To solve this problem, we need to determine how many bags of flour Carmen will need to make 3 batches of her homemade cookies, given that [tex]\( \frac{2}{7} \)[/tex] of a bag is used to make [tex]\( \frac{1}{2} \)[/tex] of a batch of cookies.
### Step-by-Step Solution
1. Determine the amount of flour needed for one full batch:
Carmen uses [tex]\( \frac{2}{7} \)[/tex] of a bag of flour to make [tex]\( \frac{1}{2} \)[/tex] of a batch. To find out how much flour is needed for a full batch, we can set up a proportion. If [tex]\( \frac{2}{7} \)[/tex] bag is used for [tex]\( \frac{1}{2} \)[/tex] batch, then for a full batch (which is double the amount), the amount of flour needed will be:
[tex]\[
\frac{\frac{2}{7}}{\frac{1}{2}} = \frac{2}{7} \times \frac{2}{1} = \frac{2 \times 2}{7 \times 1} = \frac{4}{7}
\][/tex]
So, Carmen needs [tex]\( \frac{4}{7} \)[/tex] of a bag of flour to make one full batch of cookies.
2. Calculate the amount of flour needed for 3 batches:
Now, we want to find out how much flour is needed to make 3 full batches. If one batch requires [tex]\( \frac{4}{7} \)[/tex] of a bag of flour, then 3 batches will need:
[tex]\[
3 \times \frac{4}{7} = \frac{3 \times 4}{7} = \frac{12}{7}
\][/tex]
Simplify [tex]\( \frac{12}{7} \)[/tex]:
[tex]\[
\frac{12}{7} \approx 1.7142857142857142
\][/tex]
Hence, Carmen needs approximately [tex]\( 1.7142857142857142 \)[/tex] bags of flour to make 3 batches of her special homemade cookies.
