Question

Are these ratios equivalent?

$82 for every 24 pages

$41 for every 12 pages

Answer 1

To determine if the ratios \( \frac{82}{24} \) and \( \frac{41}{12} \) are equivalent, we can simplify each ratio and compare their values.

First, let's simplify \( \frac{82}{24} \).

We can simplify by finding the greatest common divisor (GCD) of the numerator and the denominator.

The GCD of 82 and 24 can be determined by prime factorization:

- \( 82 = 2 \times 41 \)
- \( 24 = 2^3 \times 3 \)

The only common factor is \( 2 \). Therefore, we divide both by \( 2 \):

\[
\frac{82 \div 2}{24 \div 2} = \frac{41}{12}
\]

Next, we look at the second ratio \( \frac{41}{12} \).

Since we have already simplified \( \frac{82}{24} \) to \( \frac{41}{12} \), the ratios clearly match:

\[
\frac{82}{24} = \frac{41}{12}
\]

Thus, both ratios are equivalent. Therefore, we conclude:

\[
\text{Yes, the ratios are equivalent.}
\]