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The trickiest part of this problem is transforming the words into a mathematical equation. In our equations, we'll let P represent the number of boxes of pecans, and W represent the number of boxes of walnuts.
First, we know that there were 462 total pounds, and each pecan box weighs 3 pounds, while walnuts weigh 2. This gives us the following equation:
(1) 3P + 2W = 462
Next, we are told there are 24 fewer boxes of walnuts than pecans, giving us the following equation:
(2) P - W = 24.
We now have to equations and two unknows, so we can proceed to solve for P and W. To do this, we'll start by rewriting (2) in terms of P:
(3) P = 24 + W
We now plug this value for P back into (1). This will give us an equation just in terms of W. We will then solve for W
(4) 3(24 + W) + 2W = 462
=> 72 + 3W + 2W = 462
=> 72 + 5W = 462
=> 5W = 390
=> W = 78
We now know that there are 78 total boxes of walnuts. Now that we know the total boxes of walnuts, we can find pecans using (3):
P = 24 + W
=> P = 24 + 78 = 102
We have now found that there were 102 boxes of pecans and 78 boxes of walnuts.
But the problem was asking of for the total weight of the boxes. Each pecan box weighs 3 pounds, and each walnut box weight 2 pounds, so we can find this easily:
Total weight of pecans = 102 * 3 = 306 lbs
Total weight of walnuts = 78 * 2 = 156 lbs
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