Math Problem Statement

A grocer mixes peanuts that cost $1.41 per pound and walnuts that cost $2.61 per pound to make 100 pounds of a mixture that costs $1.89 per pound. How much (in lb) of each kind of nut is in the mixture?

Solution

To solve this problem, we will set up a system of equations based on the information given:

Let:

  • xx = the number of pounds of peanuts
  • yy = the number of pounds of walnuts

The problem states:

  1. The total weight of the mixture is 100 pounds: x+y=100x + y = 100

  2. The total cost of the mixture is calculated by the sum of the individual costs of peanuts and walnuts, which equals the cost of the entire mixture: 1.41x+2.61y=1.89×1001.41x + 2.61y = 1.89 \times 100

Simplifying the second equation: 1.41x+2.61y=1891.41x + 2.61y = 189

Now, we solve this system of equations:

  1. From the first equation: y=100xy = 100 - x

  2. Substitute yy in the second equation: 1.41x+2.61(100x)=1891.41x + 2.61(100 - x) = 189

  3. Simplify and solve for xx: 1.41x+2612.61x=1891.41x + 261 - 2.61x = 189 1.2x+261=189-1.2x + 261 = 189 1.2x=189261-1.2x = 189 - 261 1.2x=72-1.2x = -72 x=60x = 60

So, x=60x = 60 pounds of peanuts.

  1. Substitute x=60x = 60 into the first equation to find yy: y=10060=40y = 100 - 60 = 40

Thus, there are 60 pounds of peanuts and 40 pounds of walnuts in the mixture.

Summary

  • Peanuts: 60 pounds
  • Walnuts: 40 pounds

Would you like more details or have any questions?

Further Questions:

  1. How would the mixture's price change if the cost of walnuts increased?
  2. What happens if the total weight of the mixture changes?
  3. Can we calculate the percentage of peanuts and walnuts in the mixture?
  4. How would the mixture change if a third nut type was introduced?
  5. What is the effect of increasing the peanut cost to $2.00 per pound?

Tip: Always double-check each step in a system of equations to avoid calculation errors!

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Math Problem Analysis

Mathematical Concepts

Algebra
System of Equations
Linear Equations

Formulas

x + y = 100
1.41x + 2.61y = 189

Theorems

Linear combination of equations

Suitable Grade Level

Grades 7-10