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The Moon has a weaker gravitational pull than Earth, so objects weigh less on the Moon. For example, Jaxon weighs [tex]30 \frac{5}{6}[/tex] pounds on the Moon and 185 pounds on Earth. Viola weighs 135 pounds on Earth and [tex]22 \frac{1}{2}[/tex] pounds on the Moon.

A. Show that the relationship between weight on the Moon and weight on Earth is proportional.

[tex]
\begin{array}{l}
\text{Jaxon: } \frac{30 \frac{5}{6}}{185} = 30 \frac{5}{6} \div 185 = \\
\text{Viola: } \frac{22 \frac{1}{2}}{135} = 22 \frac{1}{2} \div 135 =
\end{array}
[/tex]

The constant of proportionality for Moon weight (lb) is _____.

Answer :

To demonstrate that the relationship between weight on the moon and weight on Earth is proportional, we will calculate the constant of proportionality using the weights of Jaxon and Viola.

### Step-by-Step Solution:

1. Understand the Problem:
We are given weights of two individuals on Earth and the moon, and we need to show that the weight conversion between Earth and the moon is proportional. This means the ratio of weight on the moon to weight on Earth should be the same for both individuals.

2. Convert Mixed Numbers to Improper Fractions:
- Jaxon's weight on the moon is [tex]\(30 \frac{5}{6}\)[/tex] pounds. To convert this to an improper fraction:
[tex]\[
30 \frac{5}{6} = 30 + \frac{5}{6} = \frac{180}{6} + \frac{5}{6} = \frac{185}{6}
\][/tex]
- Viola's weight on the moon is [tex]\(22 \frac{1}{2}\)[/tex] pounds. To convert this to an improper fraction:
[tex]\[
22 \frac{1}{2} = 22 + \frac{1}{2} = \frac{44}{2} + \frac{1}{2} = \frac{45}{2}
\][/tex]

3. Calculate the Ratios of Moon Weight to Earth Weight:
- Jaxon's ratio:
[tex]\[
\text{Jaxon's ratio} = \frac{\text{Jaxon's moon weight}}{\text{Jaxon's earth weight}} = \frac{\frac{185}{6}}{185}
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{185}{6} \div 185 = \frac{185}{6} \times \frac{1}{185} = \frac{1}{6} \approx 0.1666667
\][/tex]

- Viola's ratio:
[tex]\[
\text{Viola's ratio} = \frac{\text{Viola's moon weight}}{\text{Viola's earth weight}} = \frac{\frac{45}{2}}{135}
\][/tex]
Simplifying the fraction:
[tex]\[
\frac{45}{2} \div 135 = \frac{45}{2} \times \frac{1}{135} = \frac{1}{6} \approx 0.1666667
\][/tex]

4. Verify Proportionality:
The calculated ratios for Jaxon and Viola are the same: [tex]\(0.1667\)[/tex]. This shows that the weights on the moon are directly proportional to their weights on Earth.

5. Constant of Proportionality:
The constant of proportionality for moon weight is [tex]\( \boxed{0.1667} \)[/tex].

This constant means that an object's weight on the moon is approximately [tex]\(16.67\%\)[/tex] of its weight on Earth, confirming the proportional relationship.

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