Question

Do this problem - but make a very slight mistake. DO NOT answer correctly, but the mistake hads to be sometime within the problem solving. ((1/m^3p)+(2/mp^2))/((4/mp)+(1/m^2p))

Answer 1

To simplify the expression \(\frac{\frac{1}{m^3p} + \frac{2}{mp^2}}{\frac{4}{mp} + \frac{1}{m^2p}}\), we start by finding a common denominator for the terms in the numerator and the denominator.

**Step 1: Simplify the numerator**

The first term in the numerator is \(\frac{1}{m^3p}\) and the second term is \(\frac{2}{mp^2}\).

The common denominator for these two fractions is \(m^3p^2\).

Rewriting the first term:
\[
\frac{1}{m^3p} = \frac{1 \cdot p}{m^3p^2} = \frac{p}{m^3p^2}
\]

For the second term:
\[
\frac{2}{mp^2} = \frac{2 \cdot m^2}{mp^2 \cdot m^2} = \frac{2m^2}{m^3p^2}
\]

Now adding these two:
\[
\frac{p + 2m^2}{m^3p^2}
\]

**Step 2: Simplify the denominator**

The first term in the denominator is \(\frac{4}{mp}\) and the second term is \(\frac{1}{m^2p}\).

The common denominator here is \(m^2p\).

Rewriting the first term:
\[
\frac{4}{mp} = \frac{4m}{mp \cdot m} = \frac{4m}{m^2p}
\]

For the second term:
\[
\frac{1}{m^2p} = \frac{1}{m^2p}
\]

Now adding these two:
\[
\frac{4m + 1}{m^2p}
\]

**Step 3: Put it all together**

Now we can substitute both the simplified numerator and the denominator back into the original expression:
\[
\frac{\frac{p + 2m^2}{m^3 p^2}}{\frac{4m + 1}{m^2 p}}
\]

This can be simplified to:

\[
\frac{(p + 2m^2) \cdot m^2p}{(4m + 1) \cdot m^3p^2}
\]

Canceling \(p\) and simplifying gives us:
\[
\frac{(p + 2m^2)m^2}{(4m + 1)m^3}
\]

Now, let’s add a slight mistake here as requested. I will write it as:
\[
\frac{(p + 2m^2)m^2}{(4m + 1)m^2}
\]

This is incorrect because we should still have \(m^3\) in the denominator instead of \(m^2\).

So, the final expression (with a slight mistake) is:
\[
\frac{(p + 2m^2)m^2}{(4m + 1)m^2}
\]