Question

Rebecca is playing her favorite video game, Starlight: The Space Explorer. On the first level, she earned 3 energy points and 8 stars. On the second level, she earned 2 energy points and 5 stars. Did Rebecca earn the same ratio of energy points to stars on both levels?

Answer 1

To determine if Rebecca earned the same ratio of energy points to stars on both levels, we can calculate the ratios for each level and compare them.

**For the first level:**
- Energy points earned: 3
- Stars earned: 8

The ratio of energy points to stars for the first level is:
\[
\text{Ratio} = \frac{\text{Energy points}}{\text{Stars}} = \frac{3}{8}
\]

**For the second level:**
- Energy points earned: 2
- Stars earned: 5

The ratio of energy points to stars for the second level is:
\[
\text{Ratio} = \frac{\text{Energy points}}{\text{Stars}} = \frac{2}{5}
\]

Now we compare \(\frac{3}{8}\) and \(\frac{2}{5}\). To do this, we can find a common denominator or convert them to decimal form.

**Finding a common denominator:**
The least common multiple of 8 and 5 is 40. We can convert both ratios.

For \(\frac{3}{8}\):
\[
\frac{3}{8} = \frac{3 \times 5}{8 \times 5} = \frac{15}{40}
\]

For \(\frac{2}{5}\):
\[
\frac{2}{5} = \frac{2 \times 8}{5 \times 8} = \frac{16}{40}
\]

Now we can compare:
- First level ratio: \(\frac{15}{40}\)
- Second level ratio: \(\frac{16}{40}\)

Since \( \frac{15}{40} \neq \frac{16}{40} \), the ratios are not equal.

Thus, **Rebecca did not earn the same ratio of energy points to stars on both levels.**