https://file.aiquickdraw.com/mathbot/file/17297107324900yfi5pa2.jpg

Let's break down the problem step by step based on the image.

### Problem Explanation:
Abraham has pictures, and the ratio of the height to the width of each picture is \( 7 \) inches to \( 4 \) inches.

#### Define variables:
- Let \( x \) represent the height of each picture (in inches).
- Let \( y \) represent the width of each picture (in inches).

### Step 1: Find the constant of proportionality
The ratio of height to width is given by \( \frac{7}{4} \). This is the constant of proportionality between height and width.

So, the constant of proportionality is **7/4**.

### Step 2: Write the proportion
Using the relationship between height and width:
\[
\frac{x}{y} = \frac{7}{4}
\]
This shows that the height \( x \) is proportional to the width \( y \) with a ratio of \( \frac{7}{4} \).

### Step 3: Rewrite the proportion as an equation
Now we want to solve for \( x \) (the height) in terms of \( y \) (the width). From the proportion:
\[
\frac{x}{y} = \frac{7}{4}
\]
Multiply both sides by \( y \) to isolate \( x \):
\[
x = \frac{7}{4} \cdot y
\]
This equation represents the height of each picture in terms of the width.

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Would you like more details on this solution, or do you have any other questions?

Here are five follow-up questions that could expand on this information:

1. How would you solve the proportion for \( y \) instead of \( x \)?
2. If the width of a picture is 12 inches, what is the height using the proportion?
3. Can you describe how this constant of proportionality would change if the ratio was 9:5 instead of 7:4?
4. How can the concept of proportionality help in scaling pictures for larger prints?
5. What is the significance of proportionality in real-world applications such as photography or design?

**Tip**: When working with proportions, always ensure that the units of measurement are consistent throughout the equation to avoid calculation errors.

Abraham had pictures taken for his football team. The pictures come in different sizes. The ratio of each picture is a height of 7 inches by a width of 4 inches. Define variables to represent the height and width, find the constant of proportionality, write a proportion, and solve for the height in terms of the width.

Learn how to solve ratio and proportion problems involving the constant of proportionality. Understand how to express the relationship between the height and width of pictures with a ratio of 7:4, and solve for height in terms of width. Suitable for students in Grades 6-8.

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