Macy was asked to compare the two ratios 4:5 and 19 to 30. Her work is shown below. Which is true?


4:5 ? 19 to 30

Four-fifths question mark StartFraction 19 Over 30 EndFraction. Common denominator is 30. StartFraction 18 Over 30 EndFraction less-than StartFraction 19 Over 30 EndFraction
Macy did not correctly write the ratios as fractions.
Macy did not find a common denominator for the fractions.
Macy did not find the correct equivalent fraction with the denominator of 30.
Macy did not correctly compare the numerators.

1 answer

Let's analyze Macy's work step by step to determine which statement is true.

Macy is comparing the ratios 4:5 and 19:30. To do this, we convert the ratios to fractions:

  • The ratio 4:5 can be written as the fraction \( \frac{4}{5} \).
  • The ratio 19:30 can be written as the fraction \( \frac{19}{30} \).

Next, let's compare these two fractions. To do this, we can find a common denominator. The common denominator for 5 and 30 is 30.

Now, we convert \( \frac{4}{5} \) to have a denominator of 30:

\[ \frac{4}{5} = \frac{4 \times 6}{5 \times 6} = \frac{24}{30} \]

Now we have:

  • \( \frac{4}{5} = \frac{24}{30} \)
  • \( \frac{19}{30} \)

Next, we compare \( \frac{24}{30} \) and \( \frac{19}{30} \):

\[ \frac{24}{30} \text{ compared to } \frac{19}{30} \]

Since the denominators are the same, we can compare the numerators directly:

24 is greater than 19.

Thus,

\[ \frac{24}{30} > \frac{19}{30} \]

Now, let's evaluate the original statements about Macy's work:

  1. Macy did not correctly write the ratios as fractions. - This is not true, as she did write the ratios as fractions correctly (as \( \frac{4}{5} \) and \( \frac{19}{30} \)).

  2. Macy did not find a common denominator for the fractions. - This is not true either; she found a common denominator of 30.

  3. Macy did not find the correct equivalent fraction with the denominator of 30. - This is true, as she incorrectly stated that \( \frac{4}{5} \) is \( \frac{18}{30} \) instead of \( \frac{24}{30} \).

  4. Macy did not correctly compare the numerators. - This is true as well; because her final comparison was incorrect based on her incorrect equivalent fraction.

Given the analysis, the most accurate statement regarding Macy's work is:

Macy did not find the correct equivalent fraction with the denominator of 30.