Chain Rule
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Table of Content:
Chain Rule Formulas
Chain Rule Basics
The chain rule is used to solve problems involving multiple quantities that are related to each other.
Direct Proportion
If two quantities \( A \) and \( B \) are directly proportional, then:
\[
\frac{A_1}{B_1} = \frac{A_2}{B_2}
\]
Indirect (Inverse) Proportion
If two quantities \( A \) and \( B \) are inversely proportional, then:
\[
A_1 \times B_1 = A_2 \times B_2
\]
Combined Proportion
For combined proportion, where a quantity \( Q \) depends on multiple directly or inversely proportional quantities:
\[
Q \propto \frac{A \times B}{C \times D}
\]
where \( A \) and \( B \) are directly proportional to \( Q \), and \( C \) and \( D \) are inversely proportional to \( Q \).
Problem-Solving Using Chain Rule
Steps to solve problems using the chain rule:
- Identify the quantities and their relationships (direct or inverse).
- Set up the proportion equations based on the relationships.
- Solve the equations to find the required quantity.
Example Problem
Suppose \( A \) varies directly as \( B \) and inversely as \( C \). If \( A_1 = 6 \) when \( B_1 = 4 \) and \( C_1 = 2 \), find \( A_2 \) when \( B_2 = 8 \) and \( C_2 = 3 \).
\[
\frac{A_1}{A_2} = \frac{B_1 \times C_2}{B_2 \times C_1}
\]
\[
\frac{6}{A_2} = \frac{4 \times 3}{8 \times 2}
\]
\[
\frac{6}{A_2} = \frac{12}{16} = \frac{3}{4}
\]
\[
A_2 = 6 \times \frac{4}{3} = 8
\]