Orly uses 2 cups of raisins for every 9 cups of trail mix she makes. How many cups of trail mix will she make if she uses 12 cups of raisins? 2(2)/(3) cups 54 cups 9 cups 1(1)/(2) cups
The correct answer and explanation is:
To solve the problem, we use the ratio Orly follows: 2 cups of raisins for every 9 cups of trail mix. This can be written as: 2 cups of raisins9 cups of trail mix\frac{2 \, \text{cups of raisins}}{9 \, \text{cups of trail mix}}
We need to determine how many cups of trail mix she can make if she uses 12 cups of raisins. To do this, set up a proportion: 29=12x\frac{2}{9} = \frac{12}{x}
Here, xx represents the number of cups of trail mix. Solve for xx by cross-multiplying: 2⋅x=12⋅92 \cdot x = 12 \cdot 9 2x=1082x = 108 x=1082=54x = \frac{108}{2} = 54
Thus, Orly can make 54 cups of trail mix if she uses 12 cups of raisins.
Explanation:
The relationship between the ingredients is proportional, meaning the ratio of raisins to trail mix remains constant no matter how much she makes. If 2 cups of raisins are needed for every 9 cups of trail mix, then multiplying both amounts by the same factor will maintain this ratio.
In this case, 12 cups of raisins is 6 times the amount of raisins in the original ratio (since 12÷2=612 \div 2 = 6). Therefore, the amount of trail mix must also increase by a factor of 6. Since 9 cups of trail mix are made with 2 cups of raisins, multiplying 9 by 6 gives: 9⋅6=54 cups of trail mix.9 \cdot 6 = 54 \, \text{cups of trail mix}.
By solving proportionally, we ensure the quantities scale correctly. The answer is 54 cups of trail mix.