Learn on PengiAoPS: Introduction to Algebra (AMC 8 & 10)Chapter 7: Proportion

Lesson 2: Inverse Proportion

In this Grade 4 AoPS Introduction to Algebra lesson, students learn the concept of inverse proportion, where two quantities have a constant product rather than a constant quotient. Using worked problems from Chapter 7, students practice finding unknown values when quantities are inversely proportional, including cases involving higher powers like w² and z³, and real-world scenarios such as workers and time. The lesson also introduces a proof connecting inverse and direct proportion, building algebraic reasoning skills aligned with AMC 8 and AMC 10 preparation.

Section 1

Definition of Inverse Proportion

Property

$y$ varies inversely with $x$ if

y = \frac{k}{x}, x \neq 0

where $k$ is a positive constant. This relationship implies that the product of the variables is constant: $xy = k$ .

Section 2

Finding the Constant of Proportionality

Property

If you know that two variables vary inversely and have one corresponding pair of values, you can find the constant of variation, $k$ . Given a point $(x_1, y_1)$ , you can calculate $k = x_1 y_1$ and write the specific formula $y = \frac{k}{x}$ .

Examples

The current, $I$ , in a circuit varies inversely with resistance, $R$ . An iron with 12 ohms of resistance draws 10 amps. First, find $k = I \cdot R = 10 \cdot 12 = 120$ . The formula is $I = \frac{120}{R}$ .

Using the formula $I = \frac{120}{R}$ , how much current is drawn by a device with 20 ohms of resistance? Substitute $R=20$ to get $I = \frac{120}{20} = 6$ amps.

Lesson overview

Expand to review the lesson summary and core properties.

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Section 1

Definition of Inverse Proportion

Property

$y$ varies inversely with $x$ if

y = \frac{k}{x}, x \neq 0

where $k$ is a positive constant. This relationship implies that the product of the variables is constant: $xy = k$ .

Section 2

Finding the Constant of Proportionality

Property

Examples

The current, $I$ , in a circuit varies inversely with resistance, $R$ . An iron with 12 ohms of resistance draws 10 amps. First, find $k = I \cdot R = 10 \cdot 12 = 120$ . The formula is $I = \frac{120}{R}$ .

Using the formula $I = \frac{120}{R}$ , how much current is drawn by a device with 20 ohms of resistance? Substitute $R=20$ to get $I = \frac{120}{20} = 6$ amps.

Overview

Lesson overview

Review the lesson summary and core properties without leaving the current page.

Overview

Section 1

Definition of Inverse Proportion

Property

$y$ varies inversely with $x$ if

y = \frac{k}{x}, x \neq 0

where $k$ is a positive constant. This relationship implies that the product of the variables is constant: $xy = k$ .

Section 2

Finding the Constant of Proportionality

Property

Examples

The current, $I$ , in a circuit varies inversely with resistance, $R$ . An iron with 12 ohms of resistance draws 10 amps. First, find $k = I \cdot R = 10 \cdot 12 = 120$ . The formula is $I = \frac{120}{R}$ .

Using the formula $I = \frac{120}{R}$ , how much current is drawn by a device with 20 ohms of resistance? Substitute $R=20$ to get $I = \frac{120}{20} = 6$ amps.