In what proportion must rice at $0.8 per pound be mixed with rice at $0.9 per pound so that the mixture cost $0.825 per pound?

Options

A. `1 : 3`
B. `1 : 2`
C. `1 :1`
D. `3 : 1`

Answer

To solve the problem of mixing rice at different prices to achieve a specific mixture price, we can follow these steps:

### Step-by-Step Solution:

1. **Define Variables:**
Let \( X \) be the amount of rice costing $0.8 per pound, and \( Y \) be the amount of rice costing $0.9 per pound.

2. **Calculate Total Cost:**
The total cost of the rice can be expressed as:
\[ \text{Total Cost} = 0.8X + 0.9Y \]

3. **Calculate Total Weight:**
The total weight of the mixture is:
\[ \text{Total Weight} = X + Y \]

4. **Calculate Cost per Pound of Mixture:**
The cost per pound of the mixture is given by:
\[ \text{Cost per Pound} = \frac{0.8X + 0.9Y}{X + Y} \]

5. **Set the Cost per Pound to Desired Value:**
According to the problem, we want the mixture to cost $0.825 per pound:
\[ \frac{0.8X + 0.9Y}{X + Y} = 0.825 \]

6. **Cross-Multiply to Eliminate the Fraction:**
By cross-multiplying, we get:
\[ 0.8X + 0.9Y = 0.825(X + Y) \]

7. **Expand the Right Side:**
Expanding the right side gives:
\[ 0.8X + 0.9Y = 0.825X + 0.825Y \]

8. **Rearrange the Equation:**
Rearranging the equation to group like terms yields:
\[ 0.8X - 0.825X + 0.9Y - 0.825Y = 0 \]
Simplifying this gives:
\[ -0.025X + 0.075Y = 0 \]

9. **Express \( X \) in terms of \( Y \):**
Rearranging the equation gives:
\[ 0.075Y = 0.025X \]
Dividing both sides by \( 0.025 \) gives:
\[ 3Y = X \]

10. **Find the Ratio of \( X \) to \( Y \):**
This can be expressed as:
\[ \frac{X}{Y} = 3 \]
Thus, the ratio of \( X \) to \( Y \) is:
\[ X:Y = 3:1 \]

### Conclusion:
The required proportion in which rice at $0.8 per pound must be mixed with rice at $0.9 per pound to achieve a mixture costing $0.825 per pound is **3:1**.