Percent and Percentage

Q1: Multiple Choice Type

i. A bag contains 8 white and 12 black balls, the percentage of black balls in the bag is:

Step 1: Total number of balls = 8 white + 12 black = 20
Step 2: Black balls = 12
Step 3: Percentage of black balls = (12 / 20) × 100 = 60%
Answer: c. 60%

ii. A number is first increased by 20%, then the resulting number is decreased by 20%. On the whole the original number is increased/decreased by:

Step 1: Let original number = 100
Step 2: After 20% increase → 100 + 20% of 100 = 100 + 20 = 120
Step 3: After 20% decrease → 120 – 20% of 120 = 120 – 24 = 96
Step 4: Final value = 96, original = 100 ⇒ decrease of 4
Step 5: % Decrease = (4 / 100) × 100 = 4%
Answer: b. 4% decreased

iii. After paying 20% of the income, a man is left with ₹160; then the income of the man is:

Step 1: Let income = ₹x
Step 2: Paid 20%, so remaining = 80% of x = ₹160
Step 3: (80 / 100) × x = 160 ⇒ x = (160 × 100) / 80 = ₹200
Answer: c. ₹200

iv. If A + B + C = 400 in which A is 40% and B is 45%, then the exact quantity of C in the whole is:

Step 1: A = 40% of 400 = (40/100) × 400 = 160
Step 2: B = 45% of 400 = (45/100) × 400 = 180
Step 3: C = 400 – (A + B) = 400 – (160 + 180) = 60
Answer: b. 60

v. A number is decreased by 20%. if the resulting number is 800, the original number is:

Step 1: Let original number = x
Step 2: Decreased by 20%, so new number = 80% of x
Step 3: (80/100) × x = 800 ⇒ x = (800 × 100) / 80 = 1000
Answer: c. 1000

Q2: Evaluate

i. 55% of 160 + 24% of 50 − 36% of 150

Step 1: Convert percentages to fractions and multiply:
55% of 160 = (55 / 100) × 160 = 88
24% of 50 = (24 / 100) × 50 = 12
36% of 150 = (36 / 100) × 150 = 54
Step 2: Add and subtract the values:
= 88 + 12 − 54
= 100 − 54
= 46
Answer: 46

ii. 9.3% of 500 − 4.8% of 250 − 2.5% of 240

Step 1: Convert percentages to fractions and multiply:
9.3% of 500 = (9.3 / 100) × 500 = 46.5
4.8% of 250 = (4.8 / 100) × 250 = 12
2.5% of 240 = (2.5 / 100) × 240 = 6
Step 2: Perform the operations:
= 46.5 − 12 − 6
= 34.5 − 6
= 28.5
Answer: 28.5

Q3:

i. A number is increased from 125 to 150; find the percentage increase.

Step 1: Increase = New value − Original value = 150 − 125 = 25
Step 2: Percentage increase = (Increase / Original value) × 100
= (25 / 125) × 100
= 20%
Answer: 20%

ii. A number is decreased from 125 to 100; find the percentage decrease.

Step 1: Decrease = Original value − New value = 125 − 100 = 25
Step 2: Percentage decrease = (Decrease / Original value) × 100
= (25 / 125) × 100
= 20%
Answer: 20%

Q4: Find:

i. 45 is what percent of 54?

Step 1: Use the formula:
Percentage = (Part / Whole) × 100
Step 2: Part = 45, Whole = 54
Percentage = (45 / 54) × 100
Step 3: Simplify:
\( \frac{5}{6} \times 100 = \frac{250}{3} = 83 \frac{1}{3}\) %
Answer: \(83 \frac{1}{3}\)%

ii. 2.7 is what percent of 18?

Step 1: Use the same formula:
Percentage = (Part / Whole) × 100
Step 2: Part = 2.7, Whole = 18
= (2.7 / 18) × 100
Step 3: Simplify:
= 0.15 × 100 = 15%
Answer: 15%

Q5:

i. 252 is 35% of a certain number. Find the number.

Step 1: Let the number be x.
According to the question: 35% of x = 252
Step 2: Convert percentage to fraction:
(35 / 100) × x = 252
Step 3: Solve for x:
x = (252 × 100) / 35
x = 25200 / 35
x = 720
Answer: 720

ii. If 14% of a number is 315, find the number.

Step 1: Let the number be x.
14% of x = 315
Step 2: Convert percentage to fraction:
(14 / 100) × x = 315
Step 3: Solve for x:
x = (315 × 100) / 14
x = 31500 / 14
x = 2250
Answer: 2250

Q6: Find the percentage change, when a number is changed from:

i. 80 to 100

Step 1: Change = 100 − 80 = 20
Step 2: Percentage change = (Change / Original) × 100
= (20 / 80) × 100
= 25%
Answer: 25% increase

ii. 100 to 80

Step 1: Change = 100 − 80 = 20
Step 2: Percentage change = (20 / 100) × 100
= 20%
Answer: 20% decrease

iii. 6.25 to 7.50

Step 1: Change = 7.50 − 6.25 = 1.25
Step 2: Percentage change = (1.25 / 6.25) × 100
= (125 / 625) × 100
= 0.2 × 100 = 20%
Answer: 20% increase

Q7: An auctioneer charges 8% for selling a house. If the house is sold for ₹2,30,500. Find the charges of the auctioneer.

Step 1: Selling price of the house = ₹2,30,500
Step 2: Commission rate = 8%
Step 3: Auctioneer’s charges = 8% of ₹2,30,500
= (8 / 100) × 230500
Step 4: Calculate:
= ₹(230500 × 8) / 100
= ₹1844000 / 100
= ₹18,440
Answer: ₹18,440

Q8: Out of 800 oranges, 50 are found rotten. Find the percentage of good oranges.

Step 1: Total number of oranges = 800
Step 2: Number of rotten oranges = 50
Step 3: Number of good oranges = 800 − 50 = 750
Step 4: Percentage of good oranges = (750 / 800) × 100
Step 5: Simplify:
\( = \frac{15}{16} \times 100 = \frac{375}{4} = 93 \frac{3}{4}\) %
Answer: \(93 \frac{3}{4}\) %

Q9: A cistern contains 5 thousand litres of water. If 6% water is leaked, find how many litres would be left in the cistern.

Step 1: Total water in the cistern = 5000 litres
Step 2: Percentage of water leaked = 6%
Step 3: Amount of water leaked = (6 / 100) × 5000 = 300 litres
Step 4: Water left in the cistern = 5000 − 300 = 4700 litres
Answer: 4700 litres

Q10: A man spends 87% of his salary. If he saves ₹325, find his salary.

Step 1: Let the total salary be ₹x.
Step 2: The man spends 87% of his salary, so savings = 100% − 87% = 13% of salary.
Step 3: According to the question:
13% of x = 325
(13 / 100) × x = 325
Step 4: Multiply both sides by 100:
13x = 32500
Step 5: Divide both sides by 13:
x = 32500 ÷ 13 = 2500
Answer: ₹2500

Q11:

i. A number 3.625 is wrongly read as 3.265; find the percentage error.

Step 1: True value = 3.625
Step 2: Measured (wrong) value = 3.265
Step 3: Error = True value − Measured value = 3.625 − 3.265 = 0.36
Step 4: Percentage error = (Error / True value) × 100
= (0.36 / 3.625) × 100
Step 5: Calculate:
≈ 9.93% (rounded to two decimal places)
Answer: 9.93%

ii. A number \(5.78 \times 10^3\) is wrongly written as \(5.87 \times 10^3\); find the percentage error.

Step 1: True value = \(5.78 \times 10^3 = 5780\)
Step 2: Measured value = \(5.87 \times 10^3 = 5870\)
Step 3: Error = Measured value − True value = 5870 − 5780 = 90
Step 4: Percentage error = (Error / True value) × 100
= (90 / 5780) × 100
Step 5: Calculate:
≈ 1.56% (rounded to two decimal places)
Answer: 1.56%

Q12: In an election between two candidates, one candidate secured 58% of the votes polled and won the election by 18,336 votes. Find the total number of votes polled and the votes secured by each candidate.

Step 1: Let total votes polled be x.
Step 2: Winner got 58% of votes = (58 / 100) × x = 0.58x
Loser got (100% − 58%) = 42% of votes = 0.42x
Step 3: Difference in votes = 0.58x − 0.42x = 0.16x
Given: 0.16x = 18336
Step 4: Solve for x:
x = 18336 ÷ 0.16 = 114600
Step 5: Total votes polled = 114600
Votes secured by winning candidate = 58% of 114600 = (58 / 100) × 114600 = 66468
Votes secured by losing candidate = 114600 − 66468 = 48132
Answer: Total votes = 114600
Winning candidate = 66468 votes
Losing candidate = 48132 votes

Q13: In an election between two candidates, one candidate secured 47% of the votes polled and lost the election by 12,366 votes. Find the total number of votes polled and the votes secured by the winning candidate.

Step 1: Let total votes polled be x.
Step 2: Loser got 47% of x = (47 / 100) × x = 0.47x
Winner got 53% of x = (53 / 100) × x = 0.53x
Step 3: Difference in votes = 0.53x − 0.47x = 0.06x
Given: 0.06x = 12366
Step 4: Solve for x:
x = 12366 ÷ 0.06 = 206100
Step 5: Total votes polled = 206100
Votes secured by winning candidate = 53% of 206100 = (53 / 100) × 206100 = 109233
Answer: Total votes = 206100
Winning candidate = 109233 votes

Q14: The cost of a scooter depreciates every year by 15% of its value at the beginning of the year. If the present cost of the scooter is ₹8,000, find the cost:

i. After one year

Step 1: Present cost = ₹8000
Step 2: Depreciation rate = 15%
Step 3: Depreciated value after 1 year = 85% of ₹8000
= (85 / 100) × 8000
= ₹6800
Answer: ₹6800

ii. After two years

Step 1: Cost after 1st year = ₹6800
Step 2: Again depreciate by 15%:
= 85% of ₹6800 = (85 / 100) × 6800
= ₹5780
Answer: ₹5780

Q15: In an examination, the pass mark is 40%. If a candidate gets 65 marks and fails by 3 marks; find the maximum marks.

Step 1: The candidate got 65 marks but failed by 3 marks.
Step 2: So, the pass mark = 65 + 3 = 68
Step 3: Let the maximum marks be x.
Given: 40% of x = 68
Step 4: Convert percentage and solve:
(40 / 100) × x = 68
x = (68 × 100) / 40
x = 6800 / 40
x = 170
Answer: Maximum Marks = 170

Q16: In an examination, a candidate secures 125 marks and fails by 15 marks. If the pass percentage was 35%, find the maximum marks.

Step 1: The candidate scored 125 marks and failed by 15 marks.
Step 2: So, the pass mark = 125 + 15 = 140
Step 3: Let the maximum marks be x.
Given: 35% of x = 140
Step 4: Convert percentage and solve:
(35 / 100) × x = 140
x = (140 × 100) / 35
x = 14000 / 35
x = 400
Answer: Maximum Marks = 400

Q17: In an objective type paper of 150 questions, John got 80% correct answers and Mohan got 64% correct answers.

i. How many correct answers did each get?

Step 1: Total number of questions = 150
Step 2: John’s correct answers = 80% of 150 = (80 / 100) × 150
= 120
Step 3: Mohan’s correct answers = 64% of 150 = (64 / 100) × 150
= 96
Answer: John = 120 correct answers
Mohan = 96 correct answers

ii. What percent is Mohan’s correct answers to John’s correct answers?

Step 1: Mohan’s correct = 96
John’s correct = 120
Step 2: Required percentage = (Mohan / John) × 100
= (96 / 120) × 100
= 80%
Answer: 80%

Q18: The number 8,000 is first increased by 20% and then decreased by 20%. Find the resulting number.

Step 1: Original number = 8000
Step 2: Increase it by 20%:
= 8000 + (20 / 100) × 8000
= 8000 + 1600 = 9600
Step 3: Now decrease 9600 by 20%:
= 9600 − (20 / 100) × 9600
= 9600 − 1920 = 7680
Answer: Resulting number = 7680

Q19: The number 12,000 is first decreased by 25% and then increased by 25%. Find the resulting number.

Step 1: Original number = 12000
Step 2: First decrease by 25%:
= 12000 − (25 / 100) × 12000
= 12000 − 3000 = 9000
Step 3: Now increase 9000 by 25%:
= 9000 + (25 / 100) × 9000
= 9000 + 2250 = 11250
Answer: Resulting number = 11250

Q20: The cost of an article is first increased by 20% and then decreased by 30%. Find the percentage change in the cost of the article.

Step 1: Let the original cost be ₹100 (for simplicity).
Step 2: Increase by 20%:
New cost = 100 + (20 / 100) × 100 = 100 + 20 = ₹120
Step 3: Decrease this new cost by 30%:
Final cost = 120 − (30 / 100) × 120 = 120 − 36 = ₹84
Step 4: Calculate percentage change:
Change = Final cost − Original cost = 84 − 100 = −16
Percentage change = (Change / Original cost) × 100 = (−16 / 100) × 100 = −16%
Answer: The cost decreases by 16%

Q21: The cost of an article is first decreased by 25% and then further decreased by 40%. Find the percentage change in the cost of the article.

Step 1: Let the original cost be ₹100.
Step 2: First decrease by 25%:
New cost = 100 − (25 / 100) × 100 = 100 − 25 = ₹75
Step 3: Further decrease by 40%:
Final cost = 75 − (40 / 100) × 75 = 75 − 30 = ₹45
Step 4: Calculate overall percentage change:
Change = Final cost − Original cost = 45 − 100 = −55
Percentage change = (Change / Original cost) × 100 = (−55 / 100) × 100 = −55%
Answer: The cost decreases by 55%