Brandon wants to reduce a figure that is 9 inches tall and 16 inches wide so that it will fit on a 9-inch wide by 12-inch tall piece of paper. If he reduces the figure proportionally, what is the maximum size the reduced figure could measure?

To solve this, we need to maintain the proportional dimensions of the original figure while ensuring it fits within the new dimensions (9 inches wide by 12 inches tall).

### Step 1: Find the aspect ratio of the original figure
The original figure has dimensions:
- Height: 9 inches
- Width: 16 inches

The aspect ratio of the original figure is:
\[
\text{Aspect ratio} = \frac{\text{Width}}{\text{Height}} = \frac{16}{9}
\]

### Step 2: Fit the figure into the new dimensions
The new paper dimensions are 9 inches wide by 12 inches tall. Since we want to reduce the figure proportionally, we need to check how it fits both width-wise and height-wise:

#### Fit based on width:
Since the width of the new space is 9 inches, we scale the width of the original figure down to 9 inches and find the corresponding height. The scaling factor based on width is:
\[
\text{Scaling factor} = \frac{\text{New width}}{\text{Original width}} = \frac{9}{16}
\]
The corresponding height would be:
\[
\text{New height} = \text{Original height} \times \text{Scaling factor} = 9 \times \frac{9}{16} = 5.0625 \, \text{inches}
\]
So, if we scale by width, the reduced figure would measure **9 inches wide by 5.0625 inches tall**.

#### Fit based on height:
Next, we check if scaling by height would fit. The height of the new space is 12 inches, so we scale the height of the original figure down to 12 inches and find the corresponding width. The scaling factor based on height is:
\[
\text{Scaling factor} = \frac{\text{New height}}{\text{Original height}} = \frac{12}{9} = \frac{4}{3}
\]
The corresponding width would be:
\[
\text{New width} = \text{Original width} \times \text{Scaling factor} = 16 \times \frac{4}{3} = 21.33 \, \text{inches}
\]
This width exceeds the available 9 inches, so scaling by height is not feasible.

### Step 3: Conclusion
The maximum size the reduced figure could measure, while fitting within the new dimensions and keeping the proportions, is **9 inches wide by 5.0625 inches tall**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How do we calculate aspect ratios for other shapes?
2. What happens if the paper's dimensions were reversed (12 inches wide by 9 inches tall)?
3. How do we handle scaling when both dimensions exceed the new limits?
4. Could we fit an irregular shape with a non-rectangular aspect ratio on the paper?
5. How would the solution change if the original figure were larger?

**Tip**: Always check both width and height when scaling proportions, to avoid distortion or improper fitting.

Learn how to reduce a 9-inch tall and 16-inch wide figure proportionally to fit on a 9x12-inch paper. This problem covers key geometric concepts, including proportional scaling and aspect ratios, providing a step-by-step solution for students in Grades 6-8.

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