Question

a grocer wants to mix two kinds of coffee. one kind sells for $2.15 per pound, and the other sells for $2.45 per pound. he wants to mix a total of 18 pounds and sell it for $2.30 per pound. how many pounds of each kind should he use in the new mix? (round off the answers to the nearest hundredth.) answer how to enter your answer (opens in new window) 2 points number of pounds of coffee that sells for $2.15 number of pounds of coffee that sells for $2.45

Explanation:

Step1: Define Variables

Let \( x \) be the pounds of \(\$2.15\) coffee, \( y \) be pounds of \(\$2.45\) coffee. We know \( x + y = 18 \) (total pounds) and \( 2.15x + 2.45y = 2.30 \times 18 \) (total cost).

Step2: Simplify Total Cost Equation

Calculate \( 2.30 \times 18 = 41.4 \). So equation: \( 2.15x + 2.45y = 41.4 \).

Step3: Express \( y \) from First Equation

From \( x + y = 18 \), we get \( y = 18 - x \).

Step4: Substitute \( y \) into Cost Equation

Substitute \( y = 18 - x \) into \( 2.15x + 2.45y = 41.4 \):
\( 2.15x + 2.45(18 - x) = 41.4 \).

Step5: Solve for \( x \)

Expand: \( 2.15x + 44.1 - 2.45x = 41.4 \).
Combine like terms: \( -0.3x + 44.1 = 41.4 \).
Subtract 44.1: \( -0.3x = 41.4 - 44.1 = -2.7 \).
Divide by \(-0.3\): \( x = \frac{-2.7}{-0.3} = 9 \). Wait, no—wait, miscalculation? Wait, \( 2.45 \times 18 = 44.1 \), yes. Then \( 2.15x + 44.1 - 2.45x = 41.4 \) → \( -0.3x = 41.4 - 44.1 = -2.7 \) → \( x = \frac{-2.7}{-0.3} = 9 \)? Wait, that can't be. Wait, no, wait: \( 2.30 \) is the average. Wait, maybe miscalculation. Wait, let's recheck. Wait, \( 2.15 \) and \( 2.45 \), average is \( (2.15 + 2.45)/2 = 2.30 \). Oh! So they are mixed in equal amounts? Wait, \( x + y = 18 \), so \( x = y = 9 \)? Wait, but let's check: \( 2.15 \times 9 + 2.45 \times 9 = (2.15 + 2.45) \times 9 = 4.6 \times 9 = 41.4 \), and \( 2.30 \times 18 = 41.4 \). Yes! So \( x = 9 \), \( y = 9 \). Wait, but that seems symmetric. Wait, the difference between 2.30 and 2.15 is 0.15, between 2.45 and 2.30 is 0.15. So equal amounts. So \( x = 9 \), \( y = 9 \). Wait, but let's do the algebra again. Wait, \( 2.15x + 2.45(18 - x) = 41.4 \)
\( 2.15x + 44.1 - 2.45x = 41.4 \)
\( -0.3x = 41.4 - 44.1 = -2.7 \)
\( x = (-2.7)/(-0.3) = 9 \). Then \( y = 18 - 9 = 9 \). Yes, that's correct.

Answer:

number of pounds of coffee that sells for \$2.15: \( 9.00 \)
number of pounds of coffee that sells for \$2.45: \( 9.00 \)