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Chapter 41: Problem 1183

A grocer mixes two grades of coffee which sell for 70 cents and 80 cents per pound, respectively. How much of each must he take to make a mixture of 50 pounds which he can sell for 76 cents per pound?

Short Answer

Expert verified
The grocer must mix 20 pounds of the 70 cents per pound coffee and 30 pounds of the 80 cents per pound coffee to create a 50-pound mixture that can be sold for 76 cents per pound.

Step by step solution

01

Set up the equations

First, we know that the total weight of the mixture is 50 pounds, so we have: \( x + y = 50 \) Next, we set up an equation for the cost of the mixture. The total cost should be equal to the 50 pounds of coffee mixture's cost (76 cents per pound): \( 0.70x + 0.80y = 50 * 0.76 \)
02

Simplify the equations

First, we will multiply the cost equation with 100 to simplify the calculations: \( 70x + 80y = 50 * 76 \) Now our system of equations is: \( x + y = 50 \) \( 70x + 80y = 3800 \)
03

Solve the system of equations

We will now solve this system of equations using the substitution method. From the first equation, we have \( y = 50 - x \). We will substitute this expression into the second equation. \( 70x + 80(50 - x) = 3800 \) Now, we solve for x: \( 70x + 4000 - 80x = 3800 \) \( -10x = -200 \) \( x = 20 \) Now that we have the value of x, we can find the value of y: \( y = 50 - x = 50 - 20 = 30 \)
04

Interpret the solution

So, the grocer needs 20 pounds of the 70 cents per pound coffee and 30 pounds of the 80 cents per pound coffee to create a 50-pound mixture that can be sold for 76 cents per pound.

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