Question

Saleem wants to create a table of ratios that are equivalent to 2
3
. He includes the ratios 4
5
and 5
6
.

Are these ratios equivalent to 2
3
? Explain your reasoning.

Answer 1

To determine if the ratios \( \frac{4}{5} \) and \( \frac{5}{6} \) are equivalent to \( \frac{2}{3} \), we can compare each ratio to \( \frac{2}{3} \) by cross-multiplying.

1. **Checking \( \frac{4}{5} \):**

We want to see if:
\[
\frac{4}{5} = \frac{2}{3}
\]

Cross-multiplying gives us:
\[
4 \cdot 3 = 2 \cdot 5
\]
Calculating both sides:
\[
12 = 10
\]
Since \( 12 \neq 10 \), the ratio \( \frac{4}{5} \) is **not equivalent** to \( \frac{2}{3} \).

2. **Checking \( \frac{5}{6} \):**

Next, we check:
\[
\frac{5}{6} = \frac{2}{3}
\]

Cross-multiplying gives us:
\[
5 \cdot 3 = 2 \cdot 6
\]
Calculating both sides:
\[
15 = 12
\]
Since \( 15 \neq 12 \), the ratio \( \frac{5}{6} \) is also **not equivalent** to \( \frac{2}{3} \).

In conclusion, neither \( \frac{4}{5} \) nor \( \frac{5}{6} \) are equivalent to \( \frac{2}{3} \).