Question

One serving of trail mix has 1/2 cups of almonds and 1 1/2 cups of dried fruit. How many almonds are needed for 2 cups of dried fruit?

a) 1 cup
b) 1 1/3 cups
c) 1 1/2 cups
d) 2 cups

Answer 1

To find how many almonds are needed for 2 cups of dried fruit, we can use the given ratio of almonds to dried fruit. If one serving of trail mix has 1/2 cup of almonds and 1 1/2 cups of dried fruit, the ratio is 1:3 (almonds to dried fruit). Therefore, for 2 cups of dried fruit, you would need 2/3 cups of almonds.

Step-by-step explanation:

The given problem involves finding the proportional amount of almonds for a given quantity of dried fruit in the trail mix. Since the ratio of almonds to dried fruit is 1:3 (1/2 cup almonds to 1 1/2 cups dried fruit), we can establish the ratio as 1:3. To find the amount of almonds for 2 cups of dried fruit, we set up a proportion:

`\[ (1/2)/(1.5) = (x)/(2) \]`

Solving for \( x \), which represents the amount of almonds needed for 2 cups of dried fruit, we get \( x = 2/3 \) cups. Therefore, 2/3 cups of almonds are required.

This approach is based on the concept of setting up and solving ratios to determine the proportional relationship between the quantities of almonds and dried fruit. It's a fundamental application of proportional reasoning in solving real-world problems involving ratios and proportions.

Answer 2

To find the amount of almonds needed for 2 cups of dried fruit, the 1:3 ratio of almonds to dried fruit is maintained. Multiplying 1/2 cup almonds by 4/3 (since 2 cups is 4/3 of 1 1/2 cups), we get 1 1/3 cups of almonds.

Step-by-step explanation:

The student is asking how many cups of almonds are needed if there are 2 cups of dried fruit, given that one serving (which has 1/2 cup of almonds and 1 1/2 cups of dried fruit) is the standard proportion. First, we must find the ratio of almonds to dried fruit in one serving. The ratio is 1/2 cup almonds to 1 1/2 cups dried fruit, which simplifies to 1:3 when we divide both sides by 1/2. To maintain this ratio for 2 cups of dried fruit, we multiply the amount of almonds (1/2 cup) by the number of times 2 cups of dried fruit contains the standard 1 1/2 cups, which is 2 / (3/2) = 4/3 times. Thus, 1/2 cup of almonds × 4/3 = 2/3 × 4/3 = 8/9, which simplifies to 4/3 or 1 1/3 cups of almonds. Therefore, the correct answer is: b) 1 1/3 cups.