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 find out which batch had a greater ratio of sugar to berries, we need to calculate the ratio of sugar to berries for both batches.

Last week's batch:

• Sugar: 2 cups
• Berries: 18 cups

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

This week's batch:

• Sugar: 4 cups
• Berries: 30 cups

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

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

To compare these fractions, we can find a common denominator or convert them to decimals.

Finding a common denominator: The least common multiple of 9 and 15 is 45.

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

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

Now we can compare:

• Last week's ratio: \(\frac{5}{45}\)
• This week's ratio: \(\frac{6}{45}\)

Since \(\frac{6}{45} > \frac{5}{45}\), this means that:

\[ \text{This week's batch} > \text{Last week's batch} \]

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