Question

Are these ratios equivalent? $56 per 52 centiliters $28 per 26 centiliters

Answer 1

To determine if the ratios \( \frac{56}{52} \) and \( \frac{28}{26} \) are equivalent, we need to simplify both fractions and compare them.

First, let's simplify \( \frac{56}{52} \):

1. Calculate the greatest common divisor (gcd) of 56 and 52.
The factors of 56 are \( 1, 2, 4, 7, 8, 14, 28, 56 \).
The factors of 52 are \( 1, 2, 4, 13, 26, 52 \).
The common factors are \( 1, 2, 4 \), and the greatest is \( 4 \).

2. Now, divide both the numerator and the denominator by their gcd:
\[
\frac{56}{52} = \frac{56 \div 4}{52 \div 4} = \frac{14}{13}.
\]

Next, let’s simplify \( \frac{28}{26} \):

1. The gcd of 28 and 26 can be found as follows.
The factors of 28 are \( 1, 2, 4, 7, 14, 28 \).
The factors of 26 are \( 1, 2, 13, 26 \).
The common factors are \( 1, 2 \), and the greatest is \( 2 \).

2. Now, divide both the numerator and the denominator by their gcd:
\[
\frac{28}{26} = \frac{28 \div 2}{26 \div 2} = \frac{14}{13}.
\]

Now we compare the simplified ratios:
\[
\frac{56}{52} = \frac{14}{13}, \quad \frac{28}{26} = \frac{14}{13}.
\]

Since both ratios simplify to \( \frac{14}{13} \), we conclude that the ratios \( \frac{56}{52} \) and \( \frac{28}{26} \) are indeed equivalent.

Thus, the answer is:
\[
\text{Yes, the ratios are equivalent.}
\]