Alec makes jam to sell at a local farmers' market. To make last week's batch, he used 2 cups of sugar and 18 cups of berries. It was so popular that he made a larger batch this week. To make this week's batch, Alec used 4 cups of sugar and 30 cups of berries, and the jam sold out even faster. Which batch had a greater ratio of sugar to berries?

Answer 1

To compare the ratio of sugar to berries for both batches, we need to calculate the ratio for each batch.

For last week's batch:

• Sugar: 2 cups
• Berries: 18 cups

The ratio of sugar to berries is: \[ \text{Ratio}_{\text{last week}} = \frac{\text{sugar}}{\text{berries}} = \frac{2}{18} = \frac{1}{9} \]

For this week's batch:

• Sugar: 4 cups
• Berries: 30 cups

The ratio of sugar to berries is: \[ \text{Ratio}_{\text{this week}} = \frac{\text{sugar}}{\text{berries}} = \frac{4}{30} = \frac{2}{15} \]

Next, we need to compare the two ratios: \(\frac{1}{9}\) and \(\frac{2}{15}\).

To compare \(\frac{1}{9}\) and \(\frac{2}{15}\), we can find a common denominator. The least common multiple of 9 and 15 is 45.

Convert \(\frac{1}{9}\) to a denominator of 45: \[ \frac{1}{9} = \frac{1 \times 5}{9 \times 5} = \frac{5}{45} \]

Convert \(\frac{2}{15}\) to a denominator of 45: \[ \frac{2}{15} = \frac{2 \times 3}{15 \times 3} = \frac{6}{45} \]

Now we can compare \(\frac{5}{45}\) and \(\frac{6}{45}\): \[ \frac{5}{45} < \frac{6}{45} \]

This shows that the ratio of sugar to berries in last week's batch (\(\frac{1}{9}\)) is less than the ratio of sugar to berries in this week's batch (\(\frac{2}{15}\)).

Therefore, this week's batch had a greater ratio of sugar to berries.