1 Mark for the final answer (75%).[cite: 5]
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3 Paint Mixture (Yellow) Pg 2
Following the previous example, can you tell what percentage of the colour was made using yellow paint?[cite: 5]
Concept:
1
The entire mixture represents the "whole," which is always $100\%$ (or $1$). If the mixture is only red and yellow, then the percentage of yellow is $100\% - (\text{percentage of red})$.[cite: 5]
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Hint:
1
Red paint is $75\%$. Subtract this from $100\%$.
2
Alternatively, yellow paint is the remaining fraction: $1 - \frac{3}{4} = \frac{1}{4}$. Convert $\frac{1}{4}$ to a percentage.
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Mistakes to Avoid:
!
Assuming the remaining ratio directly: Don't assume the remaining is $\frac{1}{3}$ just because there's 3 parts red. There are 4 parts *total*. Yellow is 1 part out of 4.
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Solution:
Method 1 (Subtraction):
Total mixture = $100\%$
Red paint = $75\%$
Yellow paint = $100\% - 75\% = 25\%$
Method 2 (Fractions):
Fraction of yellow = $1 - \frac{3}{4} = \frac{1}{4}$
Percentage = $\frac{1}{4} \times 100 = 25\%$
Ans: 25% of the colour is yellow paint.
🎯 Marking Scheme:
• 1 Mark for determining the remaining percentage or fraction.
• 1 Mark for the final answer (25%).[cite: 5]
• 1 Mark for determining the remaining percentage or fraction.
• 1 Mark for the final answer (25%).[cite: 5]
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4 Prize Money Canvas Pg 2
Surya won some prize money in a contest. He wants to save $\frac{2}{5}$ of the money to purchase a new canvas. Express this quantity as a percentage.[cite: 5]
Concept:
1
To convert $\frac{2}{5}$ into a percentage, multiply by 100. This is the same as finding an equivalent fraction with 100 as the denominator: $\frac{2 \times 20}{5 \times 20}$.[cite: 5]
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Hint:
1
Calculate $\frac{2}{5} \times 100$.
2
What is $100 \div 5$? It's 20. Now multiply 20 by 2.
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Mistakes to Avoid:
!
Adding zero error: Do not just add a zero to make it 20/50 and declare 20%. The denominator must reach 100!
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Solution:
Fraction of money saved = $\frac{2}{5}$
Method 1: Multiply by 100
$\frac{2}{5} \times 100 = 2 \times 20 = 40\%$
Method 2: Equivalent Fraction
$\frac{2 \times 20}{5 \times 20} = \frac{40}{100} = 40\%$
Ans: Surya saves 40% of his money.
🎯 Marking Scheme:
• 1 Mark for showing the correct calculation ($\times 100$).
• 1 Mark for the final answer (40%).[cite: 5]
• 1 Mark for showing the correct calculation ($\times 100$).
• 1 Mark for the final answer (40%).[cite: 5]
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5 Percentage to Fraction Pg 3
Given a percentage, can you express it as a fraction? For example, express 24% as a fraction.[cite: 6]
Concept:
1
Since a percentage $z\%$ means "$z$ out of 100", it is exactly the same as the fraction $\frac{z}{100}$. You can then simplify this fraction to its lowest terms.[cite: 6]
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Hint:
1
Write $24\%$ as $\frac{24}{100}$.
2
Both 24 and 100 are divisible by 4. Divide them to simplify.
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Mistakes to Avoid:
!
Not simplifying: Leaving the answer as $\frac{24}{100}$ is mathematically correct, but usually exams require the "lowest terms" form.
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Solution:
Percentage given = $24\%$
$24\% = \frac{24}{100}$
Divide numerator and denominator by 4:
$= \frac{24 \div 4}{100 \div 4}$
$= \frac{6}{25}$
Ans: 24% as a fraction is $\frac{6}{25}$.
🎯 Marking Scheme:
• 1 Mark for writing $\frac{24}{100}$.
• 1 Mark for simplifying to $\frac{6}{25}$.[cite: 6]
• 1 Mark for writing $\frac{24}{100}$.
• 1 Mark for simplifying to $\frac{6}{25}$.[cite: 6]
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6 Figure it Out 1 Pg 3
Express the following fractions as percentages.
(i) $\frac{3}{5}$
(ii) $\frac{7}{14}$
(iii) $\frac{9}{20}$
(iv) $\frac{72}{150}$
(v) $\frac{1}{3}$
(vi) $\frac{5}{11}$[cite: 6]
(i) $\frac{3}{5}$
(ii) $\frac{7}{14}$
(iii) $\frac{9}{20}$
(iv) $\frac{72}{150}$
(v) $\frac{1}{3}$
(vi) $\frac{5}{11}$[cite: 6]
Concept:
1
Multiply any fraction by 100 to convert it directly into a percentage.[cite: 6]
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Hint:
1
For (ii), simplify $\frac{7}{14}$ to $\frac{1}{2}$ first, then multiply by 100.
2
For (v) and (vi), the divisions won't be perfect integers. They will result in repeating decimals.
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Mistakes to Avoid:
!
Rounding too early: For $\frac{1}{3}$, do not just write $33\%$. It is exactly $33.33...\%$ or $33 \frac{1}{3}\%$.
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Solution:
(i) $\frac{3}{5} \times 100 = 3 \times 20 = 60\%$
(ii) $\frac{7}{14} = \frac{1}{2} \implies \frac{1}{2} \times 100 = 50\%$
(iii) $\frac{9}{20} \times 100 = 9 \times 5 = 45\%$
(iv) $\frac{72}{150} \times 100 = \frac{72}{1.5} = \frac{720}{15} = 48\%$
(v) $\frac{1}{3} \times 100 = \frac{100}{3} = 33.33\%$ (or $33\frac{1}{3}\%$)
(vi) $\frac{5}{11} \times 100 = \frac{500}{11} \approx 45.45\%$
🎯 Marking Scheme:
• 1/2 Mark for each correct conversion.[cite: 6]
• 1/2 Mark for each correct conversion.[cite: 6]
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7 Figure it Out 2 Pg 3
Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?
(i) 10%
(ii) 15%
(iii) 25%
(iv) 60%
(v) 40%
(vi) None of these[cite: 6]
(i) 10%
(ii) 15%
(iii) 25%
(iv) 60%
(v) 40%
(vi) None of these[cite: 6]
Concept:
1
The fraction is $\frac{\text{Part}}{\text{Whole}}$. Here, the part is 15 and the whole is 25. Multiply the fraction by 100 to get the percentage.[cite: 6]
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Hint:
1
Calculate $\frac{15}{25} \times 100$.
2
$100 \div 25 = 4$. Multiply $15 \times 4$.
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Mistakes to Avoid:
!
Choosing (ii) 15%: Do not just look at the number "15" and assume it means 15%. Percentages are relative to 100, not 25.
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Solution:
Total marbles = 25
White marbles = 15
Percentage = $\frac{15}{25} \times 100$
$= 15 \times 4 = 60\%$
Ans: Option (iv) 60%
🎯 Marking Scheme:
• 1 Mark for selecting the correct option (60%).[cite: 6]
• 1 Mark for selecting the correct option (60%).[cite: 6]
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8 Figure it Out 3 Pg 4
In a school, 15 of the 80 students come to school by walking. What percentage of the students come by walking?[cite: 7]
Concept:
1
Calculate the fraction $\frac{\text{Walking Students}}{\text{Total Students}}$ and multiply by 100.[cite: 7]
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Hint:
1
Setup: $\frac{15}{80} \times 100$.
2
Simplify $\frac{15}{80}$ by dividing both by 5 to get $\frac{3}{16}$. Then solve $\frac{300}{16}$.
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Mistakes to Avoid:
!
Division errors: $\frac{300}{16}$ will result in a decimal. Don't panic if it's not a whole number.
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Solution:
Total students = 80
Walking students = 15
Percentage = $\frac{15}{80} \times 100$
$= \frac{15 \times 10}{8}$
$= \frac{150}{8}$
$= 18.75\%$
Ans: 18.75% students walk to school.
🎯 Marking Scheme:
• 1 Mark for setting up $\frac{15}{80} \times 100$.
• 1 Mark for correct calculation ($18.75\%$).[cite: 7]
• 1 Mark for setting up $\frac{15}{80} \times 100$.
• 1 Mark for correct calculation ($18.75\%$).[cite: 7]
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9 Figure it Out 4 (Race Visual) Pg 4
A group of friends is participating in a long-distance run. The positions of each of them after 15 minutes are shown in the picture (Start to Finish). Match the runners (A, B, C, D) with their approximate completed percentages: 55%, 20%, 38%, 72%, 84%, 93%.[cite: 7]
Concept:
1
Visualize the race line as a scale from 0% (Start) to 100% (Finish). The middle is 50%. A quarter way is 25%. Estimate their positions relative to these anchors.[cite: 7]
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Hint:
1
Runner A is slightly less than the 50% mark. Which option fits?
2
Runner D is almost at the finish line (100%). Which option is closest?
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Mistakes to Avoid:
!
Assuming equal spacing: The runners are not equally spaced. You must gauge their position relative to the absolute start and finish limits.
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Solution:
Based on visual estimation of the line:
Runner A is just before the half-way point. So, A $\rightarrow$ 38%.
Runner B is just past the half-way point. So, B $\rightarrow$ 55%.
Runner C is roughly three-quarters of the way. So, C $\rightarrow$ 72%.
Runner D is extremely close to the finish line. So, D $\rightarrow$ 93%.
Ans: A=38%, B=55%, C=72%, D=93%
🎯 Marking Scheme:
• 1/2 Mark for each correct match.[cite: 7]
• 1/2 Mark for each correct match.[cite: 7]
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10 Figure it Out 5 (Comparisons) Pg 4
Identify and write appropriate symbols '>', '<', '=' in the blanks. Try to do it without calculations.
(i) 50% ___ 5%
(ii) 50% ___ $\frac{5}{10}$
(iii) $\frac{3}{11}$ ___ $\frac{1}{3}$
(iv) 30% ___ 61%[cite: 7]
(i) 50% ___ 5%
(ii) 50% ___ $\frac{5}{10}$
(iii) $\frac{3}{11}$ ___ $\frac{1}{3}$
(iv) 30% ___ 61%[cite: 7]
Concept:
1
Convert mental fractions to percentages or vice versa to compare. For example, $\frac{5}{10} = \frac{1}{2} = 50\%$.[cite: 7]
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Hint:
1
For (iii), cross multiply to compare fractions quickly: $3 \times 3$ vs $11 \times 1$. $9 < 11$, so $\frac{3}{11} < \frac{1}{3}$.
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Mistakes to Avoid:
!
Assuming fractions are larger based on denominators: $\frac{3}{11}$ has an 11, but it is actually *smaller* than $\frac{1}{3}$ (which is $\frac{3}{9}$).
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Solution:
(i) $50\% \mathbf{>} 5\%$
(ii) $50\% \mathbf{=} \frac{5}{10}$ (since $\frac{5}{10} = \frac{1}{2} = 50\%$)
(iii) $\frac{3}{11} \mathbf{<} \frac{1}{3}$ (since $\frac{1}{3} = \frac{3}{9}$, and $\frac{3}{11}$ is smaller than $\frac{3}{9}$)
(iv) $30\% \mathbf{<} 61\%$
🎯 Marking Scheme:
• 1/2 Mark for each correct symbol.[cite: 7]
• 1/2 Mark for each correct symbol.[cite: 7]
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1 Convert Fraction to %
What is $\frac{17}{25}$ expressed as a percentage?
A. 34%
B. 51%
C. 68%
D. 85%
Concept:
1
Multiply the fraction by 100 to convert it directly into a percentage format.
Hint:
1
$100 \div 25 = 4$. Multiply $17 \times 4$.
Mistakes to Avoid:
!
Adding zeroes: Do not just append a zero to make it 170/250. The goal is to reach a denominator of 100.
Solution:
Percentage = $\frac{17}{25} \times 100$
$= 17 \times 4 = 68\%$
Ans: Option C (68%)
2 Convert % to Fraction
What is 45% expressed as a fraction in its lowest terms?
A. $\frac{9}{20}$
B. $\frac{45}{100}$
C. $\frac{7}{20}$
D. $\frac{9}{25}$
Concept:
1
$z\% = \frac{z}{100}$. Write it over 100 and then find the greatest common divisor to simplify.
Hint:
1
Write $\frac{45}{100}$ and divide both numbers by 5.
Mistakes to Avoid:
!
Not fully simplifying: Option B is technically 45%, but the question asks for "lowest terms".
Solution:
Fraction = $\frac{45}{100}$
Divide by 5: $\frac{45 \div 5}{100 \div 5} = \frac{9}{20}$
Ans: Option A ($\frac{9}{20}$)
3 Decimal Percentages
Convert the fraction $\frac{5}{8}$ into a percentage.
A. 58%
B. 62.5%
C. 65%
D. 72.5%
Concept:
1
Multiply by 100. If the division isn't perfect, use decimals.
Hint:
1
Calculate $\frac{500}{8}$.
2
Alternatively, remember that $\frac{1}{8} = 12.5\%$. Multiply that by 5.
Mistakes to Avoid:
!
Rounding off: $62.5\%$ is exact. Don't round it down to 62%.
Solution:
Percentage = $\frac{5}{8} \times 100 = \frac{500}{8}$
$\frac{500}{8} = 62.5\%$
Ans: Option B (62.5%)
4 Exceeding 100%
What is 150% represented as a fraction in lowest terms?
A. $\frac{15}{10}$
B. $\frac{3}{2}$
C. $\frac{5}{4}$
D. $1 \frac{1}{4}$
Concept:
1
Any percentage over 100% means the fraction will be an improper fraction (numerator > denominator). Follow the same rule: place over 100 and simplify.
Hint:
1
Write $\frac{150}{100}$ and divide both numbers by 50.
Mistakes to Avoid:
!
Stopping at 15/10: Option A is correct mathematically but it is not in its "lowest terms".
Solution:
$150\% = \frac{150}{100}$
Divide by 50: $\frac{150 \div 50}{100 \div 50} = \frac{3}{2}$
Ans: Option B ($\frac{3}{2}$)
5 Proportional Fractions
If 7 out of 10 blocks are painted red, what percentage is NOT painted red?
A. 30%
B. 70%
C. 3%
D. 7%
Concept:
1
Calculate the remaining parts first ($10 - 7 = 3$), then convert that fraction to a percentage.
Hint:
1
3 out of 10 are NOT red. Multiply $\frac{3}{10}$ by 100.
Mistakes to Avoid:
!
Solving the wrong question: Finding that 70% are red, and accidentally selecting 70% as the final answer instead of the remaining 30%.
Solution:
Not red = $10 - 7 = 3$ blocks.
Percentage = $\frac{3}{10} \times 100 = 30\%$
Ans: Option A (30%)
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
