Problem Analysis:
The task involves solving a problem related to
direct and inverse proportion. The provided text explains the concepts of direct and inverse proportionality and provides an example involving the number of eggs (\( p \)) and the number of pancakes (\( n \)). The goal is to determine the constant of proportionality (\( k \)) and use it to solve for \( n \) when \( p = 12 \).
#### Key Information from the Image:
1.
Direct Proportionality: Two variables are directly proportional if one increases as the other increases, or decreases as the other decreases. Mathematically, this is expressed as:
\[
y \propto x \quad \text{or} \quad y = kx
\]
where \( k \) is the constant of proportionality.
2.
Inverse Proportionality: Two variables are inversely proportional if one increases as the other decreases, or vice versa. Mathematically, this is expressed as:
\[
y \propto \frac{1}{x} \quad \text{or} \quad y = \frac{k}{x}
\]
3.
Example Provided:
- Mary Jennings holds a recipe that uses 2 eggs to make 5 pancakes.
- We need to find the constant of proportionality (\( k \)) and use it to determine how many pancakes can be made with 12 eggs.
#### Steps to Solve the Problem:
---
Step 1: Identify the Relationship
From the problem, the number of pancakes (\( n \)) is directly proportional to the number of eggs (\( p \)). This means:
\[
n \propto p \quad \text{or} \quad n = kp
\]
where \( k \) is the constant of proportionality.
---
Step 2: Use the Given Data to Find \( k \)
We are given that 2 eggs (\( p = 2 \)) make 5 pancakes (\( n = 5 \)). Substitute these values into the equation \( n = kp \):
\[
5 = k \cdot 2
\]
Solve for \( k \):
\[
k = \frac{5}{2} = 2.5
\]
So, the constant of proportionality is:
\[
k = 2.5
\]
---
Step 3: Write the Equation Relating \( n \) and \( p \)
Using the value of \( k \), the equation becomes:
\[
n = 2.5p
\]
---
Step 4: Solve for \( n \) When \( p = 12 \)
Substitute \( p = 12 \) into the equation \( n = 2.5p \):
\[
n = 2.5 \cdot 12
\]
Calculate:
\[
n = 30
\]
---
Final Answer:
The number of pancakes that can be made with 12 eggs is:
\[
\boxed{30}
\]
---
Explanation:
1.
Direct Proportionality: Since the number of pancakes increases as the number of eggs increases, we used the formula \( n = kp \).
2.
Finding \( k \): Using the given data (\( p = 2 \) and \( n = 5 \)), we solved for \( k \) to get \( k = 2.5 \).
3.
General Equation: With \( k \) known, the relationship between \( n \) and \( p \) is \( n = 2.5p \).
4.
Final Calculation: Substituting \( p = 12 \) into the equation gave us \( n = 30 \).
This method ensures that the solution is consistent with the principles of direct proportionality.
Parent Tip: Review the logic above to help your child master the concept of inverse proportion worksheet.
