CHAPTER 11 - DIRECT AND INVERSE PROPORTIONS

Proportion refers to the equality of two ratios. When two ratios are equal, we say they are in proportion.

\[ \text{If } \frac{a}{b} = \frac{c}{d}, \text{ then } a, b, c, d \text{ are in proportion} \]

Example: If 2:3 = 4:6, then these ratios are in proportion.

Proportions help us:

• Solve problems involving ratios and rates
• Understand relationships between quantities
• Make predictions based on known relationships
• Solve real-world problems in cooking, construction, finance, and science

Two quantities are in direct proportion if they increase or decrease at the same rate.

If quantity \(y\) is directly proportional to quantity \(x\), then:

\[ y = kx \]

Where \(k\) is the constant of proportionality.

Example: The cost of apples is directly proportional to the number of apples purchased. If 1 apple costs ₹10, then 5 apples cost ₹50.

• As one quantity increases, the other increases at the same rate
• As one quantity decreases, the other decreases at the same rate
• The ratio between corresponding values remains constant
• The graph of a direct proportion is a straight line through the origin

To solve direct proportion problems:

1. Identify that two quantities are directly proportional
2. Set up a proportion equation: \(\frac{y_1}{x_1} = \frac{y_2}{x_2}\)
3. Cross-multiply to solve for the unknown value

Example: If 5 pencils cost ₹25, how much will 8 pencils cost?

\[ \frac{25}{5} = \frac{x}{8} \Rightarrow 5 = \frac{x}{8} \Rightarrow x = 5 \times 8 = 40 \]

So, 8 pencils cost ₹40.

Two quantities are in inverse proportion if one increases while the other decreases at the same rate.

If quantity \(y\) is inversely proportional to quantity \(x\), then:

\[ y = \frac{k}{x} \]

Where \(k\) is the constant of proportionality.

Example: The time taken to complete a task is inversely proportional to the number of workers. If 4 workers take 6 hours, 8 workers would take 3 hours.

• As one quantity increases, the other decreases
• As one quantity decreases, the other increases
• The product of corresponding values remains constant
• The graph of an inverse proportion is a curve called a hyperbola

To solve inverse proportion problems:

1. Identify that two quantities are inversely proportional
2. Set up an equation: \(x_1 \times y_1 = x_2 \times y_2\)
3. Solve for the unknown value

Example: If 6 workers can build a wall in 10 days, how long will it take 15 workers?

\[ 6 \times 10 = 15 \times x \Rightarrow 60 = 15x \Rightarrow x = \frac{60}{15} = 4 \]

So, 15 workers can build the wall in 4 days.

Direct and inverse proportions are used in many real-world situations:

• Direct Proportion:
  • Cost and quantity of items
  • Distance traveled and time (at constant speed)
  • Work done and time (at constant rate)
• Inverse Proportion:
  • Speed and time (for a fixed distance)
  • Number of workers and time to complete a task
  • Brightness of light and distance from the source

Proportions are used in various fields:

• Cooking: Adjusting recipes for more or fewer people
• Construction: Calculating materials needed for projects
• Finance: Calculating interest, exchange rates
• Science: Boyle's Law (pressure and volume), Ohm's Law (voltage and current)
• Maps and scales: Converting between actual distances and map distances

In this chapter, you will learn:

• To identify direct and inverse proportions
• To solve problems involving direct proportion
• To solve problems involving inverse proportion
• To distinguish between direct and inverse proportions
• To apply proportional reasoning to real-world situations

This chapter includes exercises to help you practice:

• Identifying whether two quantities are in direct or inverse proportion
• Solving word problems using proportions
• Finding unknown values in proportional relationships
• Applying proportions to practical situations