Question

Compare the ratios 9 to 12 and 4 to 6.

One model has 12 sections. The second model has 6 sections.

Which statement correctly compares the ratios?
The ratio 9 to 12 is greater than 4 to 6.
The ratio 9 to 12 is less than 4 to 6.
The ratio 9 to 12 is equal to 4 to 6.
The ratios cannot be compared.

Answer 1

To compare the ratios 9 to 12 and 4 to 6, we can convert both ratios to their simplest forms.

1. For the ratio 9 to 12:
\[
\frac{9}{12} = \frac{3}{4} \quad \text{(by dividing both numerator and denominator by 3)}
\]

2. For the ratio 4 to 6:
\[
\frac{4}{6} = \frac{2}{3} \quad \text{(by dividing both numerator and denominator by 2)}
\]

Now we will compare \(\frac{3}{4}\) and \(\frac{2}{3}\).

To do this, we can find a common denominator, which is 12:

- Convert \(\frac{3}{4}\) to have a denominator of 12:
\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\]

- Convert \(\frac{2}{3}\) to have a denominator of 12:
\[
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\]

Now we can compare:
- \(\frac{9}{12}\) (which is \(\frac{3}{4}\)) and
- \(\frac{8}{12}\) (which is \(\frac{2}{3}\)).

Since \(9 > 8\), we find that:
\[
\frac{3}{4} > \frac{2}{3}
\]

Thus, the ratio \(9\) to \(12\) is greater than the ratio \(4\) to \(6\).

Therefore, the correct statement is:
**The ratio 9 to 12 is greater than 4 to 6.**