Question

The following three proportions can all be used to solve the problem about Caroline's salary. Explain why the solutions to each proportion are the same.


36
4
=
x
7

4
36
=
7
x

36
x
=
4
7

Answer 1

To understand why the solutions to the three proportions \( \frac{36}{4} = \frac{x}{7} \), \( \frac{4}{36} = \frac{7}{x} \), and \( \frac{36}{x} = \frac{4}{7} \) are the same, we will analyze each proportion step by step.

### Proportion 1: \( \frac{36}{4} = \frac{x}{7} \)

1. Cross-multiply:
\[
36 \cdot 7 = 4 \cdot x
\]
This simplifies to:
\[
252 = 4x
\]

2. Solve for \( x \):
\[
x = \frac{252}{4} = 63
\]

### Proportion 2: \( \frac{4}{36} = \frac{7}{x} \)

1. Cross-multiply:
\[
4x = 36 \cdot 7
\]
This simplifies to:
\[
4x = 252
\]

2. Solve for \( x \):
\[
x = \frac{252}{4} = 63
\]

### Proportion 3: \( \frac{36}{x} = \frac{4}{7} \)

1. Cross-multiply:
\[
36 \cdot 7 = 4 \cdot x
\]
This simplifies to:
\[
252 = 4x
\]

2. Solve for \( x \):
\[
x = \frac{252}{4} = 63
\]

### Conclusion

In each case, we end up with the same equation, \( 4x = 252 \), irrespective of how we set up the proportion. Each proportion represents the same relationship between the values involved. Therefore, we find that the solutions for \( x \) from all three proportions are equal, yielding \( x = 63 \) consistently across each proportion.

This demonstrates that the ratios are equivalent; hence, the value of \( x \) remains unchanged across different proportional setups.