Question
Su Li is going to make a mixture of almonds and cashews. She wants the mixture to cost $9 per kilogram. Su Li has 8 kilograms of cashews that cost $6 per kilogram. How many kilograms of almonds that cost $4 per kilogram does Su Li need to mix with the cashews so that the mixture costs $9 per kilogram?
Answers
Answered by
GPT-4o mini
Let \( x \) be the number of kilograms of almonds that Su Li needs to mix with the cashews.
The cost of the cashews is calculated as follows:
\[
\text{Cost of cashews} = 8 \text{ kg} \times 6 \text{ dollars/kg} = 48 \text{ dollars}
\]
The cost of the almonds is:
\[
\text{Cost of almonds} = x \text{ kg} \times 4 \text{ dollars/kg} = 4x \text{ dollars}
\]
The total weight of the mixture will be the sum of the weights of cashews and almonds:
\[
\text{Total weight} = 8 \text{ kg} + x \text{ kg} = 8 + x \text{ kg}
\]
The total cost of the mixture is:
\[
\text{Total cost} = \text{Cost of cashews} + \text{Cost of almonds} = 48 + 4x \text{ dollars}
\]
We want the mixture to cost \( 9 \) dollars per kilogram, so we set up the equation:
\[
\frac{\text{Total cost}}{\text{Total weight}} = 9
\]
Substituting in the expressions for total cost and total weight:
\[
\frac{48 + 4x}{8 + x} = 9
\]
To eliminate the fraction, multiply both sides by \( 8 + x \):
\[
48 + 4x = 9(8 + x)
\]
Expanding the right side:
\[
48 + 4x = 72 + 9x
\]
Next, we will move all terms involving \( x \) to one side and constant terms to the other:
\[
48 - 72 = 9x - 4x
\]
\[
-24 = 5x
\]
Now, solving for \( x \):
\[
x = \frac{-24}{5} = -4.8
\]
Since \( x \) needs to be a positive quantity (the number of kilograms of almonds cannot be negative), there must be an error in assuming that we can achieve a mixture worth \( 9 \) dollars using these prices.
Let's reason through this:
- The cashews cost \( 6 \) dollars per kilogram and we have \( 8 \) kg of them (totaling \( 48 \) dollars).
- The almonds cost \( 4 \) dollars per kilogram, which is less than the cashew price.
- Mixing a cheaper product (almonds) with a more expensive product (cashews) cannot yield a final mixture value higher than the more expensive ingredient.
Given that the aim was a \( 9 \) dollars per kilogram mixture, which exceeds the costliest ingredient's price, it's impossible to achieve that mixture.
Therefore, there is no amount of almonds that Su Li can use to create a mixture costing \( 9 \) dollars per kilogram. The result indicates that it is not feasible to achieve \( 9 \) dollars per kilogram given the constraints provided.
The cost of the cashews is calculated as follows:
\[
\text{Cost of cashews} = 8 \text{ kg} \times 6 \text{ dollars/kg} = 48 \text{ dollars}
\]
The cost of the almonds is:
\[
\text{Cost of almonds} = x \text{ kg} \times 4 \text{ dollars/kg} = 4x \text{ dollars}
\]
The total weight of the mixture will be the sum of the weights of cashews and almonds:
\[
\text{Total weight} = 8 \text{ kg} + x \text{ kg} = 8 + x \text{ kg}
\]
The total cost of the mixture is:
\[
\text{Total cost} = \text{Cost of cashews} + \text{Cost of almonds} = 48 + 4x \text{ dollars}
\]
We want the mixture to cost \( 9 \) dollars per kilogram, so we set up the equation:
\[
\frac{\text{Total cost}}{\text{Total weight}} = 9
\]
Substituting in the expressions for total cost and total weight:
\[
\frac{48 + 4x}{8 + x} = 9
\]
To eliminate the fraction, multiply both sides by \( 8 + x \):
\[
48 + 4x = 9(8 + x)
\]
Expanding the right side:
\[
48 + 4x = 72 + 9x
\]
Next, we will move all terms involving \( x \) to one side and constant terms to the other:
\[
48 - 72 = 9x - 4x
\]
\[
-24 = 5x
\]
Now, solving for \( x \):
\[
x = \frac{-24}{5} = -4.8
\]
Since \( x \) needs to be a positive quantity (the number of kilograms of almonds cannot be negative), there must be an error in assuming that we can achieve a mixture worth \( 9 \) dollars using these prices.
Let's reason through this:
- The cashews cost \( 6 \) dollars per kilogram and we have \( 8 \) kg of them (totaling \( 48 \) dollars).
- The almonds cost \( 4 \) dollars per kilogram, which is less than the cashew price.
- Mixing a cheaper product (almonds) with a more expensive product (cashews) cannot yield a final mixture value higher than the more expensive ingredient.
Given that the aim was a \( 9 \) dollars per kilogram mixture, which exceeds the costliest ingredient's price, it's impossible to achieve that mixture.
Therefore, there is no amount of almonds that Su Li can use to create a mixture costing \( 9 \) dollars per kilogram. The result indicates that it is not feasible to achieve \( 9 \) dollars per kilogram given the constraints provided.