Question
An ice cream shop uses a mix of blueberries and cherries on its ice cream sundaes. The shop has
5 3/4 pounds of blueberries and 4 1/2
of cherries. The shop mixes the blueberries and cherries and uses 1/16
pound of the mix on each sundae. Which expression represents the total number of sundaes that the shop can make using all of the blueberries and cherries?
A
(534÷116)+412\left(5\frac{3}{4}\div \frac{1}{16}\right)+4\frac{1}{2}(5
4
3
÷
16
1
)+4
2
1
B
534+(412÷116)5\frac{3}{4}+\left(4\frac{1}{2}\div \frac{1}{16}\right)5
4
3
+(4
2
1
÷
16
1
)
C
116÷(534+412)\frac{1}{16}\div \left(5\frac{3}{4}+4\frac{1}{2}\right)
16
1
÷(5
4
3
+4
2
1
)
D
(534+412)÷116\left(5\frac{3}{4}+4\frac{1}{2}\right)\div \frac{1}{16}(5
4
3
+4
2
1
)÷
16
1
Answers
GPT-4o mini
To determine how many sundaes the shop can make, we need to first find the total weight of the blueberries and cherries combined and then divide that total by the amount of mix used per sundae.
### Step 1: Convert mixed numbers to improper fractions.
- Blueberries:
\( 5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4} \) pounds.
- Cherries:
\( 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \) pounds.
### Step 2: Convert the cherries to have a common denominator with the blueberries.
To add these two fractions, we need a common denominator. The least common multiple of 4 and 2 is 4.
- **Cherries**:
\( \frac{9}{2} = \frac{9 \times 2}{2 \times 2} = \frac{18}{4} \).
### Step 3: Add the weights of blueberries and cherries.
Now we add the blueberries and cherries:
\[
\frac{23}{4} + \frac{18}{4} = \frac{23 + 18}{4} = \frac{41}{4} \quad \text{pounds}.
\]
### Step 4: Determine how many sundaes can be made.
Each sundae uses \( \frac{1}{16} \) pounds of the mix. To find out how many sundaes can be made with \( \frac{41}{4} \) pounds, we divide:
\[
\frac{41}{4} \div \frac{1}{16} = \frac{41}{4} \times 16 = \frac{41 \times 16}{4} = \frac{656}{4} = 164.
\]
So, the shop can make 164 sundaes.
### Choice Analysis
Now we look at the given options:
- **A**: \( \left(5\frac{3}{4} \div \frac{1}{16}\right) + 4\frac{1}{2} \) is not correct because it doesn’t correctly sum the total weight before dividing.
- **B**: Includes the division and addition in an incorrect manner.
- **C**: Seems to get multiplication and division mixed up; \( 16 \) in the denominator with a direct division does not apply here.
- **D**: Would potentially express the desired calculation.
The correct approach was primarily to add the two amounts (weights of blueberries and cherries) and then divide by the amount used per sundae. Thus none of the options correctly convey that detail.
So the correct representation for calculating how many sundaes can be made is \( \frac{41}{4} \div \frac{1}{16} = \frac{41}{4} \times 16 \). A correct expression based on our work would be:
\[
\left(5 \frac{3}{4} + 4 \frac{1}{2}\right) \div \frac{1}{16}.
\]
None of the provided answer choices represents the calculation accurately.
### Step 1: Convert mixed numbers to improper fractions.
- Blueberries:
\( 5 \frac{3}{4} = \frac{5 \times 4 + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4} \) pounds.
- Cherries:
\( 4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{8 + 1}{2} = \frac{9}{2} \) pounds.
### Step 2: Convert the cherries to have a common denominator with the blueberries.
To add these two fractions, we need a common denominator. The least common multiple of 4 and 2 is 4.
- **Cherries**:
\( \frac{9}{2} = \frac{9 \times 2}{2 \times 2} = \frac{18}{4} \).
### Step 3: Add the weights of blueberries and cherries.
Now we add the blueberries and cherries:
\[
\frac{23}{4} + \frac{18}{4} = \frac{23 + 18}{4} = \frac{41}{4} \quad \text{pounds}.
\]
### Step 4: Determine how many sundaes can be made.
Each sundae uses \( \frac{1}{16} \) pounds of the mix. To find out how many sundaes can be made with \( \frac{41}{4} \) pounds, we divide:
\[
\frac{41}{4} \div \frac{1}{16} = \frac{41}{4} \times 16 = \frac{41 \times 16}{4} = \frac{656}{4} = 164.
\]
So, the shop can make 164 sundaes.
### Choice Analysis
Now we look at the given options:
- **A**: \( \left(5\frac{3}{4} \div \frac{1}{16}\right) + 4\frac{1}{2} \) is not correct because it doesn’t correctly sum the total weight before dividing.
- **B**: Includes the division and addition in an incorrect manner.
- **C**: Seems to get multiplication and division mixed up; \( 16 \) in the denominator with a direct division does not apply here.
- **D**: Would potentially express the desired calculation.
The correct approach was primarily to add the two amounts (weights of blueberries and cherries) and then divide by the amount used per sundae. Thus none of the options correctly convey that detail.
So the correct representation for calculating how many sundaes can be made is \( \frac{41}{4} \div \frac{1}{16} = \frac{41}{4} \times 16 \). A correct expression based on our work would be:
\[
\left(5 \frac{3}{4} + 4 \frac{1}{2}\right) \div \frac{1}{16}.
\]
None of the provided answer choices represents the calculation accurately.