To solve the problem, we need to first express the amounts of chocolate chips and pecans in fractional form.
Given:
- 5 4/7 cups of chocolate chips
- 7 1/4 cups of pecans
We can convert these mixed numbers into improper fractions:
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Convert 5 4/7 to an improper fraction: \[ 5 \frac{4}{7} = \frac{5 \times 7 + 4}{7} = \frac{35 + 4}{7} = \frac{39}{7} \]
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Convert 7 1/4 to an improper fraction: \[ 7 \frac{1}{4} = \frac{7 \times 4 + 1}{4} = \frac{28 + 1}{4} = \frac{29}{4} \]
Now we see that:
- For every \(\frac{39}{7}\) cups of chocolate chips, we need \(\frac{29}{4}\) cups of pecans.
Next, let's set up the ratio: \[ \frac{39/7 \text{ cups of chocolate chips}}{29/4 \text{ cups of pecans}} \rightarrow \text{Let x be the number of pecans for 1 cup of chocolate chips.} \]
To find how many cups of pecans correspond to 1 cup of chocolate chips, we can set up a proportion: \[ \frac{39/7 \text{ cups of chocolate chips}}{29/4 \text{ cups of pecans}} = \frac{1 \text{ cup of chocolate chips}}{x \text{ cups of pecans}} \]
Cross-multiplying gives us: \[ (39/7) \cdot x = (29/4) \cdot 1 \]
Now solve for \(x\): \[ x = \frac{29/4}{39/7} \] To divide fractions, we multiply by the reciprocal: \[ x = \frac{29}{4} \cdot \frac{7}{39} = \frac{29 \cdot 7}{4 \cdot 39} \]
Calculating: \[ x = \frac{203}{156} \]
This fraction can be simplified: \[ 203 \div 13 = 15.615 (approximately) \text{ and } 156 \div 13 = 12 \Rightarrow x = \frac{29}{4} \]
From that, we have: \[ x \approx 1.3 \text{ cups of pecans (rounded to a decimal)} \]
Thus, for 1 cup of chocolate chips, you would require approximately 1.3 cups of pecans.
Here’s the completed table:
| Chocolate chips (cups) | Pecans (cups) | |-------------------------|----------------| | 5 4/7 | 7 1/4 | | 1 | 1.3 |