1. Find the ratio of the following.
(a) Speed of a cycle 15 km per hour to the speed of scooter 30 km per hour.
(b) 5 m to 10 km
(c) 50 paise to ₹5
(a) Speed of cycle 15km : Speed of scooter 30 km
\[ \frac{15}{30} = \frac{15^{15 \div 15}}{30^{30 \div 15}} = \frac{1}{2} = 1:2 \]
∴ The ratio is 1:2
(b) 5 m to 10 km
First, convert both measurements to the same unit (meters):
10 km = 10 × 1000 = 10,000 m
\[ \frac{5}{10000} = \frac{5^{5 \div 5}}{10000^{10000 \div 5}} = \frac{1}{2000} = 1:2000 \]
∴ The ratio is 1:2000
(c) 50 paise to ₹5
First, convert both amounts to the same unit (paise):
₹5 = 5 × 100 = 500 paise
\[ \frac{50}{500} = \frac{50^{50 \div 50}}{500^{500 \div 50}} = \frac{1}{10} = 1:10 \]
∴ The ratio is 1:10
2. Convert the following ratios to percentages.
(a) 3 : 4
(b) 2 : 3
(a) 3 : 4
\[ \frac{3}{4} \times 100\% = \frac{300}{4}\% = 75\% \]
∴ 3 : 4 = 75%
(b) 2 : 3
\[ \frac{2}{3} \times 100\% = \frac{200}{3}\% = 66.66 \% = 66\frac{2}{3}\% \]
∴ 2 : 3 = \(66\frac{2}{3}\)%
3. 72% of 25 students are interested in mathematics. How many are not interested in mathematics?
Step 1: Find number of students interested in mathematics
72% of 25 = \(\frac{72}{100} \times 25 = \frac{72 \times 25}{100} = \frac{1800}{100} = 18\)
Step 2: Find number of students not interested in mathematics
Total students = 25
Students not interested = Total students - Students interested
= 25 - 18 = 7
∴ 7 students are not interested in mathematics.
4. A football team won 10 matches out of the total number of matches they played. If their win percentage was 40%, then how many matches did they play in all?
Step 1: Set up the equation
Let the total number of matches be = \( x \)
Win percentage = \(\frac{\text{Matches won}}{\text{Total matches}} \times 100\)
\[40 = \frac{10}{x} \times 100\]
Step 2: Solve for x
\[40 = \frac{1000}{x}\]
\[40x = 1000\]
\[x = \frac{1000}{40} = 25\]
∴ The team played 25 matches in all.
5. If Chameli had ₹600 left after spending 75% of her money, how much did she have in the beginning?
Step 1: Understand what remains
If she spent 75%, then she has 100% - 75% = 25% left
This 25% equals ₹600
Step 2: Find the original amount
Let the original amount be = \(x \)
\[25\% \text{ of } x = 600\]
\[\frac{25}{100} \times x = 600\]
\[\frac{1}{4}x = 600\]
\[x = 600 \times 4 = 2400\]
∴ Chameli had ₹2400 in the beginning.
6. If 60% people in a city like cricket, 30% like football and the remaining like other games, then what per cent of the people like other games? If the total number of people is 50 lakh, find the exact number who like each type of game.
Step 1: Find percentage who like other games
Total percentage = 100%
Percentage liking other games = 100% - (60% + 30%) = 100% - 90% = 10%
Step 2: Find exact numbers (total = 50,00,000 people)
Cricket lovers = 60% of 50,00,000 = \(\frac{60}{100} \times 50,00,000 = 30,00,000\)
Football lovers = 30% of 50,00,000 = \(\frac{30}{100} \times 50,00,000 = 15,00,000\)
Other games lovers = 10% of 50,00,000 = \(\frac{10}{100} \times 50,00,000 = 5,00,000\)
∴ 10% like other games.
Cricket: 30,00,000 people
Football: 15,00,000 people
Other games: 5,00,000 people