Ms. Jones is buying raisins and peanuts. She needs 8 pounds total, and she wants to have 1.5 more pounds of peanuts than raisins. The solution to the equation below shows how she can determine the solution for n, the number of pounds of peanuts she should buy.

Question

What explanations can be made for each of the three steps in the procedure?
Tell the correct explanation next to each step.
Given: n + (n - 1.5) = 8
Step 1: 2n - 1.5 = 8 Rational:
Step 2: 2n = 9.5 Rational:
Step 3: n = 4.75 Rational:

Answer 1

Let's break down each step in the procedure with corresponding explanations for the operations being performed.

Step 1: 2n - 1.5 = 8
Rational: This step combines the expressions for the total weight of the raisins and peanuts. We know that the amount of raisins is represented as (n - 1.5), where n is the amount of peanuts. By substituting this expression into the equation n + (n - 1.5) = 8, we group the peanuts and the raisins together, arriving at 2n - 1.5 = 8, which simplifies the total pounds equation.

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Step 2: 2n = 9.5
Rational: In this step, we add 1.5 to both sides of the equation to isolate the term with n. We are trying to get the expression with n on one side of the equation by eliminating the constant term on the left side. When we do this, we get 2n = 8 + 1.5, which simplifies to 2n = 9.5.

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Step 3: n = 4.75
Rational: Finally, in this step, we divide both sides of the equation by 2 to solve for n. This gives us the greater number of pounds of peanuts, n. Thus, we simplify 2n = 9.5 to find that n equals 4.75, indicating that Ms. Jones should buy 4.75 pounds of peanuts.

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Combining all the steps provides a clear solution to Ms. Jones's problem by understanding the relationships in the quantities of raisins and peanuts through algebraic manipulation.