Question

Emilia and Darnell are arguing over who makes the best fruit punch. Emilia likes to use 2 cups of grape juice for every 5 cups of pineapple juice. Darnell prefers to use 1 cup of grape juice for every 4 cups of pineapple juice. Do Emilia and Darnell use the same ratio of grape juice to pineapple juice?

Answer 1

To determine if Emilia and Darnell use the same ratio of grape juice to pineapple juice, we can compare their respective ratios.

**Emilia's ratio:**
- Grape juice = 2 cups
- Pineapple juice = 5 cups

The ratio of grape juice to pineapple juice for Emilia is:
\[
\text{Ratio} = \frac{2 \text{ cups of grape juice}}{5 \text{ cups of pineapple juice}} = \frac{2}{5}
\]

**Darnell's ratio:**
- Grape juice = 1 cup
- Pineapple juice = 4 cups

The ratio of grape juice to pineapple juice for Darnell is:
\[
\text{Ratio} = \frac{1 \text{ cup of grape juice}}{4 \text{ cups of pineapple juice}} = \frac{1}{4}
\]

Now, we can compare the two ratios, \(\frac{2}{5}\) and \(\frac{1}{4}\).

To compare the fractions, we can find a common denominator. The least common multiple of 5 and 4 is 20.

Converting both fractions to have a denominator of 20:
\[
\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}
\]
\[
\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}
\]

Now we can see that:
- Emilia's ratio is \(\frac{8}{20}\).
- Darnell's ratio is \(\frac{5}{20}\).

Since \(\frac{8}{20} \neq \frac{5}{20}\), the ratios are different.

Thus, **Emilia and Darnell do not use the same ratio of grape juice to pineapple juice.**