Chapter 1: Fractions in Disguise

Figure it Out 1

1. Express the following fractions as percentages.
(i) 3/5   (ii) 7/14   (iii) 9/20   (iv) 72/150   (v) 1/3   (vi) 5/11
Solution:
Formula: Percentage = Fraction × 100%
Calculation:
(i) (3 / 5) × 100% = 3 × (100 / 5)% = 3 × 20% = 60%
(ii) (7 / 14) × 100% = (1 / 2) × 100% = 100 / 2% = 50%
(iii) (9 / 20) × 100% = 9 × (100 / 20)% = 9 × 5% = 45%
(iv) (72 / 150) × 100% = (72 / 1.5)% = 48%
(v) (1 / 3) × 100% = 100 / 3% = 33.33%
(vi) (5 / 11) × 100% = 500 / 11% = 45.45%
Ans: 60%, 50%, 45%, 48%, 33.33%, 45.45%
2. Nandini has 25 marbles, of which 15 are white. What percentage of her marbles are white?
Solution:
Formula: Percentage = (Part / Total) × 100%
Given Values:
• Total number of marbles = 25
• Number of white marbles = 15
Calculation:
Percentage = (15 / 25) × 100%
Percentage = (3 / 5) × 100%
Percentage = 3 × 20% = 60%
Ans: (iv) 60%
3. In a school, 15 of the 80 students come to school by walking. What percentage of the students come by walking?
Solution:
Formula: Percentage = (Part / Total) × 100%
Given Values:
• Total students = 80
• Students walking = 15
Calculation:
Percentage = (15 / 80) × 100%
Percentage = (3 / 16) × 100%
Percentage = 300 / 16 %
Percentage = 18.75%
Ans: 18.75%
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. Match what percentage of the race each of them has approximately completed.
Options: 55%, 20%, 38%, 72%, 84%, 93%
Solution:
Formula: Position Percentage ≈ (Distance from Start / Total Distance) × 100%
Visual Estimation:
• Point A: Positioned before the halfway (50%) mark, roughly one-third. Closest logical match from options = 38%.
• Point B: Positioned slightly past the halfway (50%) midpoint. Closest match = 55%.
• Point C: Positioned approximately three-quarters (75%) of the way. Closest match = 72%.
• Point D: Positioned extremely close to the finish line (100%). Closest match = 93%.
Ans: A = 38%, B = 55%, C = 72%, D = 93%
5. Pairs of quantities are shown below. Identify and write appropriate symbols '>', '<', '=' in the blanks.
Solution:
Formula: Convert all terms to Percentage (Fraction × 100%) to compare.
Calculation:
(i) 50% vs 5%. Since 50 > 5, then 50% > 5%
(ii) 5/10 vs 50%. Convert LHS: (5 / 10) × 100% = (1 / 2) × 100% = 50%. Since 50% = 50%, then =
(iii) 3/11 vs 61%. Convert LHS: (3 / 11) × 100% ≈ 27.27%. Since 27.27% < 61%, then <
(iv) 30% vs 1/3. Convert RHS: (1 / 3) × 100% ≈ 33.33%. Since 30% < 33.33%, then <
Ans: (i) >, (ii) =, (iii) <, (iv) <

Figure it Out 2

1. Find the missing numbers. (i) 100% bar split in 5 parts. (ii) 100% bar for 60, 75, 90. (iii) 100% bar for 140.
Solution:
Formula: Value per part = Total Value / Number of Equal Parts
Calculation:
(i) Total Percentage = 100%. Parts = 5. Value per part = 100% / 5 = 20%
(ii) The bar represents a total. Let the total be x. If split into 5 parts:
For Total = 60: Value per part = 60 / 5 = 12
For Total = 75: Value per part = 75 / 5 = 15
For Total = 90: Value per part = 90 / 5 = 18
(iii) For Total = 140. Split into 5 equal parts: Value per part = 140 / 5 = 28
Ans: Missing numbers are 12, 15, 18, and 28.
2. Find the value of the following.
(i) 25% of 160   (ii) 16% of 250   (iii) 62% of 360   (iv) 140% of 40   (v) 1% of 1 hour   (vi) 7% of 10 kg
Solution:
Formula: Value = (Percentage / 100) × Total Base
Calculation:
(i) (25 / 100) × 160 = (1 / 4) × 160 = 40
(ii) (16 / 100) × 250 = (16 / 10) × 25 = 1.6 × 25 = 40
(iii) (62 / 100) × 360 = 0.62 × 360 = 223.2
(iv) (140 / 100) × 40 = 1.4 × 40 = 56
(v) Total = 1 hour = 60 minutes = 3600 seconds. Value = (1 / 100) × 3600 = 36 seconds
(vi) Total = 10 kg = 10,000 g. Value = (7 / 100) × 10000 = 7 × 100 = 700 g (or 0.7 kg)
Ans: (i) 40, (ii) 40, (iii) 223.2, (iv) 56, (v) 36s, (vi) 700g
3. 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:
Formula: Value of Part = Fraction × Total Quantity
Given Values:
• Total paint volume = 60 ml
• Fraction of red paint = 3 / 4
Calculation:
Red paint volume = (3 / 4) × 60
Red paint volume = 3 × (60 / 4)
Red paint volume = 3 × 15
Red paint volume = 45 ml
Ans: 45 ml
4. Pairs of quantities are shown below. Identify and write appropriate symbols '>', '<', '='.
Solution:
Formula: Value = (Percentage / 100) × Base
Calculation:
(i) 50% of 510 ___ 50% of 515. LHS: 255. RHS: 257.5. 255 < 257.5 → <
(ii) 37% of 148 ___ 73% of 148. Base is same (148), 37% < 73% → <
(iii) 29% of 43 ___ 92% of 110. LHS: 12.47. RHS: 101.2. 12.47 < 101.2 → <
(iv) 30% of 40 ___ 40% of 50. LHS: 12. RHS: 20. 12 < 20 → <
(v) 45% of 200 ___ 10% of 490. LHS: 90. RHS: 49. 90 > 49 → >
(vi) 30% of 80 ___ 24% of 64. LHS: 24. RHS: 15.36. 24 > 15.36 → >
Ans: (i) <, (ii) <, (iii) <, (iv) <, (v) >, (vi) >
5. Fill in the blanks appropriately:
Solution:
Formula: Value = (Percentage / 100) × Base
(i) Solving in terms of k:
Given: 30% of k = 70 ⇒ 0.3k = 70 ⇒ k = 700 / 3
• 60% of k = (60 / 100) × (700 / 3) = 0.6 × 233.33 = 140
• 90% of k = (90 / 100) × (700 / 3) = 0.9 × 233.33 = 210
• 120% of k = (120 / 100) × (700 / 3) = 1.2 × 233.33 = 280
(ii) Solving in terms of m:
Given: 100% of m = 215 ⇒ 1 × m = 215 ⇒ m = 215
• 10% of m = (10 / 100) × 215 = 21.5
• 1% of m = (1 / 100) × 215 = 2.15
• 6% of m = (6 / 100) × 215 = 12.9
(iii) Solving in terms of n:
Given: 90% of n = 270 ⇒ 0.9n = 270 ⇒ n = 300
• 9% of n = (9 / 100) × 300 = 27
• 18% of n = (18 / 100) × 300 = 54
• 100% of n = (100 / 100) × 300 = 300
Ans: (i) 140, 210, 280 (ii) 21.5, 2.15, 12.9 (iii) 27, 54, 300
6. Fill in the blanks:
Solution:
Formula: Value = (Percentage / 100) × Base
(i) 3 is ___ % of 300.
Let the missing percentage be x.
⇒ (x / 100) × 300 = 3
⇒ 3x = 3 ⇒ x = 1
(ii) ___ is 40% of 4.
Let the missing value be x.
⇒ x = (40 / 100) × 4
⇒ x = 0.40 × 4 ⇒ x = 1.6
(iii) 40 is 80% of ___.
Let the missing base value be x.
⇒ 40 = (80 / 100) × x
⇒ 40 = 0.80x ⇒ x = 40 / 0.80 ⇒ x = 50
Ans: (i) 1, (ii) 1.6, (iii) 50
7. Is 10% of a day longer than 1% of a week?
Solution:
Formula: Time Value = (Percentage / 100) × Total Time (in hours)
Given Values:
• 1 day = 24 hours
• 1 week = 7 days = 7 × 24 = 168 hours
Calculation:
10% of a day = (10 / 100) × 24 = 0.10 × 24 = 2.4 hours.
1% of a week = (1 / 100) × 168 = 0.01 × 168 = 1.68 hours.
Comparing values: 2.4 hours > 1.68 hours.
Ans: Yes, 10% of a day is longer.
8. Mariam's farm has a peculiar bull. Day 1: gave 2 ate 1. Day 2: gave 3 ate 2... Day 99: gave 100 ate 99. Represent as percentages.
Solution:
Formula: Percentage = (Amount Eaten / Amount Given) × 100%
Calculation:
Day 1: Amount eaten = 1, Given = 2. Percentage = (1 / 2) × 100% = 50%
Day 2: Amount eaten = 2, Given = 3. Percentage = (2 / 3) × 100% = 66.67%
Day 3: Amount eaten = 3, Given = 4. Percentage = (3 / 4) × 100% = 75%
... (This pattern continues) ...
Day n: Amount eaten = n, Given = n + 1. Percentage = (n / (n+1)) × 100%
Day 99: Amount eaten = 99, Given = 100. Percentage = (99 / 100) × 100% = 99%
Ans: The percentages are 50%, 66.67%, 75%, ... up to 99%.
9. Workers in a coffee plantation take 18 days to pick coffee berries in 20% of the plantation. How many days will they take to complete the picking work for the entire plantation?
Solution:
Formula: Total Time = Time for Part × (100% / Percentage of Part)
Given Values:
• Percentage of work done = 20%
• Time taken for 20% work = 18 days
Calculation:
Let total time required be x.
20% of x = 18
(20 / 100) × x = 18
0.20x = 18
x = 18 / 0.20
x = 90
Ans: 90 days
10. The badminton coach has planned the training sessions such that the ratio of warm up: play: cool down is 10%: 80%: 10%. If he wants to conduct a training of 90 minutes. How long should each activity be done?
Solution:
Formula: Time for Activity = (Percentage / 100) × Total Time
Given Values:
• Total Time = 90 minutes
• Warm up % = 10%
• Play % = 80%
• Cool down % = 10%
Calculation:
Warm up duration = (10 / 100) × 90 = 0.10 × 90 = 9 minutes.
Play duration = (80 / 100) × 90 = 0.80 × 90 = 72 minutes.
Cool down duration = (10 / 100) × 90 = 0.10 × 90 = 9 minutes.
Ans: Warm up: 9 min, Play: 72 min, Cool down: 9 min
11. An estimated 90% of the world's population lives in the Northern Hemisphere. Find the (approximate) number of people living in the Northern Hemisphere based on this year's worldwide population.
Solution:
Formula: Hemisphere Population = (Percentage / 100) × Total World Population
Given Values:
• Estimated World Population = ~8.2 billion
• Percentage in Northern Hemisphere = 90%
Calculation:
Hemisphere Population = (90 / 100) × 8.2 billion
Hemisphere Population = 0.90 × 8.2 billion
Hemisphere Population = 7.38 billion
Ans: Approximately 7.38 billion people
12. A recipe for halwa for 4 people has: Rava 40%, Sugar 40%, Ghee 20%. (i) For 8 people, what is the proportion? (ii) If total weight is 2 kg, how much of each?
Solution:
Formula: Ingredient Weight = (Percentage / 100) × Total Weight
Given Values:
• Base proportions: Rava=40%, Sugar=40%, Ghee=20%
• Total Weight = 2 kg = 2000 grams
Calculation:
(i) Proportion for 8 people: The percentage breakdown relates to the recipe ratio, which remains fixed regardless of the number of people. The proportion remains 40%, 40%, 20%.
(ii) Weights for 2 kg total:
Rava Weight = (40 / 100) × 2000 = 0.40 × 2000 = 800 g = 0.8 kg.
Sugar Weight = (40 / 100) × 2000 = 0.40 × 2000 = 800 g = 0.8 kg.
Ghee Weight = (20 / 100) × 2000 = 0.20 × 2000 = 400 g = 0.4 kg.
Ans: (i) Rava 40%, Sugar 40%, Ghee 20% (ii) Rava 0.8kg, Sugar 0.8kg, Ghee 0.4kg

Figure it Out 3

1. If a shopkeeper buys a geometry box for ₹75 and sells it for ₹110, what is his profit margin with respect to the cost?
Solution:
Formula: Profit = SP - CP
Profit Margin % = (Profit / CP) × 100%
Given Values:
• Cost Price (CP) = ₹75
• Selling Price (SP) = ₹110
Calculation:
Profit = 110 - 75 = 35
Profit Margin = (35 / 75) × 100%
Profit Margin = (7 / 15) × 100%
Profit Margin = 700 / 15 % = 46.666...%
Ans: 46.67%
2. I am a carpenter and I make chairs. The cost of materials for a chair is ₹475 and I want to have a profit margin of 50%. At what price should I sell a chair?
Solution:
Formula: SP = CP × (1 + Profit Margin % / 100)
Given Values:
• Cost Price (CP) = ₹475
• Profit Margin = 50%
Calculation:
SP = 475 × (1 + 50 / 100)
SP = 475 × (1 + 0.50)
SP = 475 × 1.50
SP = 712.5
Ans: ₹712.5
3. The total sales of a company was ₹2.5 crore last year. They had a healthy profit margin of 25%. What was the total expenditure (costs) of the company last year?
Solution:
Formula: SP = CP × (1 + Profit Margin % / 100)
Given Values:
• Selling Price (SP) = ₹2.5 crore
• Profit Margin = 25%
Calculation:
2.5 = CP × (1 + 25 / 100)
2.5 = CP × (1 + 0.25)
2.5 = CP × 1.25
CP = 2.5 / 1.25 = 2.0
Ans: ₹2 crore
4. A clothing shop offers a 25% discount on all shirts. If the original price of a shirt is ₹300, how much will Anwar have to pay to buy this shirt?
Solution:
Formula: SP = Original Price × (1 - Discount % / 100)
Given Values:
• Original Price = ₹300
• Discount = 25%
Calculation:
SP = 300 × (1 - 25 / 100)
SP = 300 × (1 - 0.25)
SP = 300 × 0.75
SP = 225
Ans: ₹225
5. The petrol price in 2015 was ₹60 and ₹100 in 2025. What is the percentage increase in the price of petrol?
Solution:
Formula: Increase % = ((New Price - Old Price) / Old Price) × 100%
Given Values:
• Old Price = ₹60
• New Price = ₹100
Calculation:
Increase Amount = 100 - 60 = 40
Increase % = (40 / 60) × 100%
Increase % = (2 / 3) × 100%
Increase % = 66.666...%
Ans: 66.67%
6. Samson bought a car for ₹4,40,000 after getting a 15% discount from the car dealer. What was the original price of the car?
Solution:
Formula: Original Price = SP / (1 - Discount % / 100)
Given Values:
• Selling Price (SP) = ₹4,40,000
• Discount = 15%
Calculation:
Original Price = 440000 / (1 - 15 / 100)
Original Price = 440000 / (1 - 0.15)
Original Price = 440000 / 0.85
Original Price ≈ 517,647.058
Ans: ₹517,647.06
7. 1600 people voted in an election and the winner got 500 votes. What percent of the total votes did the winner get? Can you guess the minimum number of candidates who stood for the election?
Solution:
Formula: Percentage = (Votes Obtained / Total Votes) × 100%
Given Values:
• Total Votes = 1600
• Winner Votes = 500
Calculation:
Winner % = (500 / 1600) × 100% = (5 / 16) × 100% = 31.25%.
To find the minimum number of candidates, we must maximize the votes of the losing candidates such that no one beats the winner (max 499 votes each).
Remaining votes pool = 1600 - 500 = 1100.
Candidate 2 gets 499 votes. Remaining = 1100 - 499 = 601.
Candidate 3 gets 499 votes. Remaining = 601 - 499 = 102.
Candidate 4 gets 102 votes.
Therefore, 1 winner + 3 losers = 4 candidates minimum.
Ans: 31.25%. Minimum candidates = 4.
8. The price of 1 kg of rice was ₹38 in 2024. It is ₹42 in 2025. What is the rate of inflation?
Solution:
Formula: Inflation Rate % = ((New Price - Old Price) / Old Price) × 100%
Given Values:
• Old Price = ₹38
• New Price = ₹42
Calculation:
Increase Amount = 42 - 38 = 4
Inflation Rate = (4 / 38) × 100%
Inflation Rate = (2 / 19) × 100% = 200 / 19 %
Inflation Rate ≈ 10.526%
Ans: 10.53%
9. A number increased by 20% becomes 90. What is the number?
Solution:
Formula: Final Number = Original Number × (1 + Increase % / 100)
Given Values:
• Final Number = 90
• Increase % = 20%
Calculation:
Let Original Number be x.
90 = x × (1 + 20 / 100)
90 = x × (1 + 0.20)
90 = 1.2x
x = 90 / 1.2
x = 75
Ans: 75
10. A milkman sold two buffaloes for ₹80,000 each. On one of them, he made a profit of 5% and on the other a loss of 10%. Find his overall profit or loss.
Solution:
Formula: CP = SP / (1 + Profit % / 100)
CP = SP / (1 - Loss % / 100)
Given Values:
• SP for Buffalo 1 = ₹80,000. Profit = 5%
• SP for Buffalo 2 = ₹80,000. Loss = 10%
Calculation:
Total Selling Price (SP) = 80000 + 80000 = 160,000.
CP for Buffalo 1 = 80000 / (1 + 0.05) = 80000 / 1.05 ≈ 76,190.48.
CP for Buffalo 2 = 80000 / (1 - 0.10) = 80000 / 0.90 ≈ 88,888.89.
Total Cost Price (CP) = 76,190.48 + 88,888.89 = 165,079.37.
Overall Result = Total CP - Total SP = 165,079.37 - 160,000 = 5,079.37. (Since CP > SP, it's a loss).
Ans: Overall loss of ₹5,079.37
11. The population of elephants in a national park increased by 5% in the last decade. If the population of the elephants last decade is p, the population now is...
Solution:
Formula: New Population = Old Population × (1 + Increase % / 100)
Given Values:
• Old Population = p
• Increase % = 5%
Calculation:
New Population = p × (1 + 5 / 100)
New Population = p × (1 + 0.05)
New Population = p × 1.05
Ans: (iv) p × 1.05
12. Which of the following statement(s) mean the same as "The demand for cameras has fallen by 85% in the last decade"?
Solution:
Formula: New Demand = Old Demand × (1 - Decrease % / 100)
Given Values:
• Decrease % = 85%
Calculation:
New Demand = Old Demand × (1 - 85 / 100)
New Demand = Old Demand × (1 - 0.85)
New Demand = Old Demand × 0.15
This represents 15% of the original demand.
Ans: (iii) The demand now is 15% of the demand a decade ago.

Figure it Out 4

1. Bank of Yahapur offers an interest of 10% p.a. Compare how much one gets if they deposit ₹20,000 for a period of 2 years with compounding and without compounding annually.
Solution:
Formula Simple: A = P × (1 + r×t)
Formula Compound: A = P × (1 + r)t
Given Values:
• Principal (P) = ₹20,000
• Rate (r) = 10% = 0.10
• Time (t) = 2 years
Calculation:
Without Compounding (Simple Interest): A = 20000 × (1 + 0.10 × 2) = 20000 × 1.20 = 24,000.
With Compounding: A = 20000 × (1 + 0.10)2 = 20000 × 1.21 = 24,200.
Ans: Without = ₹24,000, With = ₹24,200
2. Bank of Wahapur offers an interest of 5% p.a. Compare how much one gets if one deposits ₹20,000 for a period of 4 years with compounding and without compounding annually.
Solution:
Formula Simple: A = P × (1 + r×t)
Formula Compound: A = P × (1 + r)t
Given Values:
• Principal (P) = ₹20,000
• Rate (r) = 5% = 0.05
• Time (t) = 4 years
Calculation:
Without Compounding (Simple Interest): A = 20000 × (1 + 0.05 × 4) = 20000 × 1.20 = 24,000.
With Compounding: A = 20000 × (1 + 0.05)4 = 20000 × 1.21550625 = 24,310.125.
Ans: Without = ₹24,000, With ≈ ₹24,310.13
3. Do you observe anything interesting in the solutions of the two questions above?
Solution:
Observation based on mathematical principles
Calculation / Analysis:
In Simple Interest, multiplying the rate by time (10% × 2 vs 5% × 4) yields the exact same total multiplier (20%). Both result in ₹24,000.
However, in Compound Interest, generating the same 20% base linear growth over a longer duration (4 years) allows the interest-on-interest mechanic more cycles to trigger, producing a higher final return (₹24,310.13 > ₹24,200).
Ans: Compounding over a longer duration produces a higher yield despite equivalent simple interest rates.
4. Jasmine invests amount 'p' for 4 years at an interest of 6% p.a. Which expression describes the total amount she will get after 4 years when compounding is not done?
Solution:
Formula: Amount = P + (P × r × t)
Given Values:
• Principal = p
• Rate (r) = 6% = 0.06
• Time (t) = 4 years
Calculation:
Amount = p + (p × 0.06 × 4)
Ans: (vii) p + (p × 0.06 × 4)
5. The post office offers an interest of 7% p.a. How much interest would one get if one invests ₹50,000 for 3 years without compounding? How much more would one get if it was compounded?
Solution:
Formula: SI = P × r × t
CI = [P × (1 + r)t] - P
Given Values:
• Principal (P) = ₹50,000
• Rate (r) = 7% = 0.07
• Time (t) = 3 years
Calculation:
Simple Interest (SI) = 50000 × 0.07 × 3 = 10,500.
Compound Amount = 50000 × (1 + 0.07)3 = 50000 × 1.225043 = 61,252.15.
Compound Interest (CI) = 61252.15 - 50000 = 11,252.15.
Difference (More gained) = 11,252.15 - 10,500 = 752.15.
Ans: Simple Interest = ₹10,500. Compounding yields ₹752.15 more.
6. Giridhar borrows a loan of ₹12,500 at 12% per annum for 3 years without compounding and Raghava borrows the same amount for the same time period at 10% per annum, compounded annually. Who pays more interest and by how much?
Solution:
Formula: SI = P × r × t
CI = [P × (1 + r)t] - P
Given Values:
• Principal (P) = ₹12,500
• Time (t) = 3 years
• Giridhar Rate = 12% (Simple)
• Raghava Rate = 10% (Compound)
Calculation:
Giridhar Interest = 12500 × 0.12 × 3 = 4,500.
Raghava Amount = 12500 × (1 + 0.10)3 = 12500 × 1.331 = 16,637.50.
Raghava Interest = 16637.50 - 12500 = 4,137.50.
Difference = 4500 - 4137.50 = 362.50.
Ans: Giridhar pays more by ₹362.50.
7. Consider an amount ₹1000. If this grows at 10% p.a., how long will it take to double when compounding is done vs. when compounding is not done? Is it linear vs exponential?
Solution:
Formula: Double SI (P × (1 + rt) = 2P)
Double CI (P × (1 + r)t = 2P)
Given Values:
• Principal (P) = 1000, Target = 2000
• Rate (r) = 10% = 0.10
Calculation:
Without Compounding (Simple): 1000 × (1 + 0.10t) = 2000 ⇒ 1 + 0.10t = 2 ⇒ 0.10t = 1 ⇒ t = 10 years.
With Compounding (Compound): 1000 × (1 + 0.10)t = 2000 ⇒ 1.10t = 2.
Using logarithms or estimation: 1.107 ≈ 1.948, 1.108 ≈ 2.143. Using rule of 72: 72/10 = 7.2 years. Exact t ≈ 7.27 years.
Simple interest charts a straight line (linear growth). Compound interest curves upwards (exponential growth).
Ans: 10 years (linear) vs ~7.27 years (exponential).
8. The population of a city is rising by about 3% every year. If the current population is 1.5 crore, what is the expected population after 3 years?
Solution:
Formula: Population = P × (1 + r)t
Given Values:
• Initial Population (P) = 1.5 crore
• Rate (r) = 3% = 0.03
• Time (t) = 3 years
Calculation:
Expected Pop = 1.5 × (1 + 0.03)3
Expected Pop = 1.5 × 1.092727
Expected Pop = 1.63909
Ans: Approximately 1.639 crore.
9. In a laboratory, the number of bacteria in a certain experiment increases at the rate of 2.5% per hour. Find the number of bacteria at the end of 2 hours if the initial count is 5,06,000.
Solution:
Formula: Final Count = Initial Count × (1 + r)t
Given Values:
• Initial Count (P) = 506,000
• Rate (r) = 2.5% = 0.025
• Time (t) = 2 hours
Calculation:
Final Count = 506000 × (1 + 0.025)2
Final Count = 506000 × (1.025)2
Final Count = 506000 × 1.050625
Final Count = 531,616.25
Ans: 531,616 bacteria

Figure it Out 5

1. The population of Bengaluru in 2025 is about 250% of its population in 2000. If the population in 2000 was 50 lakhs, what is the population in 2025?
Solution:
Formula: New Population = (Percentage / 100) × Old Population
Given Values:
• Old Population = 50 lakhs
• Percentage scaling = 250%
Calculation:
New Population = (250 / 100) × 50
New Population = 2.5 × 50
New Population = 125 lakhs
Ans: 125 lakhs (or 1.25 crore)
2. The population of the world in 2025 is about 8.2 billion. Match the countries with their approximate percentage share.
Solution:
Formula: Percentage = (Country Pop / World Pop) × 100%
Given Values:
• World Population = 8.2 billion = 8200 million
Calculation:
Germany (83m): (83 / 8200) × 100% = 1.01%. Rounds to 1%.
India (1460m): (1460 / 8200) × 100% = 17.80%. Rounds to 18%.
Bangladesh (175m): (175 / 8200) × 100% = 2.13%. Rounds to 2%.
Ans: Germany → 1%, India → 18%, Bangladesh → 2%
3. The price of a mobile phone is ₹8,250. A GST of 18% is added to the price. Which of the following gives the final price of the phone including the GST?
Solution:
Formula: Final Price = Base Price × (1 + GST % / 100)
Given Values:
• Base Price = ₹8250
• GST = 18%
Calculation:
Final Price = 8250 × (1 + 18 / 100) = 8250 × (1 + 0.18) = 8250 × 1.18.
Using distributive property: 8250 × 1 + 8250 × 0.18 = 8250 + 8250 × 0.18.
Ans: (v) 8250 × 1.18 and (vi) 8250 + 8250 × 0.18
4. The monthly percentage change in population of mice in a lab: Month 1 change +5%, Month 2 change -2%, Month 3 change -3%. Which statement(s) are true? The initial population is p.
Solution:
Formula: Final = Initial × (1 + r1) × (1 + r2) × (1 + r3)
Given Values:
• Initial Population = p
• Changes: +5%, -2%, -3%
Calculation:
Final = p × (1 + 0.05) × (1 - 0.02) × (1 - 0.03)
Final = p × 1.05 × 0.98 × 0.97. (This matches Statement ii).
Multiplying out: 1.05 × 0.98 × 0.97 = 0.99813.
Since 0.99813 < 1, the final population is less than p. (This matches Statement vi).
Ans: (ii) and (vi) are true.
5. A shopkeeper initially set the price of a product with a 35% profit margin. Due to poor sales, he decided to offer a 30% discount on the selling price. Will he make a profit or a loss?
Solution:
Formula: Marked Price = CP × (1 + Margin % / 100)
Final SP = Marked Price × (1 - Discount % / 100)
Given Values:
• Profit Margin = 35%
• Discount = 30%
Calculation:
Let Cost Price (CP) be 100.
Marked Price = 100 × (1 + 0.35) = 135.
Final SP = 135 × (1 - 0.30) = 135 × 0.70 = 94.5.
Final SP (94.5) < CP (100). The shopkeeper makes a loss.
Loss % = ((100 - 94.5) / 100) × 100% = 5.5%.
Ans: Loss of 5.5%.
6. What percentage of area is occupied by the region marked 'E' in the figure?
Solution:
Formula: Percentage = (Area of Region / Total Area) × 100%
Visual Analysis:
The square is split into 4 equal quadrants (25% each). Regions D and E together form one quadrant.
A diagonal line perfectly bisects this quadrant, creating two equal triangles, D and E.
Therefore, Region E = (1/2) × 25% = 12.5%.
Ans: 12.5%
7. What is 5% of 40? What is 40% of 5? ... What do you notice? Generalize algebraically.
Solution:
Formula: x% of y = (x / 100) × y
Calculation:
5% of 40 = (5 / 100) × 40 = 2.
40% of 5 = (40 / 100) × 5 = 2.
25% of 12 = 3. 12% of 25 = 3.
Algebraic Generalization:
LHS: x% of y = (x / 100) × y = xy / 100.
RHS: y% of x = (y / 100) × x = yx / 100.
Since xy = yx, the two are identical.
Ans: x% of y equals y% of x.
8. A school excursion: 40% are Grade 8 students. Among these, 60% are girls. (i) What percentage of the students going to the excursion are Grade 8 girls? (ii) If total is 160, how many are Grade 8 girls?
Solution:
Formula: Part of a Part = Percentage 1 × Percentage 2
Count = Percentage × Total
Given Values:
• Total Excursion = 100%
• Grade 8 = 40% of total
• Girls = 60% of Grade 8
Calculation:
(i) Percentage: 60% of 40% = (60 / 100) × (40 / 100) = 0.60 × 0.40 = 0.24 = 24%.
(ii) Count: Total = 160. Number of girls = 24% of 160 = 0.24 × 160 = 38.4. Round to nearest whole person = 38.
Ans: (i) 24% (ii) 38 girls
9. A shopkeeper sells pencils at a price such that the selling price of 3 pencils is equal to the cost of 5 pencils. Does he make a profit or a loss? What is his profit or loss percentage?
Solution:
Formula: Profit % = ((SP - CP) / CP) × 100%
Given Values:
• SP of 3 = CP of 5
Calculation:
3 × SP = 5 × CP
SP / CP = 5 / 3
Since SP (5 units) > CP (3 units), he makes a profit.
Profit % = ((5 - 3) / 3) × 100% = (2 / 3) × 100% = 66.666...%
Ans: Profit of 66.67%.
10. The bus fares were increased by 3% last year and by 4% this year. What is the overall percentage price increase in the last 2 years?
Solution:
Formula: Overall Multiplier = (1 + r1) × (1 + r2)
Given Values:
• Increase 1 = 3% = 0.03
• Increase 2 = 4% = 0.04
Calculation:
Multiplier = (1 + 0.03) × (1 + 0.04)
Multiplier = 1.03 × 1.04 = 1.0712
Overall increase % = (1.0712 - 1) × 100% = 0.0712 × 100% = 7.12%
Ans: Overall increase is 7.12%.
11. If the length of a rectangle is increased by 10% and the area is unchanged, by what percentage (exactly) does the breadth decrease by?
Solution:
Formula: Area = L × B. Constant Area requires Lnew × Bnew = L × B.
Given Values:
• Length increase = 10%
Calculation:
Lnew = L × (1 + 0.10) = 1.1L
Since Area is unchanged: 1.1L × Bnew = L × B
Bnew = B / 1.1 = (10 / 11)B
Decrease amount = B - (10 / 11)B = (1 / 11)B
Percentage Decrease = (1 / 11) × 100% = 100 / 11 % = 9 1/11 % ≈ 9.09%
Ans: Exactly 9 1/11% (approx 9.09%).
12. The percentage of ingredients in a 65 g chips packet is shown: Potato: 70%, Vegetable oil: 24%, Salt: 3%, Spices: 3%. Find out the weight each ingredient makes up in this packet.
Solution:
Formula: Ingredient Weight = (Percentage / 100) × Total Weight
Given Values:
• Total Weight = 65 g
Calculation:
Potato = (70 / 100) × 65 = 0.70 × 65 = 45.5 g.
Vegetable Oil = (24 / 100) × 65 = 0.24 × 65 = 15.6 g.
Salt = (3 / 100) × 65 = 0.03 × 65 = 1.95 g.
Spices = (3 / 100) × 65 = 0.03 × 65 = 1.95 g.
Ans: Potato 45.5g, Veg Oil 15.6g, Salt 1.95g, Spices 1.95g.
13. Three shops sell the same items. Shop A: "Buy 1 get 1 free". Shop B: "Buy 2 get 1 free". Shop C: "Buy 3 get 1 free". (i) If price=100, effective price? (ii) % discount? (iii) Which for 4 items?
Solution:
Formula: Effective Price = Total Cost Paid / Total Items Received
Discount % = (Free Items / Total Items) × 100%
Given Values:
• Base item price = 100
Calculation:
(i) Effective Price:
Shop A: Pay for 1, get 2. Total cost = 100. Effective = 100 / 2 = ₹50.
Shop B: Pay for 2, get 3. Total cost = 200. Effective = 200 / 3 = ₹66.67.
Shop C: Pay for 3, get 4. Total cost = 300. Effective = 300 / 4 = ₹75.
(ii) Percentage Discount:
Shop A = 1 free / 2 total = 50%.
Shop B = 1 free / 3 total = 33.33%.
Shop C = 1 free / 4 total = 25%.
(iii) Buying exactly 4 items:
Shop A: Buy 2, get 2 free = 4 items. Cost = ₹200.
Shop B: Buy 2, get 1 free (3 items) + Buy 1 full price = 4 items. Cost = 200 + 100 = ₹300.
Shop C: Buy 3, get 1 free = 4 items. Cost = ₹300.
Ans: Shop A is the best deal (50% discount, ₹200 total for 4).
14. In a room of 100 people, 99% are left-handed. How many left-handed people have to leave the room to bring that percentage down to 98%?
Solution:
Formula: Percentage = (Target Part / New Total) × 100%
Given Values:
• Initial Total = 100
• Initial Left-handed = 99% (99 people)
• Initial Right-handed = 1% (1 person)
Calculation:
When left-handed people leave, the number of right-handed people remains exactly 1.
We want the left-handed percentage to be 98%. This means the right-handed percentage must be 2%.
Therefore, 1 person = 2% of the New Total.
New Total = 1 / 0.02 = 50 people.
If the new total must be 50, then 100 - 50 = 50 people must have left the room.
Ans: 50 people must leave.
15. Look at the graph Ability to use computer by age and gender (2023). Based on the graph, which statement(s) are valid?
Solution:
Formula: Visual Data Interpretation
Analysis of Statements:
(i) Twenties most computer literate: True. The bars for Male (37%) and Female (26%) in the twenties bracket are the highest peaks on the graph.
(ii) Women lag behind across age groups: True. The Female percentage bar is visually shorter in every single age category presented.
(iii) More people in twenties than teenagers: False. The graph depicts percentage capability within demographics, not absolute total population counts.
(iv) More than a quarter in thirties: False. Male is 25%, Female is 14%. The aggregate across both genders is below 25%.
(v) Less than 1 in 10 seniors: True. Male 4%, Female 2% are both below the 10% (1 in 10) threshold.
(vi) Half of people in twenties: False. The rates are 37% and 26%, both of which fall short of 50%.
Ans: Statements (i), (ii), and (v) are valid.