Saleem wants to create a table of ratios that are equivalent to 2

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.

GPT-4o mini · Answer

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} $.