Question

1. are the two statements equivalent? explain your reasoning. - the ratio of orange to blue is 3 to 4. - the ratio of blue to orange is 12 to 9.

Explanation:

Step1: Simplify the first ratio

The ratio of orange to blue is \( 3:4 \). This means for every 3 units of orange, there are 4 units of blue.

Step2: Simplify the second ratio

The ratio of blue to orange is \( 12:9 \). We can simplify this ratio by dividing both numbers by their greatest common divisor, which is 3. So, \( \frac{12}{3}:\frac{9}{3}=4:3 \).

Step3: Analyze the relationship

The first ratio (orange to blue) is \( 3:4 \), which is equivalent to saying the ratio of blue to orange is \( 4:3 \) (since if orange : blue = \( a:b \), then blue : orange = \( b:a \)). The simplified second ratio is also \( 4:3 \).

Answer:

Yes, the two statements are equivalent. The ratio of orange to blue being \( 3:4 \) implies the ratio of blue to orange is \( 4:3 \), and the ratio \( 12:9 \) simplifies to \( 4:3 \), so they represent the same relationship between the quantities of blue and orange.