Question

1. xavier’s problem:

\\(\frac{4w - 8}{3} + w = 2w\\)
a. complete the steps below to show the mental calculations that were made.
“i think of a number (call it w)
step 1:
step 2:
step 3:
step 4:
the result is double the number i started with.”

Explanation:

Step 1: Analyze the first operation in the equation

The equation is \(\frac{4w - 8}{3}+w = 2w\). The first part of the left - hand side is \(\frac{4w - 8}{3}\), which can be thought of as: take the number \(w\), multiply it by \(4\) to get \(4w\), then subtract \(8\) from \(4w\) to get \(4w - 8\), and then divide the result (\(4w - 8\)) by \(3\). So Step 1: Multiply the number \(w\) by \(4\), subtract \(8\), then divide by \(3\) (i.e., perform \(\frac{4w - 8}{3}\)).

Step 2: Add the original number \(w\)

After Step 1, we have \(\frac{4w - 8}{3}\), and then we add the original number \(w\) to it. So Step 2: Add the number \(w\) to the result of Step 1 (i.e., \(\frac{4w - 8}{3}+w\)).

Step 3: The result equals \(2w\)

The problem states that "The result is double the number I started with", and double the number \(w\) is \(2w\). So Step 3: The result of Step 2 is equal to \(2w\) (i.e., \(\frac{4w - 8}{3}+w=2w\)).

If we were to solve the equation:

Step 1: Eliminate the fraction

Multiply each term in the equation \(\frac{4w - 8}{3}+w = 2w\) by \(3\) to get rid of the denominator.
\(3\times\frac{4w - 8}{3}+3\times w=3\times2w\)
Simplify each term: \(4w - 8 + 3w=6w\)

Step 2: Combine like terms

Combine the \(w\) terms on the left - hand side: \((4w + 3w)-8=6w\), so \(7w-8 = 6w\)

Step 3: Solve for \(w\)

Subtract \(6w\) from both sides: \(7w-6w-8=6w - 6w\), which gives \(w - 8=0\)
Then add \(8\) to both sides: \(w-8 + 8=0 + 8\), so \(w = 8\)

But for the mental calculation steps as per the question:
Step 1: Multiply \(w\) by \(4\), subtract \(8\), divide by \(3\) (perform \(\frac{4w - 8}{3}\))
Step 2: Add \(w\) to the result of Step 1 (perform \(\frac{4w - 8}{3}+w\))
Step 3: The result is equal to \(2w\) (recognize that \(\frac{4w - 8}{3}+w = 2w\))

Answer:

Step 1: Multiply \(w\) by \(4\), subtract \(8\), divide by \(3\) (or "I take \(4\) times the number \(w\), subtract \(8\), and then divide the result by \(3\)")
Step 2: I add the number \(w\) to the result of Step 1
Step 3: I find that the result is equal to \(2w\) (or "The result of Step 2 is double the number \(w\)")