To find the equation that represents the relationship between the number of cups of bananas [tex]\( c \)[/tex] and the number of loaves of bread [tex]\( b \)[/tex], we need to determine the unit rate of cups of bananas per loaf of bread.
Step-by-step solution:
1. Identify the data given:
- Paul uses [tex]\( 11 \frac{1}{4} \)[/tex] cups of bananas.
- These [tex]\( 11 \frac{1}{4} \)[/tex] cups are used to make 5 loaves of bread.
2. Convert the mixed number to an improper fraction:
- [tex]\( 11 \frac{1}{4} = 11 + \frac{1}{4} = \frac{44}{4} + \frac{1}{4} = \frac{45}{4} \)[/tex] cups of bananas.
3. Determine the unit rate (the number of cups of bananas per loaf of bread):
- The number of cups per loaf of bread is found by dividing the total number of cups by the number of loaves:
[tex]\[
\text{unit rate} = \frac{\text{cups of bananas}}{\text{loaves of bread}} = \frac{\frac{45}{4}}{5}
\][/tex]
4. Perform the division:
- Dividing a fraction by a whole number is equivalent to multiplying the fraction by the reciprocal of the whole number:
[tex]\[
\frac{\frac{45}{4}}{5} = \frac{45}{4} \times \frac{1}{5} = \frac{45 \times 1}{4 \times 5} = \frac{45}{20}
\][/tex]
5. Simplify the fraction:
- [tex]\(\frac{45}{20}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 5:
[tex]\[
\frac{45 \div 5}{20 \div 5} = \frac{9}{4}
\][/tex]
6. Convert the improper fraction back to a mixed number:
- [tex]\(\frac{9}{4} = 2 \frac{1}{4}\)[/tex]
Hence, the equation that represents the relationship between [tex]\( c \)[/tex], the number of cups of bananas, and [tex]\( b \)[/tex], the number of loaves of bread, is:
[tex]\[ c = 2 \frac{1}{4} b \][/tex]
This corresponds to option C.
