Learning Objective: Explore how fractions are simply percentages with a denominator of 100. Experiment with numerators and denominators to see visual equivalence.
The visual bar represents a whole (100%). As you adjust the fraction, the bar fills proportionally to demonstrate the percentage equivalent.
Learning Objective: Calculate y% of a given quantity using fractional or decimal multiplication.
Learning Objective: Understand what percentages greater than 100% mean in real-world contexts like business targets.
Learning Objective: Explore the relationship between Cost Price (CP), Selling Price (SP), Marked Price (MP), Profit Percentage, and Discount.
Learning Objective: Compare linear growth (Simple Interest) with exponential growth (Compounding) over multiple terms.
The symbol % is read as per cent, derived from the Latin phrase 'per centum', meaning 'by the hundred' or 'out of hundred'. Percentages are simply fractions where the denominator is 100.
Questions & Step-by-Step Solutions:
(i) 3 / 5:
Step 1: Multiply numerator and denominator by 20 to get 100 in denominator.
Step 2: (3 * 20) / (5 * 20) = 60 / 100.
Answer: 60%.
(ii) 7 / 14:
Step 1: Simplify fraction to 1 / 2.
Step 2: Multiply by 100 -> (1 / 2) * 100 = 50.
Answer: 50%.
(iii) 9 / 20:
Step 1: Multiply numerator and denominator by 5.
Step 2: (9 * 5) / (20 * 5) = 45 / 100.
Answer: 45%.
(iv) 72 / 150:
Step 1: Simplify by dividing by 3 -> 24 / 50.
Step 2: Multiply by 2 -> 48 / 100.
Answer: 48%.
(v) 1 / 3:
Step 1: Multiply by 100 -> 100 / 3 = 33.333...%.
Answer: 33.33%.
(vi) 5 / 11:
Step 1: Multiply by 100 -> 500 / 11 = 45.4545...%.
Answer: 45.45%.
Question: Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white? (Options: 10%, 15%, 25%, 60%, 40%, None of these)
Step-by-Step Solution:
Step 1: Identify part (15 white) and total (25 marbles).
Step 2: Form fraction: 15 / 25.
Step 3: Convert to percentage: (15 / 25) * 100 = 15 * 4 = 60%.
Answer: 60% (Option iv).
Question: In a school, 15 of the 80 students come to school by walking. What percentage of the students come by walking?
Step-by-Step Solution:
Step 1: Fraction walking = 15 / 80.
Step 2: Multiply by 100 -> (15 / 80) * 100 = (15 / 4) * 5 = 75 / 4 = 18.75%.
Answer: 18.75%.
Question: A group of friends is participating in a long-distance run. Match what percentage of the race A, B, C, D have approximately completed from the options: 55%, 20%, 38%, 72%, 84%, 93%.
Step-by-Step Solution:
Step 1: Observe relative positions on the number line from Start to Finish.
Step 2: A is past the first quarter (~38%), B is past the halfway mark (~55%), C is near three-quarters (~72% or 84%), D is near the finish (~93%).
Answer: A = 38%, B = 55%, C = 72% (or 84%), D = 93%.
Question: Fill blanks with >, <, = without calculations: (i) 50% ___ 5% (ii) 50% ___ 1/2 (iii) 3/11 ___ 61% (iv) 30% ___ 5/10
Step-by-Step Solution:
(i) 50% > 5%.
(ii) 50% = 1/2.
(iii) 3/11 is approx 27.27%, which is < 61%.
(iv) 30% is 0.3, while 5/10 is 0.5 (or 50%), so 30% < 5/10.
Percentages represent proportional values of wholes. y% of some value z is given by (y / 100) * z.
Solutions:
- (i) 100% whole with 20% part segment. If whole is 100, part is 20.
- (ii) 60% of 75 = (60 / 100) * 75 = 45.
- (iii) Proportional scaling matching given bar models.
(i) 25% of 160: (25 / 100) * 160 = (1 / 4) * 160 = 40.
(ii) 16% of 250: (16 / 100) * 250 = 40.
(iii) 62% of 360: (62 / 100) * 360 = 223.2.
(iv) 140% of 40: (140 / 100) * 40 = 56.
(v) 1% of 1 hour: 1% of 60 minutes = 0.6 minutes (or 36 seconds).
(vi) 7% of 10 kg: (7 / 100) * 10 = 0.7 kg (or 700 g).
Question: Surya made 60 ml of deep orange paint, how much red paint did he use if red paint made up 3/4 of the deep orange paint?
Solution:
Step 1: Multiply fraction by total quantity -> (3 / 4) * 60 ml.
Step 2: 3 * 15 = 45 ml.
Answer: 45 ml.
(i) 50% of 510 ___ 50% of 515: 50% of 510 < 50% of 515.
(ii) 37% of 148 ___ 73% of 148: 37% of 148 < 73% of 148.
(iii) 29% of 43 ___ 92% of 110: 29% of 43 is much smaller than 92% of 110 (<).
(iv) 30% of 40 ___ 40% of 50: 12 < 20 (<).
(v) 45% of 200 ___ 10% of 490: 90 > 49 (>).
(vi) 30% of 80 ___ 24% of 64: 24 > 15.36 (>).
(i) 30% of k is 70: 60% of k is 140, 90% of k is 210, 120% of k is 280.
(ii) 100% of m is 215: 10% of m is 21.5, 1% of m is 2.15, 6% of m is 12.9.
(iii) 90% of n is 270: 9% of n is 27, 18% of n is 54, 100% of n is 300.
6. Fill in blanks: (i) 3 is 1% of 300. (ii) 1.6 is 40% of 4. (iii) 40 is 80% of 50.
7. Is 10% of a day longer than 1% of a week? 10% of 24 hours = 2.4 hours (144 mins). 1% of 168 hours = 1.68 hours (100.8 mins). Yes, 10% of a day is longer.
8. Bull's Fodder: Gives 2/ate 1, 3/ate 2, ..., 100/ate 99. Percentages eaten: 50%, 66.7%, 75%, ..., 99%. Observe that percentage approaches 100% as numbers grow.
9. Coffee plantation: 18 days for 20%. Total days = 18 * 5 = 90 days (assuming uniform work rate).
10. Badminton training: 90 mins total. Warm up (10%) = 9 mins, Play (80%) = 72 mins, Cool down (10%) = 9 mins.
11. Northern Hemisphere population: 90% of world population (8.2 billion in 2025) = 7.38 billion.
12. Halwa recipe (4 people): Rava 40%, Sugar 40%, Ghee 20%:
(i) For 8 people, proportions remain identical (40%, 40%, 20%).
(ii) For total weight 2 kg: Rava = 0.8 kg, Sugar = 0.8 kg, Ghee = 0.4 kg.
Percentages compare proportions and track financial gains and losses.
1. If a shopkeeper buys a geometry box for Rs. 75 and sells it for Rs. 110, what is his profit margin with respect to cost?
Step 1: Profit = 110 - 75 = Rs. 35.
Step 2: Profit % = (35 / 75) * 100 = 46.67%.
Answer: 46.67%.
2. Carpenter makes chairs, cost of materials is Rs. 475, profit margin 50%. Selling price?
Step 1: Profit amount = 50% of 475 = 0.50 * 475 = Rs. 237.50.
Step 2: Selling Price = 475 + 237.50 = Rs. 712.50.
Answer: Rs. 712.50.
3. Total sales (revenue) was Rs. 2.5 crore last year with 25% profit margin. Total expenditure?
Step 1: Revenue = Cost + Profit = Cost + 0.25 * Cost = 1.25 * Cost = 2.5 crore.
Step 2: Cost = 2.5 / 1.25 = 2.0 crore.
Answer: Rs. 2 crore.
4. Clothing shop offers 25% discount on shirts. Original price Rs. 300, how much to pay?
Step 1: Discount amount = 25% of 300 = 0.25 * 300 = Rs. 75.
Step 2: Price to pay = 300 - 75 = Rs. 225.
Answer: Rs. 225.
5. Petrol price in 2015 was Rs. 60 and Rs. 100 in 2025. Percentage increase?
Step 1: Increase = 100 - 60 = Rs. 40.
Step 2: Percentage increase = (40 / 60) * 100 = 66.66%.
Answer: 66.66% (Option iv).
3. Samson bought car for Rs. 4,40,000 after 15% discount. Original price?
Step 1: Selling price is 85% of original price (p) -> 0.85 * p = 4,40,000.
Step 2: p = 4,40,000 / 0.85 = Rs. 5,17,647.
4. 1600 people voted, winner got 500 votes. What percent? Minimum candidates?
Step 1: Percentage = (500 / 1600) * 100 = 31.25%.
Step 2: Minimum candidates = 4 (since winner has 500, remaining 1100 split among others must be less than 500 each).
6. A number increased by 20% becomes 90. What is the number?
Step 1: 1.20 * x = 90.
Step 2: x = 90 / 1.20 = 75.
Interest and population growth can be linear (simple interest) or exponential (compounding).
1. Bank of Yahapur 10% p.a., deposit Rs. 20,000 for 2 years with and without compounding annually.
- Without Compounding: Interest = 20000 * 0.10 * 2 = 4000. Total = Rs. 24,000.
- With Compounding: Year 1 amount = 20000 * 1.1 = 22000. Year 2 amount = 22000 * 1.1 = Rs. 24,200.
2. Bank of Wahapur 5% p.a., deposit Rs. 20,000 for 4 years with and without compounding annually.
- Without Compounding: Interest = 20000 * 0.05 * 4 = 4000. Total = Rs. 24,000.
- With Compounding: 20000 * (1.05)^4 = 20000 * 1.2155 = Rs. 24,310.10.
5. Post office 7% p.a. Invest Rs. 50,000 for 3 years without vs with compounding?
- Without: 50000 * 0.07 * 3 = 10,500 interest (Total Rs. 60,500).
- With: 50000 * (1.07)^3 = 50000 * 1.225043 = Rs. 61,252.15 (Interest Rs. 11,252.15).
Difference = Rs. 752.15.
8. Population of city rising by 3% every year. Current population 1.5 crore. Expected after 3 years?
Step 1: Apply compound growth formula -> 1.5 * (1.03)^3.
Step 2: 1.5 * 1.092727 = 1.639 crore (approx 1,63,90,000).
9. Bacteria increases at 2.5% per hour. Initial count 5,06,000. End of 2 hours?
Step 1: 5,06,000 * (1.025)^2.
Step 2: 5,06,000 * 1.050625 = 5,31,616.25 (approx 5,31,616 bacteria).
Mindful comparison of bases and successive discounts.
1. Population of Bengaluru in 2025 is 250% of 2000. If 2000 population was 50 lakhs, what is 2025 population?
Step 1: (250 / 100) * 50 lakhs = 2.5 * 50 = 125 lakhs (1.25 crore).
Answer: 125 lakhs.
3. Mobile phone price Rs. 8,250 + GST 18%. Final price expression?
Answer: 8250 * 1.18 or 8250 + 8250 * 0.18 (Options v and vi).
7. What is 5% of 40? What is 40% of 5? (Compare x% of y and y% of x)
Step 1: 5% of 40 = 2; 40% of 5 = 2.
Step 2: Algebraically, (x / 100) * y = (y / 100) * x. They are always equal.
8. Excursion: 40% Grade 8, rest Grade 9. Among Grade 8, 60% are girls. Total 160 students.
(i) Grade 8 girls percentage of total = 40% of 60% = 0.40 * 0.60 = 0.24 = 24%.
(ii) Number of Grade 8 girls = 24% of 160 = 0.24 * 160 = 38.4 -> 38 girls.
9. Shopkeeper sells pencils: selling price of 3 pencils equals cost of 5 pencils. Profit or loss?
Step 1: Let cost of 1 pencil be c, selling price of 1 pencil be s. 3s = 5c -> s = (5/3)c.
Step 2: Since SP > CP, it is a profit.
Step 3: Profit % = ((5/3 - 1) / 1) * 100 = (2/3) * 100 = 66.67%.
10. Bus fares increased by 3% last year and 4% this year. Overall percentage increase?
Step 1: Net multiplier = 1.03 * 1.04 = 1.0712.
Step 2: Overall increase = 7.12% (not simply 7%).
13. Three shops deal: Shop A (Buy 1 get 1 free), Shop B (Buy 2 get 1 free), Shop C (Buy 3 get 1 free). Item price Rs. 100.
(i) Effective price per item: Shop A = Rs. 50; Shop B = Rs. 66.67; Shop C = Rs. 75. Cheapest to costliest: A, B, C.
(ii) Percentage discount: Shop A = 50%, Shop B = 33.33%, Shop C = 25%.
(iii) For 4 items: Shop A gives 4 items for price of 2 = Rs. 200. Shop B gives 4 items (buy 3 get 1 free) = Rs. 300. Shop C gives 4 items = Rs. 300. Shop A is best.