Understanding and Calculating Ratios

Understanding Ratios

• Ratios express the relationship between two numbers, indicating how much larger or smaller one is compared to the other
• Ratios can be used to compare quantities, determining which is larger or smaller
• The ratio is found when comparing the amounts of two items, like apples to oranges

Calculating Ratios: Apples vs. Oranges Example

• Initial setup involves writing the quantities with a ratio symbol: (20:5), where 20 is for apples and 5 for oranges
• The ratio can be written as a fraction, placing the first quantity (apples) over the second (oranges): 20/5
• Simplify the fraction by dividing both numbers by a common factor: 20/5 simplifies to 4/1 (both divisible by 5)
• Convert the simplified fraction back into a ratio: 4:1, signifying the apples are four times more abundant than oranges

Applying Ratios: Cycling vs. Scooter Speed

• A comparison between cycling speed (15 km/h) and scooter speed (30 km/h) will be used using ratios
• Write the ratio as cycling speed to scooter speed: 15:30
• The ratio is simplified by writing as a fraction: 15/30
• Simplify the fraction by dividing both numbers by a common factor: 1/2 (both divisible by 15)
• An answer shows the scooter is twice as fast as the cycle if simplified with a ratio of 1:2

Converting Units: Meters to Kilometers Ratio

• The task is to find the ratio of 5 meters to 10 kilometers
• The first step involves converting both values to the same unit
• Conversion of 10 kilometers to meters: 10 km * 1000 meters/km = 10,000 meters
• Write the ratio of meters to meters: 5:10,000
• Ratio written in fraction form: 5/10,000
• Simplify the fraction by dividing both numbers by a common factor: 1/2000 (both divisible by 5)
• The answer is equivalent to the equivalent rato, expressed as a ratio: 1:2000

Calculating Ratios: Paise vs. Rupees

• Ratio is 50 paise to 5 rupees
• Convert rupees to paise: 5 rupees * 100 paise/rupee = 500 paise
• Write the ratio of paise to paise: 50:500
• Writte the ratio in simplified fraction form: 50/500
• Simplify the fraction by dividing both numbers by a common factor: 1/10 (both divisible by 50)
• Equivalent ratio is: 1/10 converted to 1:10

Converting Ratios to Percentages

• Percentage means "out of 100," denoted by the symbol %
• Formula to calculate percentage: Multiply the fraction by 100

Converting 3:4 to a Percentage

• Given ratio: 3:4, converts to fraction 3/4
• Formula application: ((3/4) \times 100 = 75%)
• Answer is that a faction in an equivalent percentage is: (75%)

Converting 2:3 to a Percentage

• Given ratio: 2:3 converts to 2/3
• Calculation: ((2/3) \times 100 = 66.66%), rounded to (66 \frac{2}{3}%)
• Another fraction is a simplified version of its equivalent: (66 \frac{2}{3}%)

Applying Percentages: Student Interest in Mathematics

• From a group of 25 students, 72% show interest in mathematics
• Goal involves calculation of the percentage of students not interested in mathematics
• Calculation: (100% - 72% = 28%)
• Statement: In this case, 28% are not interested in mathematics

Determining Number of Students Not Interested

• With 28% of 25 students not interested, the calculation is ((28/100) \times 25 = 7)
• 7 students are not as interested in Mathematics as others are

Calculating Total Matches Played by a Football Team

• The team won 10 matches, representing a 40% win rate
• The goal is to determine the total number of matches played
• Let (x) be the total matches, then (0.40x = 10)
• Calculation: (x = 10 / 0.40 = 25)
• Interpretation: The team played a total of 25 matches

Calculating Initial Amount with Chamelee

• Chamelee spent 75% of her money and has ₹600 remaining
• Remaining amount represents 25% of her initial sum
• Let (y) be the initial amount, then (0.25y = 600)
• Calculation: (y = 600 / 0.25 = 2400)
• Initial amount with Chamelee: ₹2400

Calculating Sports Preferences in a City

• 60% prefer cricket, 30% football, and the remainder prefer other sports
• City population: 50 lakh people
• Goal: Determine the number of people preferring each sport

Percentage Preference of Cricket

• Number of cricket fans: (0.60 \times 50 \text{ lakh} = 30 \text{ lakh})

Percentage Preference of Football

• Number of football fans: (0.30 \times 50 \text{ lakh} = 15 \text{ lakh})

Calculating Percentage Preference of Other Sports

• Number of people preferring other sports: (10% \newline)Total ( = 100% - (60% + 30%) = 10%)
• Population: (0.10 \times 50 \text{ lakh} = 5 \text{ lakh})