Question

A snack mix is made of mixed nuts and raisins.
Complete the table below to show the amounts of each ingredient and the total amount of snack mix (in cups). The ratio between mixed nuts and raisins should remain the same for different amounts of snack mix.

Amount of mixed nuts (cups)
Amount of raisins (cups)
Total amount of snack mix (cups)
1.5
0.5
10
12

Answer 1

To maintain the same ratio between mixed nuts and raisins as given in the first row (1.5 cups of mixed nuts and 0.5 cups of raisins), we first need to determine the ratio of mixed nuts to raisins:

\[ \text{Ratio} = \frac{\text{Mixed nuts}}{\text{Raisins}} = \frac{1.5}{0.5} = 3 \]

This means that for every 3 parts of mixed nuts, there is 1 part of raisins (3:1).

Now, we can fill the table for the other amounts of snack mix while keeping this ratio. We can express the total amount of snack mix (in cups) as:

\[ \text{Total amount} = \text{Amount of mixed nuts} + \text{Amount of raisins} \]

1. For **10 cups** of total snack mix:
- Let the amount of mixed nuts = \( x \) and the amount of raisins = \( y \).
- We know \( x + y = 10 \).
- From the ratio: \( \frac{x}{y} = 3 \) => \( x = 3y \).
- Substituting \( x \) in the total:
\[ 3y + y = 10 \]
\[ 4y = 10 \]
\[ y = 2.5 \] (amount of raisins)
\[ x = 3(2.5) = 7.5 \] (amount of mixed nuts)
- So, for **10 cups**:
- Amount of mixed nuts: **7.5 cups**
- Amount of raisins: **2.5 cups**

2. For **12 cups** of total snack mix:
- Again, let mixed nuts = \( x \) and raisins = \( y \).
- \( x + y = 12 \).
- The ratio gives: \( x = 3y \).
- Substituting:
\[ 3y + y = 12 \]
\[ 4y = 12 \]
\[ y = 3 \] (amount of raisins)
\[ x = 3(3) = 9 \] (amount of mixed nuts)
- So, for **12 cups**:
- Amount of mixed nuts: **9 cups**
- Amount of raisins: **3 cups**

Now we can compile the completed table:

| Amount of mixed nuts (cups) | Amount of raisins (cups) | Total amount of snack mix (cups) |
|-----------------------------|---------------------------|-----------------------------------|
| 1.5 | 0.5 | 2.0 |
| 7.5 | 2.5 | 10.0 |
| 9 | 3 | 12.0 |

In summary:
- For 10 cups total: 7.5 mixed nuts, 2.5 raisins.
- For 12 cups total: 9 mixed nuts, 3 raisins.