This MCQ module is based on: Fractions as Percentages & the FDP Trio
TOPIC 1 OF 23
Fractions as Percentages & the FDP Trio
🎓 Class 8
Mathematics
CBSE
Theory
Ch 1 — Percentages
⏱ ~25 min
🌐 Language: [gtranslate]
🧠 AI-Powered MCQ Assessment
▲
📐 Maths Assessment
▲
This mathematics assessment will be based on: Fractions as Percentages & the FDP Trio
Targeting Class 8 level in General Mathematics, with Basic difficulty.
Upload images, PDFs, or Word documents to include their content in assessment generation.
1.1 Fractions as Percentages
You might have heard statements like, "Mega Sale — up to 50% off!" or "Hiya scored 83% in her board exams". Do you know what the symbol '%' means?
This symbol is read as per cent?. The word 'per cent' is derived from the Latin phrase per centum, meaning 'by the hundred' or 'out of hundred'.
Definition
25 per cent (25%) means 25 out of every 100 — like 25 people out of 100, 25 rupees out of 100 rupees, or 25 marks out of 100 marks.If we say 50% of some quantity \(s\), it means: \(50\% = 50 \times \frac{1}{100} \times s = \frac{50}{100} \times s = \frac{1}{2}s\).
Thus, percentages are simply fractions where the denominator is 100.
Some quick examples:
\(20\% = \frac{20}{100} = \frac{1}{5}\) \(33\% = \frac{33}{100}\)
We saw that percentages are just fractions. Given any fraction, can we express it as a percentage? Yes, let us see how.
Expressing Fractions as Percentages
Example 1: Surya's Paint Mixture
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?
\(\frac{3}{4}\) is 3 out of every 4. That is, 6 out of every 8 (equivalent fraction). That is, 30 out of every 40. That is, 75 out of every 100. This means 75%.
Two Methods to Convert
Method 1: Find the equivalent fraction with denominator 100.\(\frac{3}{4} = \frac{3 \times 25}{4 \times 25} = \frac{75}{100} = 75\%\)
Method 2: Multiply the fraction by 100.
\(\frac{3}{4} \times 100 = \frac{300}{4} = 75\). So \(\frac{3}{4} = 75\%\).
Bar model: \(\frac{3}{4}\) of the mixture is red = 75%
Example 2: Surya's Savings
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.
Three Methods
Method 1: \(\frac{2}{5} = \frac{2 \times 20}{5 \times 20} = \frac{40}{100} = 40\%\)Method 2: \(\frac{2}{5} \times 100 = \frac{200}{5} = 40\). So \(\frac{2}{5} = 40\%\).
Method 3 (bar model): Draw a bar split into 5 equal parts. Shade 2 parts. The shaded region is \(\frac{2}{5}\) of the total.
Bar model: \(\frac{2}{5}\) of prize money saved = 40%
Example 3: Percentage to Fraction
Given a percentage, can you express it as a fraction? For example, express 24% as a fraction.
Since a percentage is a fraction, \(24\% = \frac{24}{100} = \frac{6}{25}\).
In general, we can say that a percentage, \(x\%\), can be expressed by any of the fractions that are equivalent to \(\frac{x}{100}\).
🔵 Can you tell what percentage of the colour was made using yellow? Since 75% is red, yellow = \(100\% - 75\% = 25\%\).
Figure it Out (Section 1.1)
Q1. Express the following fractions as percentages:
(i) \(\frac{3}{5}\) (ii) \(\frac{7}{14}\) (iii) \(\frac{5}{25}\) (iv) \(\frac{7}{150}\) (v) \(\frac{1}{3}\) (vi) \(\frac{5}{11}\)
(i) \(\frac{3}{5}\) (ii) \(\frac{7}{14}\) (iii) \(\frac{5}{25}\) (iv) \(\frac{7}{150}\) (v) \(\frac{1}{3}\) (vi) \(\frac{5}{11}\)
(i) \(\frac{3}{5} \times 100 = 60\%\)
(ii) \(\frac{7}{14} = \frac{1}{2} = 50\%\)
(iii) \(\frac{5}{25} = \frac{1}{5} = 20\%\)
(iv) \(\frac{7}{150} \times 100 = \frac{700}{150} = 4\frac{2}{3}\% \approx 4.67\%\)
(v) \(\frac{1}{3} \times 100 = 33\frac{1}{3}\% \approx 33.33\%\)
(vi) \(\frac{5}{11} \times 100 = \frac{500}{11} = 45\frac{5}{11}\% \approx 45.45\%\)
(ii) \(\frac{7}{14} = \frac{1}{2} = 50\%\)
(iii) \(\frac{5}{25} = \frac{1}{5} = 20\%\)
(iv) \(\frac{7}{150} \times 100 = \frac{700}{150} = 4\frac{2}{3}\% \approx 4.67\%\)
(v) \(\frac{1}{3} \times 100 = 33\frac{1}{3}\% \approx 33.33\%\)
(vi) \(\frac{5}{11} \times 100 = \frac{500}{11} = 45\frac{5}{11}\% \approx 45.45\%\)
Q2. 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
(i) 10% (ii) 15% (iii) 25% (iv) 60% (v) 40% (vi) None of these
White marbles = 15 out of 25 = \(\frac{15}{25} = \frac{3}{5} = 60\%\). Answer: (iv) 60%
Q3. In a school, 15 of the 80 students come to school by walking. What percentage of the students come by walking?
\(\frac{15}{80} \times 100 = \frac{1500}{80} = 18.75\%\). 18.75% of students walk to school.
Q4. A group of friends is participating in a long-distance run. The positions of each of them after 15 minutes are shown below. Match the approximate percentage of the race each of them has completed.
Options: 55%, 20%, 38%, 72%, 84%, 93%
By estimating positions on the line:
A ≈ 20%, B ≈ 38%, C ≈ 72%, D ≈ 93% (approximately).
A ≈ 20%, B ≈ 38%, C ≈ 72%, D ≈ 93% (approximately).
Q5. Pairs of quantities are shown below. Identify and write appropriate symbols '>', '<', '=' in the blanks. Try to do it without calculations.
(i) 50% ___ 5% (ii) \(\frac{3}{10}\) ___ 50% (iii) \(\frac{3}{4}\) ___ 61% (iv) 30% ___ \(\frac{3}{5}\)
(i) 50% ___ 5% (ii) \(\frac{3}{10}\) ___ 50% (iii) \(\frac{3}{4}\) ___ 61% (iv) 30% ___ \(\frac{3}{5}\)
(i) 50% > 5% (obvious)
(ii) \(\frac{3}{10} = 30\%\) < 50%
(iii) \(\frac{3}{4} = 75\%\) > 61%
(iv) \(30\% = \frac{30}{100}\) and \(\frac{3}{5} = 60\%\), so 30% < \(\frac{3}{5}\)
(ii) \(\frac{3}{10} = 30\%\) < 50%
(iii) \(\frac{3}{4} = 75\%\) > 61%
(iv) \(30\% = \frac{30}{100}\) and \(\frac{3}{5} = 60\%\), so 30% < \(\frac{3}{5}\)
Historical Note
Long before the decimal fraction was introduced, the need for computing by hundredths was felt. The idea of 'per hundred' can be found as early as the 4th century BCE in Kautilya's Arthashastra: "An interest of a pana and a quarter per month per cent is just. Five panas per month per cent is commercial interest. Ten panas per month per cent prevails among forests." This shows that percentage-based calculations have been used in India for over 2,300 years!
Percentages Around Us
Percentages are widely used in a variety of contexts. Here are some interesting findings that involve percentages:
Human Body
The human body, on average, is about 60% water by weight.
Ice Cream
Ice cream is about 30-50% air by volume.
FIFA World Cup
45% of the world's population watched at least part of the 2022 FIFA World Cup.
Teenagers
Over 80% of teenagers globally fail to meet the recommended 1 hour of daily physical activity.
Solar System
About 99.86% of the Solar System's mass is contained in the Sun.
Degraded Land
An estimated 52% of the agricultural land worldwide is degraded.
The FDP Trio — Fractions, Decimals, and Percentages
We can find 50% of a value by multiplying \(\frac{1}{2}\) with the value. Will multiplying the value by 0.5 also give the answer for 50% of the value? Yes, since \(\frac{1}{2} = 0.5\).
\(50\% = \frac{50}{100} = \frac{1}{2} = 0.5\)
🔵 Similarly, to find 10% of a quantity, what decimal value should be multiplied? Since \(10\% = \frac{10}{100} = 0.1\), multiply by 0.1.
🔵 Complete the following table:
| Per cent | 50% | 100% | 25% | 75% | 10% | 1% | 5% | 43% |
|---|---|---|---|---|---|---|---|---|
| Fraction | \(\frac{50}{100}\) | \(\frac{100}{100}\) | \(\frac{25}{100}\) | \(\frac{75}{100}\) | \(\frac{10}{100}\) | \(\frac{1}{100}\) | \(\frac{5}{100}\) | \(\frac{43}{100}\) |
| Decimal | 0.5 | 1.0 | 0.25 | 0.75 | 0.1 | 0.01 | 0.05 | 0.43 |
Interactive: FDP Converter
Enter a fraction (numerator and denominator) to see its percentage and decimal equivalents
\(\frac{3}{4}\) = 75% = 0.75
Activity: How Close Can You Get?
L3 Apply
Materials: Paper, pencil, partner
Challenge: Can you estimate the percentage equivalent of a fraction in under 5 seconds?
- Make a pair. Each of you choose a number.
- Share your numbers with each other. Both of you should estimate the percentage equivalent to the fraction \(\frac{a}{b}\) (where \(a\) and \(b\) are the two numbers chosen).
- Announce your answers by a fixed time (say, 5 seconds).
- The one whose estimate is the closest wins this round. Play this for 10 rounds.
Strategy: Use benchmark fractions you know by heart:
- \(\frac{1}{2} = 50\%\), \(\frac{1}{3} \approx 33\%\), \(\frac{1}{4} = 25\%\), \(\frac{1}{5} = 20\%\), \(\frac{1}{10} = 10\%\)
- If the fraction is close to one of these, adjust up or down.
- E.g., \(\frac{3}{7}\) — since \(\frac{3}{6} = 50\%\) and \(\frac{3}{7}\) is slightly less, estimate ~43%.
Example 3: Zuhin's Grade
The maximum marks in a test are 75. If students score 80% or above, they get an A grade. How much should Zuhin score at least to get an A grade?
Solution:
80% of 75 can be found in different ways:
Method 1: \(\frac{80}{100} \times 75 = \frac{6000}{100} = 60\)
Method 2: \(0.8 \times 75 = 60\)
Method 3: 10% of 75 = 7.5, so 80% = 8 × 7.5 = 60.
Zuhin needs at least 60 marks to get an A grade.
Method 1: \(\frac{80}{100} \times 75 = \frac{6000}{100} = 60\)
Method 2: \(0.8 \times 75 = 60\)
Method 3: 10% of 75 = 7.5, so 80% = 8 × 7.5 = 60.
Zuhin needs at least 60 marks to get an A grade.
Competency-Based Questions
Scenario: A nutrition label on a cereal box states: Protein 12%, Carbohydrates 68%, Fat 8%, Fibre 5%, and the rest is moisture and other nutrients. The box contains 500 g of cereal.
Q1. How many grams of protein are in the box?
L3 ApplyAnswer: (b) 60 g. \(12\% \times 500 = \frac{12}{100} \times 500 = 60\) g.
Q2. What percentage of the cereal is moisture and other nutrients? Analyse whether this is a significant portion.
L4 AnalyseAnswer: Total known = 12 + 68 + 8 + 5 = 93%. So moisture/other = \(100 - 93 = 7\%\). In 500 g: \(7\% \times 500 = 35\) g. This is relatively small — 7% is comparable to fat (8%) and fibre (5%), so it's a minor but not negligible portion.
Q3. A competitor's cereal claims "25% more protein." Evaluate: is this claim about 25% of the total weight, or 25% more than the 12% protein content? How many grams of protein would the competitor's cereal have per 500 g box?
L5 EvaluateAnswer: "25% more protein" means 25% more than the original protein amount — NOT 25% of total weight. Original protein = 12%. 25% more = \(12 + 25\% \times 12 = 12 + 3 = 15\%\). In 500 g: \(15\% \times 500 = 75\) g. The competitor has 75 g of protein (vs 60 g in the original).
Q4. Design a nutrition label for a healthy cereal where protein is at least 15%, fat is at most 5%, and fibre is at least 10%. The remaining can be carbohydrates, moisture, and other nutrients. Present your label as percentages that sum to 100%.
L6 CreateOne possible design:
Protein: 18%, Carbohydrates: 55%, Fat: 4%, Fibre: 12%, Moisture: 6%, Other nutrients: 5%.
Check: 18 + 55 + 4 + 12 + 6 + 5 = 100% ✓
Protein ≥ 15% ✓, Fat ≤ 5% ✓, Fibre ≥ 10% ✓. Many valid answers are possible!
Protein: 18%, Carbohydrates: 55%, Fat: 4%, Fibre: 12%, Moisture: 6%, Other nutrients: 5%.
Check: 18 + 55 + 4 + 12 + 6 + 5 = 100% ✓
Protein ≥ 15% ✓, Fat ≤ 5% ✓, Fibre ≥ 10% ✓. Many valid answers are possible!
Assertion–Reason Questions
Assertion (A): \(\frac{3}{8} = 37.5\%\)
Reason (R): To convert a fraction to a percentage, multiply it by 100.
Reason (R): To convert a fraction to a percentage, multiply it by 100.
Answer: (a) — \(\frac{3}{8} \times 100 = 37.5\). Both true, R explains A.
Assertion (A): 50% of 80 is the same as 80% of 50.
Reason (R): Multiplication is commutative: \(a \times b = b \times a\).
Reason (R): Multiplication is commutative: \(a \times b = b \times a\).
Answer: (a) — \(50\% \times 80 = \frac{50 \times 80}{100} = 40\). And \(80\% \times 50 = \frac{80 \times 50}{100} = 40\). Both equal because \(\frac{a \times b}{100} = \frac{b \times a}{100}\). R explains A.
Assertion (A): \(\frac{1}{3} = 33\%\) exactly.
Reason (R): \(\frac{1}{3} \times 100 = 33.333...\), which is a non-terminating decimal.
Reason (R): \(\frac{1}{3} \times 100 = 33.333...\), which is a non-terminating decimal.
Answer: (d) — A is false: \(\frac{1}{3} = 33.\overline{3}\%\), not exactly 33%. R is true: it correctly states the non-terminating nature of the decimal. R explains why A is wrong.
Frequently Asked Questions — Chapter 1
What is Fractions as Percentages & the FDP Trio in NCERT Class 8 Mathematics?
Fractions as Percentages & the FDP Trio is a key concept covered in NCERT Class 8 Mathematics, Chapter 1: Chapter 1. This lesson builds the student's foundation in the chapter by explaining the core ideas with worked examples, definitions, and step-by-step methods aligned to the CBSE curriculum.
How do I solve problems on Fractions as Percentages & the FDP Trio step by step?
To solve problems on Fractions as Percentages & the FDP Trio, follow the NCERT method: identify the given quantities, choose the relevant formula or theorem, substitute values carefully, and simplify. Class 8 exercises gradually increase in difficulty — start with solved NCERT examples before attempting exercise questions, and always verify your answer by substitution or diagram.
What are the most important formulas for Chapter 1: Chapter 1?
The essential formulas of Chapter 1 (Chapter 1) are listed in the chapter summary and highlighted throughout the lesson in formula boxes. Memorise them and practise at least 2–3 problems per formula. CBSE board exams frequently test direct application as well as combined use of multiple formulas from this chapter.
Is Fractions as Percentages & the FDP Trio important for the Class 8 board exam?
Fractions as Percentages & the FDP Trio is part of the NCERT Class 8 Mathematics syllabus and appears in CBSE board exams. Questions typically include short-answer, long-answer, and competency-based items. Review the NCERT examples, exercise questions, and previous-year board problems on this topic to prepare confidently.
What mistakes should students avoid in Fractions as Percentages & the FDP Trio?
Common mistakes in Fractions as Percentages & the FDP Trio include skipping steps, misapplying formulas, sign errors, and losing track of units. Write each step clearly, double-check algebraic manipulations, and re-read the question after solving to verify that your answer matches what was asked.
Where can I find more NCERT practice questions on Fractions as Percentages & the FDP Trio?
End-of-chapter NCERT exercises for Fractions as Percentages & the FDP Trio cover all difficulty levels tested in CBSE exams. After completing them, try the examples again without looking at the solutions, attempt the NCERT Exemplar questions for Chapter 1, and solve at least one previous-year board paper to consolidate your understanding.
AI Tutor
Mathematics Class 8 — Ganita Prakash Part II
Ready