Patrick made a simple sugar solution using 3 parts sugar and 4 parts water. Thomas made a sugar solutions using 6 parts sugar and 7 parts water. Whose solutions was more sugary? Explain

To determine which solution is more sugary, we need to compare the ratios of sugar to the total amount of solution for each person.

### Patrick's Solution:
- **Sugar:** 3 parts
- **Water:** 4 parts
- **Total Solution:** \(3 + 4 = 7\) parts

The ratio of sugar to the total solution for Patrick is:
\[
\frac{3 \text{ parts sugar}}{7 \text{ parts total}} = \frac{3}{7}
\]

### Thomas's Solution:
- **Sugar:** 6 parts
- **Water:** 7 parts
- **Total Solution:** \(6 + 7 = 13\) parts

The ratio of sugar to the total solution for Thomas is:
\[
\frac{6 \text{ parts sugar}}{13 \text{ parts total}} = \frac{6}{13}
\]

### Comparing the Ratios:
Now, let's compare the two fractions, \(\frac{3}{7}\) and \(\frac{6}{13}\).

We can compare these fractions by cross-multiplying:

- For Patrick: \(3 \times 13 = 39\)
- For Thomas: \(6 \times 7 = 42\)

Since \(42 > 39\), this means:
\[
\frac{6}{13} > \frac{3}{7}
\]

### Conclusion:
Thomas's solution is more sugary because the ratio of sugar to the total solution (\(\frac{6}{13}\)) is greater than the ratio in Patrick's solution (\(\frac{3}{7}\)).

Would you like further details on comparing fractions or have any other questions?

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Here are five related questions for further exploration:
1. How can you determine which fraction is greater without using cross-multiplication?
2. What is the significance of the ratio of sugar to total solution in determining sweetness?
3. How would the ratios change if the amount of water was doubled in each solution?
4. How can we apply this reasoning to compare different recipes in cooking?
5. What is the importance of understanding ratios in everyday life?

**Tip:** When comparing fractions, cross-multiplication is a quick way to see which one is larger without converting to decimals.

Compare the sweetness of sugar solutions made by Patrick and Thomas using ratios. Analyze which solution has a higher sugar concentration based on the ratio of sugar to total solution parts. Suitable for students in Grades 6-8 learning about ratios and fractions in mathematics.

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