So this boils down to systems of equations and manipulating them.

We are given the times it takes both the machines to make $w$ widgets. Let's assign some variables:

Let $x$ be the rate of Machine A

Let $y$ be the rate of Machine B

Let $t$ be the time it takes Machine B to create $w$ widgets

If it takes 2 days longer for Machine A to produce $w$ widgets, then it takes Machine A $t+2$ days to create them.

Putting these together, we get:

$w = x(t+2)$

$w = yt$

Additionally, we know that working together, they create $\frac{5}{4}$$w$ widgets in 3 days, so that means:

$\frac{5}{4}$$w = (x+y)(3)$

Let's isolate the right side of this equation:

$\frac{5}{12}$$w = (x+y)$

What we should do now is take advantage of the first two equations we have so we can make the entire equation consist of only one variable, namely $x$, since we're looking for the time it takes Machine A to create widgets. In other words, we're looking for $t+2$, so lets replace $w$ and $y$ with $x$ wherever we can.

So let's look at the first two equations:

$w = x(t+2)$

$w = yt$

Since both are equal to $w$, let's set them equal to each other and solve for $y$ so we can sub in the value for it in the third equation above:

$yt = x(t+2)$

$y =$$\frac{{x(t+2)}}{t}$

Subbing this value for $y$ into the third equation, we get:

$\frac{5}{12}$$w = x +$$\frac{{x(t+2)}}{t}$

Now we know that:

$w = x(t+2)$

So let's sub this in for $w$:

$\frac{5}{12}$$x(t+2) = x +$$\frac{{x(t+2)}}{t}$

Now we have everything in terms of $x$, so we can begin to solve.

It may not be obvious at first glance because this has become one giant mess, but notice how we have $x(t+2)$ on both sides. Let's put them on the same side:

$\frac{5}{12}$$x(t+2) -$$\frac{{x(t+2)}}{t}$ = $x$

We can now factor out the $x(t+2)$ from the left side:

$x(t+2)($$\frac{5}{12}$$-$$\frac{1}{t}$$)$ = $x$

Now we can divide both sides by $x$:

$(t+2)($$\frac{5}{12}$$-$$\frac{1}{t}$$)$ = $1$

And simplify the fraction on the left side:

$(t+2)($$\frac{{5t-12}}{12t}$$)$ = $1$

Multiply both sides by $12t$:

$(t+2)(5t-12) = 12t$

Now we solve for $t$:

$5t^2 -12t + 10t - 24 = 12t$

$5t^2 -14t - 24 = 0$

Factoring this results in:

$(5t + 6)(t - 4)$

Which means $t = -$$\frac{6}{5}$ or $t = 4$

Negative days don't make sense, so we know that $t = 4$.

Now that we know $t$, we can solve the problem.

Recall that:

$w = x(t+2)$

Subbing in 4 for $t$ gives:

$w = x(4+2)$

$w = x(6)$

Multiply both sides by 2:

$2w = x(12)$

So it takes Machine A 12 days to make $2w$ widgets.

Therefore the answer is A