Let's walk through the steps one-by-one to find Catron's error and provide the correct solution.
Given expression: [tex]\((-9)\left(2 \frac{2}{5}\right)\)[/tex]
First, convert the mixed fraction to an improper fraction:
[tex]\[ 2 \frac{2}{5} = 2 + \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5} \][/tex]
So, our original expression becomes:
[tex]\[ (-9) \left( \frac{12}{5} \right) \][/tex]
Let's evaluate each given step:
Step 1: [tex]\((-9)\left(2 + \frac{2}{5}\right)\)[/tex]
[tex]\[ (-9) \left(2 + \frac{2}{5}\right) \][/tex]
This step is correctly rewriting the mixed fraction as a sum.
Step 2: [tex]\((-9)(2) + (-9)\left(\frac{2}{5}\right)\)[/tex]
[tex]\[ (-9) (2) + (-9) \left( \frac{2}{5} \right) \][/tex]
This step shows the correct application of the distributive property.
Step 3: [tex]\(-18 + (-9)\left(\frac{2}{5}\right)\)[/tex]
[tex]\[ -18 + (-9) \left( \frac{2}{5} \right) \][/tex]
This step is correct; [tex]\( (-9)(2) \)[/tex] correctly simplifies to [tex]\(-18\)[/tex].
Step 4: [tex]\((-27)\left(\frac{2}{5}\right)\)[/tex]
This is where Catron makes an error. Instead of computing [tex]\( (-9) \left( \frac{2}{5} \right) \)[/tex], he mistakenly multiplies [tex]\(-27\)[/tex] by [tex]\(\frac{2}{5}\)[/tex].
The correct step should be:
[tex]\[ -18 + (-9) \left( \frac{2}{5} \right) \][/tex]
[tex]\[ = -18 + \left( - \frac{18}{5} \right) \][/tex]
[tex]\[ = -18 - \frac{18}{5} \][/tex]
To simplify this sum:
[tex]\[ -18 - \frac{18}{5} \][/tex]
Convert [tex]\(-18\)[/tex] to a fraction with a common denominator:
[tex]\[ -18 = -\frac{90}{5} \][/tex]
So, we have:
[tex]\[ -\frac{90}{5} - \frac{18}{5} = -\frac{108}{5} = -21.6 \][/tex]
Thus, the correct solution should be:
[tex]\[ -21.6 \][/tex]
Catron's error occurred in Step 4, where he incorrectly multiplied [tex]\(-27\)[/tex] by [tex]\(\frac{2}{5}\)[/tex] instead of continuing with the correct terms obtained from the distributive property.
The final correct numerical solution is:
[tex]\[ -21.6 \][/tex]
