An orange has about
4
1
-cup of juice. How many oranges are needed to make 2
2
1
-cups of juice? Which equations represent this question?
- ?×
4
1
=2
2
1
4
1
÷2
2
1
=? 3. ?×2
4
1
=
4
1
- 2
2
1
÷
4
1
=?
The Correct Answer and Explanation is :
To find out how many oranges are needed to make (2 \frac{2}{1}) cups of juice when each orange yields (4 \frac{1}{4}) cups, we can set up a few equations.
Conversion of Mixed Numbers
First, let’s convert the mixed numbers to improper fractions for easier calculations:
- For (4 \frac{1}{4}):
[
4 \frac{1}{4} = 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4}
] - For (2 \frac{2}{1}):
[
2 \frac{2}{1} = 2 + 2 = 4
]
Setting Up the Equation
To determine the number of oranges needed ((x)), we can set up the equation:
[
x \times \frac{17}{4} = 4
]
Solving the Equation
To isolate (x), divide both sides by (\frac{17}{4}):
[
x = 4 \div \frac{17}{4} = 4 \times \frac{4}{17} = \frac{16}{17}
]
This result indicates that you need (\frac{16}{17}) of an orange to produce (4) cups of juice, which is less than one full orange.
Evaluating the Options
Now, let’s match the equations provided with our situation:
- (? \times \frac{17}{4} = 4) (Correct)
- This represents our main equation.
- (\frac{17}{4} \div 4 = ?) (Incorrect)
- This doesn’t answer the question about how many oranges are needed.
- (? \times 2 \frac{17}{4} = \frac{17}{4}) (Incorrect)
- This equation is also irrelevant to the problem.
- (4 \div \frac{17}{4} = ?) (Correct)
- This is equivalent to our main equation in a different form.
Conclusion
In conclusion, to produce (2 \frac{2}{1}) (or 4) cups of juice, you need (\frac{16}{17}) of an orange, meaning that slightly less than one full orange is required. The correct equation that represents this relationship is (? \times \frac{17}{4} = 4).
This calculation highlights the utility of understanding mixed numbers and how to set up equations to solve for unknown quantities in practical scenarios like juice extraction.