1. Basic Percentage Calculations
Question 1:
\[
\text{What is 25% of 200?}
\]
\[
\text{(a) } 25, \quad \text{(b) } 50, \quad \text{(c) } 75, \quad \text{(d) } 100
\]
Answer: B
Step by Step Solution
Solution:
\[
\text{To find 25% of 200, we convert 25% to a fraction and multiply by 200.}
\]
\[
25\% = \frac{25}{100} = 0.25
\]
\[
0.25 \times 200 = 50
\]
\[
\text{Thus, 25% of 200 is } \boxed{50}
\]
Question 2:
\[
\text{60% of a number is 90. What is the number?}
\]
\[
\text{(a) } 120, \quad \text{(b) } 130, \quad \text{(c) } 150, \quad \text{(d) } 160
\]
Answer: C
Step by Step Solution
Solution:
\[
\text{Let the number be } x.
\]
\[
\text{60% of } x \text{ means } \frac{60}{100} \times x = 90
\]
\[
x = \frac{90 \times 100}{60}
\]
\[
x = 150
\]
\[
\text{Thus, the number is } \boxed{150}
\]
2. Percentage Increase & Decrease
Question 3:
\[
\text{If the price of a product increases from } \$50 \text{ to } \$65, \text{ what is the percentage increase?}
\]
\[
\text{(a) } 20\%, \quad \text{(b) } 25\%, \quad \text{(c) } 30\%, \quad \text{(d) } 35\%
\]
Answer: c
Step by Step Solution
Solution:
\[
\text{The formula for percentage increase is:}
\]
\[
\frac{\text{New Value} – \text{Old Value}}{\text{Old Value}} \times 100
\]
\[
\frac{65 – 50}{50} \times 100
\]
\[
\frac{15}{50} \times 100 = 30\%
\]
\[
\text{Thus, the percentage increase is } \boxed{30\%}
\]
Question 4:
\[
\text{A car’s value decreased from } \$20,000 \text{ to } \$18,000. \text{ What is the percentage decrease?}
\]
\[
\text{(a) } 5\%, \quad \text{(b) } 8\%, \quad \text{(c) } 9\%, \quad \text{(d) } 10\%
\]
Answer: D
Step by Step Solution
Solution:
\[
\text{The formula for percentage decrease is:}
\]
\[
\frac{\text{Old Value} – \text{New Value}}{\text{Old Value}} \times 100
\]
\[
\frac{20000 – 18000}{20000} \times 100
\]
\[
\frac{2000}{20000} \times 100 = 10\%
\]
\[
\text{Thus, the percentage decrease is } \boxed{10\%}
\]
3. Finding the Original Value
Question 5:
\[
\text{After a 10\% discount, a shirt costs } \$45. \text{ What was the original price?}
\]
\[
\text{(a) } \$50, \quad \text{(b) } \$55, \quad \text{(c) } \$60, \quad \text{(d) } \$65
\]
Answer: A
Step by Step Solution
Alternative Method:
### Step 1: Understand the Meaning of 10% Discount
\[
\text{A 10\% discount means the customer pays 90\% of the original price.}
\]
\[
\text{So, 90\% of the original price is } 45.
\]
### Step 2: Write as a Proportion
\[
90\% \text{ of Original Price} = 45
\]
\[
\frac{90}{100} \times x = 45
\]
### Step 3: Find the Value of \( x \)
\[
x = \frac{45 \times 100}{90}
\]
\[
x = \frac{4500}{90}
\]
\[
x = 50
\]
### Step 4: Final Answer
\[
\boxed{50}
\]
Question #6:
\[
\text{A product is sold for } \$80 \text{ after a 20\% markup. What was the original cost price?}
\]
\[
\text{(a) } \$60, \quad \text{(b) } \$65, \quad \text{(c) } \$66.67, \quad \text{(d) } \$70
\]
Answer: C
Step by Step Solution
Solution:
### Step 1: Understand the Meaning of 20% Markup
\[
\text{A 20\% markup means the selling price is 120\% of the original cost price.}
\]
\[
\text{So, 120\% of the original cost price is } 80.
\]
### Step 2: Write as a Proportion
\[
120\% \text{ of Cost Price} = 80
\]
\[
\frac{120}{100} \times x = 80
\]
### Step 3: Find the Value of \( x \)
\[
x = \frac{80 \times 100}{120}
\]
\[
x = \frac{8000}{120}
\]
\[
x = 66.67
\]
### Step 4: Final Answer
\[
\boxed{66.67}
\]
4. Successive Percentage Changes
Question 7:
\[
\text{A salary increases by 10\% and then decreases by 5\%. What is the overall percentage change?}
\]
\[
\text{(a) } 4.5\% \text{ increase}, \quad \text{(b) } 5\% \text{ increase}, \quad \text{(c) } 4\% \text{ increase}, \quad \text{(d) } 4.5\% \text{ decrease}
\]
Answer: A
Step by Step Solution
Solution:
### Step 1: Use the Percentage Change Formula
\[
\text{Overall Percentage Change} = a + b + \frac{a \times b}{100}
\]
where:
– \( a = 10\% \) (increase)
– \( b = -5\% \) (decrease)
### Step 2: Substitute the Values
\[
\text{Overall Change} = 10 + (-5) + \frac{10 \times (-5)}{100}
\]
\[
= 10 – 5 + \frac{-50}{100}
\]
\[
= 10 – 5 – 0.5
\]
\[
= 4.5
\]
### Step 3: Final Answer
\[
\boxed{4.5\% \text{ increase}}
\]
Question 8:
\[
\text{A population increases by 15\% in the first year and 10\% in the second year. What is the total increase?}
\]
\[
\text{(a) } 25\%, \quad \text{(b) } 26.5\%, \quad \text{(c) } 24\%, \quad \text{(d) } 27\%
\]
Answer: B
Step by Step Solution
Solution:
### Step 1: Use the Percentage Change Formula
\[
\text{Total Percentage Increase} = a + b + \frac{a \times b}{100}
\]
where:
– \( a = 15\% \) (increase in the first year)
– \( b = 10\% \) (increase in the second year)
### Step 2: Substitute the Values
\[
\text{Total Increase} = 15 + 10 + \frac{15 \times 10}{100}
\]
\[
= 15 + 10 + \frac{150}{100}
\]
\[
= 15 + 10 + 1.5
\]
\[
= 26.5
\]
### Step 3: Final Answer
\[
\boxed{26.5\%}
\]
5. Percentage Comparisons
Question 9:
\[
\text{If A is 40\% of B, and B is 60\% of C, what percentage of C is A?}
\]
\[
\text{(a) } 20\%, \quad \text{(b) } 24\%, \quad \text{(c) } 30\%, \quad \text{(d) } 36\%
\]
Answer: B
Step by Step Solution
Solution:
\[
\text{Given: } A = 40\% \text{ of } B \text{ and } B = 60\% \text{ of } C.
\]
\[
A = \frac{40}{100} \times B
\]
\[
B = \frac{60}{100} \times C
\]
\[
\text{Substituting } B \text{ in the equation for } A:
\]
\[
A = \frac{40}{100} \times \left( \frac{60}{100} \times C \right)
\]
\[
A = \frac{40 \times 60}{100 \times 100} \times C
\]
\[
A = \frac{2400}{10000} \times C = 0.24C
\]
\[
\text{Thus, } A \text{ is } \boxed{24\%} \text{ of } C.
\]
6. Profit, Loss & Discount Percentages
Question 10:
\[
\text{A shopkeeper buys a product for } \$200 \text{ and sells it for } \$250. \text{ What is the profit percentage?}
\]
\[
\text{(a) } 20\%, \quad \text{(b) } 22\%, \quad \text{(c) } 25\%, \quad \text{(d) } 30\%
\]
Answer: C
Step by Step Solution
Solution:
\[
\text{Profit} = \text{Selling Price} – \text{Cost Price}
\]
\[
= 250 – 200 = 50
\]
\[
\text{Profit Percentage} = \frac{\text{Profit}}{\text{Cost Price}} \times 100
\]
\[
= \frac{50}{200} \times 100
\]
\[
= 25\%
\]
\[
\text{Thus, the profit percentage is } \boxed{25\%}.
\]
Question 11:
\[
\text{If a product originally priced at } \$120 \text{ is sold at a 30\% discount, what is the final selling price?}
\]
\[
\text{(a) } \$80, \quad \text{(b) } \$84, \quad \text{(c) } \$90, \quad \text{(d) } \$95
\]
Answer: C
Step by Step Solution
Solution:
\[
\text{Discount} = \frac{30}{100} \times 120
\]
\[
= 0.30 \times 120 = 36
\]
\[
\text{Selling Price} = \text{Original Price} – \text{Discount}
\]
\[
= 120 – 36 = 84
\]
\[
\text{Thus, the final selling price is } \boxed{84}.
\]
7. Percentage in Data Interpretation
Question 12:
\[
\text{In a survey of 500 people, 30\% prefer tea, 50\% prefer coffee, and the rest prefer juice. How many people prefer juice?}
\]
\[
\text{(a) } 50, \quad \text{(b) } 100, \quad \text{(c) } 150, \quad \text{(d) } 200
\]
Answer: B
Step by Step Solution
Solution:
\[
\text{Total number of people} = 500
\]
\[
\text{People who prefer tea} = \frac{30}{100} \times 500 = 150
\]
\[
\text{People who prefer coffee} = \frac{50}{100} \times 500 = 250
\]
\[
\text{People who prefer juice} = \text{Total} – (\text{Tea} + \text{Coffee})
\]
\[
= 500 – (150 + 250)
\]
\[
= 500 – 400 = 100
\]
\[
\text{Thus, the number of people who prefer juice is } \boxed{100}.
\]
Question 13:
\[
\text{A company’s revenue increased from } \$1,200,000 \text{ to } \$1,500,000 \text{ in a year. What is the percentage increase?}
\]
\[
\text{(a) } 20\%, \quad \text{(b) } 22\%, \quad \text{(c) } 25\%, \quad \text{(d) } 30\%
\]
Answer: C
Step by Step Solution
Solution:
\[
\text{Percentage Increase} = \frac{\text{New Value} – \text{Old Value}}{\text{Old Value}} \times 100
\]
\[
= \frac{1,500,000 – 1,200,000}{1,200,000} \times 100
\]
\[
= \frac{300,000}{1,200,000} \times 100
\]
\[
= 25\%
\]
\[
\text{Thus, the percentage increase in revenue is } \boxed{25\%}.
\]
9. Probability and Percentage
Question 14:
\[
\text{If 75\% of students passed an exam and there are 240 students, how many students failed?}
\]
\[
\text{(a) } 40, \quad \text{(b) } 50, \quad \text{(c) } 60, \quad \text{(d) } 70
\]
Answer: C
Step by Step Solution
Solution:
\[
\text{Students who passed} = \frac{75}{100} \times 240
\]
\[
= 0.75 \times 240
\]
\[
= 180
\]
\[
\text{Students who failed} = \text{Total students} – \text{Students who passed}
\]
\[
= 240 – 180
\]
\[
= 60
\]
\[
\text{Thus, the number of students who failed is } \boxed{60}.
\]
Question 15:
\[
\text{A bag contains 20 balls, 30\% of which are red. How many red balls are there?}
\]
\[
\text{(a) } 4, \quad \text{(b) } 5, \quad \text{(c) } 6, \quad \text{(d) } 7
\]
Answer: C
Step by Step Solution
Solution:
\[
\text{Red balls} = \frac{30}{100} \times 20
\]
\[
= 0.30 \times 20
\]
\[
= 6
\]
\[
\text{Thus, the number of red balls is } \boxed{6}.
\]
