Percentage

Q1: Convert each of the following into a fraction:

i. 68%

Step 1: Write percent as fraction over 100
68% = \(\frac{68}{100}\)
Step 2: Simplify the fraction
\(\frac{68}{100} = \frac{17}{25}\)
Answer: \(\frac{17}{25}\)

ii. \(3\frac{1}{3}\)%

Step 1: Convert mixed number to improper fraction
\(3\frac{1}{3} = \frac{10}{3}\)%
Step 2: Write percent as fraction over 100
\(\frac{10}{3}\)% = \(\frac{10}{3 \times 100} = \frac{10}{300}\)
Step 3: Simplify
\(\frac{10}{300} = \frac{1}{30}\)
Answer: \(\frac{1}{30}\)

iii. 224%

Step 1: Write percent as fraction over 100
224% = \(\frac{224}{100}\)
Step 2: Simplify
\(\frac{224}{100} = \frac{56}{25} = 2\frac{6}{25}\)
Answer: \(2 \frac{6}{25}\)

iv. 0.05%

Step 1: Write percent as fraction
0.05% = \(\frac{0.05}{100}\)
Step 2: Convert decimal to fraction
\(\frac{0.05}{100} = \frac{5}{100 \times 1000} = \frac{5}{10000} = \frac{1}{2000}\)
Answer: \(\frac{1}{2000}\)

Q2: Convert each of the following into a percentage:

i. \(\frac{2}{15}\)

Step 1: Multiply the fraction by 100
\(\frac{2}{15} \times 100 = \frac{200}{15}\)
Step 2: Simplify
\(\frac{200}{15} = \frac{40}{3} = 13 \frac{1}{3}\)%
Answer: \(13 \frac{1}{3}\)%

ii. \(\frac{9}{40}\)

Step 1: Multiply the fraction by 100
\(\frac{9}{40} \times 100 = \frac{900}{40}\)
Step 2: Simplify
\(\frac{900}{40} = \frac{45}{2} = 22 \frac{1}{2}\)%
Answer: \(22 \frac{1}{2}\)%

iii. \(1\frac{2}{3}\)

Step 1: Convert the mixed number to improper fraction
\(1\frac{2}{3} = \frac{5}{3}\)
Step 2: Multiply by 100
\(\frac{5}{3} \times 100 = \frac{500}{3}\)
Step 3: Simplify
\(\frac{500}{3} = 166 \frac{2}{3}\)%
Answer: \(166 \frac{2}{3}\)%

iv. \(2\frac{2}{5}\)

Step 1: Convert the mixed number to improper fraction
\(2\frac{2}{5} = \frac{12}{5}\)
Step 2: Multiply by 100
\(\frac{12}{5} \times 100 = \frac{1200}{5}\)
Step 3: Simplify
\(\frac{1200}{5} = 240\)%
Answer: 240%

Q3: Express each of the following ratios as a percentage:

i. 13 : 20

Step 1: Write the ratio as a fraction
\(\frac{13}{20}\)
Step 2: Multiply by 100 to convert into percentage
\(\frac{13}{20} \times 100 = \frac{1300}{20} = 65\%\)
Answer: 65%

ii. 11 : 18

Step 1: Write the ratio as a fraction
\(\frac{11}{18}\)
Step 2: Multiply by 100
\(\frac{11}{18} \times 100 = \frac{1100}{18} = \frac{550}{9} = 61 \frac{1}{9}\%\)
Answer: \(61 \frac{1}{9}\)%

iii. 87 : 25

Step 1: Write the ratio as a fraction
\(\frac{87}{25}\)
Step 2: Multiply by 100
\(\frac{87}{25} \times 100 = \frac{8700}{25} = 348\%\)
Answer: 348%

iv. \(6\frac{1}{4} : 4\frac{3}{8}\)

Step 1: Convert mixed numbers to improper fractions
\(6\frac{1}{4} = \frac{25}{4}, \quad 4\frac{3}{8} = \frac{35}{8}\)
Step 2: Write as a fraction
\(\frac{25}{4} \div \frac{35}{8} = \frac{25}{4} \times \frac{8}{35} = \frac{200}{140} = \frac{10}{7}\)
Step 3: Multiply by 100
\(\frac{10}{7} \times 100 = \frac{1000}{7} = 142 \frac{6}{7}\%\)
Answer: \(142 \frac{6}{7}\)%

Q4: Express each of the following decimals as a percentage:

i. 0.2

Step 1: Multiply the decimal by 100
0.2 × 100 = 20
Step 2: Attach the percent symbol
20%
Answer: 20%

ii. 0.06

Step 1: Multiply the decimal by 100
0.06 × 100 = 6
Step 2: Attach the percent symbol
6%
Answer: 6%

iii. 0.008

Step 1: Multiply the decimal by 100
0.008 × 100 = 0.8
Step 2: Attach the percent symbol
0.8%
Answer: 0.8%

iv. 2.4

Step 1: Multiply the decimal by 100
2.4 × 100 = 240
Step 2: Attach the percent symbol
240%
Answer: 240%

Q5: Express each of the following percentages as a decimal:

i. 25%

Step 1: Write percent as a fraction over 100
25% = \(\frac{25}{100}\)
Step 2: Convert to decimal
\(\frac{25}{100} = 0.25\)
Answer: 0.25

ii. 4%

Step 1: Write percent as a fraction over 100
4% = \(\frac{4}{100}\)
Step 2: Convert to decimal
\(\frac{4}{100} = 0.04\)
Answer: 0.04

iii. \(3\frac{1}{5}\)%

Step 1: Convert mixed number to improper fraction
\(3\frac{1}{5} = \frac{16}{5}\)%
Step 2: Write as a fraction over 100
\(\frac{16}{5} \times \frac{1}{100} = \frac{16}{500}\)
Step 3: Convert to decimal
\(\frac{16}{500} = 0.032\)
Answer: 0.032

iv. 0.3%

Step 1: Write percent as a fraction over 100
0.3% = \(\frac{0.3}{100}\)
Step 2: Convert to decimal
\(\frac{0.3}{100} = 0.003\)
Answer: 0.003

Q6: Express each of the following as a ratio:

i. 48%

Step 1: Write percentage as a fraction over 100
48% = \(\frac{48}{100}\)
Step 2: Express as a ratio
\(\frac{48}{100} = 48 : 100\)
Step 3: Simplify the ratio
48 : 100 = 12 : 25
Answer: 12 : 25

ii. \(26\frac{2}{3}\)%

Step 1: Convert mixed number to improper fraction
\(26\frac{2}{3} = \frac{80}{3}\)%
Step 2: Write as a fraction over 100
\(\frac{80}{3} \div 100 = \frac{80}{3 \times 100} = \frac{80}{300}\)
Step 3: Write as a ratio
\(\frac{80}{300} = 80 : 300 = 8 : 30 = 4 : 15\)
Answer: 4 : 15

iii. 0.06%

Step 1: Write as fraction over 100
0.06% = \(\frac{0.06}{100}\)
Step 2: Remove decimal by multiplying numerator and denominator by 100
\(\frac{0.06}{100} = \frac{6}{10000}\)
Step 3: Express as a ratio
6 : 10000 = 3 : 5000
Answer: 3 : 5000

iv. 120%

Step 1: Write percentage as a fraction over 100
120% = \(\frac{120}{100}\)
Step 2: Express as a ratio
120 : 100 = 6 : 5
Answer: 6 : 5

Q7: Find the value of:

i. 33% of ₹50

Step 1: Convert percent to fraction
33% = \(\frac{33}{100}\)
Step 2: Multiply with the given amount
\(\frac{33}{100} \times 50 = \frac{1650}{100} = 16.5\)
Answer: ₹16.50

ii. \(6\frac{2}{3}\)% of 3 m

Step 1: Convert mixed number to improper fraction
\(6\frac{2}{3} = \frac{20}{3}\)%
Step 2: Convert percent to fraction
\(\frac{20}{3} \div 100 = \frac{20}{300}\)
Step 3: Multiply with 3 m
\(\frac{20}{300} \times 3 = \frac{60}{300} = \frac{1}{5} = 0.2\)
Answer: 0.2 m

iii. 0.6% of 35 kg

Step 1: Convert percent to fraction
0.6% = \(\frac{0.6}{100} = \frac{6}{1000}\)
Step 2: Multiply with 35 kg
\(\frac{6}{1000} \times 35 = \frac{210}{1000} = 0.21\)
Answer: 0.21 kg

iv. \(3\frac{1}{4}\)% of 5 l

Step 1: Convert mixed number to improper fraction
\(3\frac{1}{4} = \frac{13}{4}\)%
Step 2: Convert percent to fraction
\(\frac{13}{4} \div 100 = \frac{13}{400}\)
Step 3: Multiply with 5 l
\(\frac{13}{400} \times 5 = \frac{65}{400} = \frac{13}{80} = 0.1625\)
Answer: 0.1625 l

Q8:

i. What per cent of ₹9 is ₹5?

Step 1: Use the formula: \[\text{Percentage} = \left(\frac{\text{Part}}{\text{Whole}}\right) \times 100\]Step 2: Apply values:
\(\frac{5}{9} \times 100 = \frac{500}{9} = 55\frac{5}{9}\)%
Answer: \( 55 \frac{5}{9}\)%

ii. What per cent of 32 m is 80 m?

Step 1: Use the formula: \[\frac{80}{32} \times 100\]Step 2: Simplify:
\(\frac{80}{32} = 2.5\), so \(2.5 \times 100 = 250\%\)
Answer: 250%

iii. What per cent of 50 kg is 65 kg?

Step 1: Apply formula:
\(\frac{65}{50} \times 100 = \frac{6500}{50} = 130\%\)
Answer: 130%

iv. What per cent of 5 litres is 400 ml?

Step 1: Convert 5 litres to millilitres:
5 litres = 5000 ml
Step 2: Apply percentage formula:
\(\frac{400}{5000} \times 100 = \frac{40000}{5000} = 8\%\)
Answer: 8%

Q9:

i. If 8% of a number is 24, find the number.

Step 1: Let the number be \(x\)
8% of \(x = 24\)
Step 2: Convert percentage to fraction
\(\frac{8}{100} \times x = 24\)
Step 3: Solve for \(x\)
\(x = \frac{24 \times 100}{8} = \frac{2400}{8} = 300\)
Answer: 300

ii. If 7.25% of a number is 2.9, find the number.

Step 1: Let the number be \(x\)
7.25% of \(x = 2.9\)
Step 2: Convert percentage to fraction
\(\frac{7.25}{100} \times x = 2.9\)
Step 3: Solve for \(x\)
\(x = \frac{2.9 \times 100}{7.25} = \frac{290}{7.25} = 40\)
Answer: 40

iii. If \(6\frac{2}{3}\)% of a number is 1, find the number.

Step 1: Convert mixed number to improper fraction
\(6\frac{2}{3} = \frac{20}{3}\)%
Step 2: Let the number be \(x\)
\(\frac{20}{3} \div 100 \times x = 1\)
\(\frac{20}{300} \times x = 1\)
Step 3: Solve for \(x\)
\(x = \frac{1 \times 300}{20} = \frac{300}{20} = 15\)
Answer: 15

Q10:

i. Increase 75 by 24%

Step 1: Find 24% of 75
\(\frac{24}{100} \times 75 = \frac{1800}{100} = 18\)
Step 2: Add the increase to original value
75 + 18 = 93
Answer: 93

ii. Decrease 375 by 8%

Step 1: Find 8% of 375
\(\frac{8}{100} \times 375 = \frac{3000}{100} = 30\)
Step 2: Subtract the decrease from original value
375 − 30 = 345
Answer: 345

Q11: What number when increased by 15% becomes 276?

Step 1: Let the required number be \(x\)
After increasing by 15%, the number becomes 276.
Step 2: Use the formula:
\(x + 15\%\text{ of }x = 276\)
\(\Rightarrow x + \frac{15}{100}x = 276\)
\(\Rightarrow x\left(1 + \frac{15}{100}\right) = 276\)
\(\Rightarrow x \times \frac{115}{100} = 276\)
Step 3: Solve for \(x\)
\(x = \frac{276 \times 100}{115} = \frac{27600}{115} = 240\)
Answer: 240

Q12: What number when decreased by 8% becomes 345?

Step 1: Let the required number be \(x\)
After decreasing by 8%, it becomes 345.
Step 2: Use the formula:
\(x – 8\%\text{ of }x = 345\)
\(\Rightarrow x – \frac{8}{100}x = 345\)
\(\Rightarrow x\left(1 – \frac{8}{100}\right) = 345\)
\(\Rightarrow x \times \frac{92}{100} = 345\)
Step 3: Solve for \(x\)
\(x = \frac{345 \times 100}{92} = \frac{34500}{92} = 375\)
Answer: 375