Question
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.
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.
Answers
GPT-4o mini
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.**
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.**