Quiz 1

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Direct and Inverse Proportions

Direct Proportion

Two quantities x and y are in direct proportion when they increase or decrease in the same ratio, written as:

\[ x \propto y \] \[ \frac{x}{y} = k \] where k is the constant of proportionality

Properties of Direct Proportion

• If x increases, y increases in the same ratio
• If x decreases, y decreases in the same ratio
• The ratio x/y remains constant

Inverse Proportion

Two quantities x and y are in inverse proportion when one increases as the other decreases in the same ratio, written as:

\[ x \propto \frac{1}{y} \] \[ xy = k \] where k is the constant of proportionality

Applications and Examples

Direct Proportion Examples

1. Distance and Time (Uniform Speed):

• Speed = 75 km/hour
• Distance in 20 minutes = (75 × 20)/60 = 25 km

2. Cost and Quantity:

• 5 meters cloth = ₹210
• 13 meters cloth = (210 × 13)/5 = ₹546

Inverse Proportion Examples

1. Workers and Time:

• 15 workers = 48 hours
• For 30 hours = (15 × 48)/30 = 24 workers

2. Food Supply and People:

• 100 students = 20 days
• 125 students = (100 × 20)/125 = 16 days

Key Formulas

Direct Proportion	Inverse Proportion
\[ \frac{x_1}{y_1} = \frac{x_2}{y_2} \]	\[ x_1y_1 = x_2y_2 \]

Practice Problems

Problem 1:

A train moves at 75 km/hour. How far will it travel in 20 minutes?

Answer: 25 km

Solution: \[ \text{Distance} = \frac{75 \text{ km}}{60 \text{ minutes}} \times 20 \text{ minutes} = 25 \text{ km} \]

Problem 2:

If 15 workers can build a wall in 48 hours, how many workers are needed to complete it in 30 hours?

Answer: 24 workers

Solution: \[ \text{Workers needed} = \frac{15 \times 48}{30} = 24 \text{ workers} \]

Problem 3:

The cost of 5 meters of cloth is ₹210. What is the cost of 13 meters?

Answer: ₹546

Solution: \[ \text{Cost} = \frac{210 \times 13}{5} = 546 \]

Problem 4:

A 14-meter pole casts a 10-meter shadow. What is the height of a tree that casts a 15-meter shadow?

Answer: 21 meters

Solution: \[ \text{Height} = \frac{14 \times 15}{10} = 21 \text{ meters} \]

Problem 5:

If x is inversely proportional to y and x = 15 when y = 6, find y when x = 9.

Answer: 10

Solution: \[ xy = k \therefore 15 \times 6 = 90 = k \] \[ \text{When } x = 9, y = \frac{90}{9} = 10 \]