Simplify this first:$$ab+(k-a)(k-b)=2ab-ak-bk+k^2$$This means we need a mental path to get either near or exactly that total.
Obviously, as k gets large, dealing with this via mental math becomes tougher. The method described below works well, assuming $a$ and $b$ are 2-digit numbers.
First, we need to establish a starting point. Knowing $k$, here's a series of mental steps. Calculate:
$$\frac{1}{2}k$$
Next, square this:
$$(\frac{1}{2}k)^{2}$$
Double this:
$$2((\frac{1}{2}k)^{2})$$
Those mental steps give you a simple path to:
$$\frac{1}{2}k^{2}$$
Mental math techniques for squaring 2-digit numbers are easily found on the internet if you don't already know them.
Example: We know that, in your example, $k=110$, so:$$\frac{1}{2}(110)=55$$ Next,$$55^{2}=3,025$$ (There happen to be specific mental math tricks for squaring 2-digit numbers ending in 5, so this is surprisingly easy). Don't forget the doubling step!$$3,025 \times 2=6,050$$
Now that we have a starting point $(\frac{1}{2}k^{2})$, we need to either work out or estimate a distance from that starting point. Calculate both:$$a-\frac{1}{2}k \\ b-\frac{1}{2}k$$ From there, you'll multiply them together: $$(a-\frac{1}{2}k)(b-\frac{1}{2}k)$$ Double this total: $$2((a-\frac{1}{2}k)(b-\frac{1}{2}k))$$What you actually wind up calculating mentally here is:$$2ab-ak-bk+\frac{1}{2}k^2$$
Your final estimate can be given by adding this number to your starting point above:$$\frac{1}{2}k^2+(2ab-ak-bk+\frac{1}{2}k^2)=2ab-ak-bk+k^2$$
$2ab-ak-bk+k^2$ should look familiar. It's the answer for which we've been searching!
If $k$ happens to be even, and both $a-\frac{1}{2}k$ and $b-\frac{1}{2}k$ happen to be small enough to easily multiply and double, then you're on your way to an exact answer!
If multiplying $a-\frac{1}{2}k$ and $b-\frac{1}{2}k$ together seems daunting, simply round both numbers to the nearest power of 10, and multiply those together and double that instead. We're just going for an approximation, after all.
Example I: In your example, $a=75$, $b=53$, and $\frac{1}{2}k=55$. We work out$$75-55=20 \\ 53-55=-2$$ Multiply:$$20(-2)=-40$$Double:$$2(-40)=-80$$ Add this to the starting point, working it out mentally: $$\\ 6,050 + (-80) \\ 6,050 - 80 \\ 6,000 - 30 \\ 5,970$$
In this case, it happens that we can easily calculate an exact answer. $$(75)(53)+(57)(35)=5,970$$
What happens when this isn't the case? Let's try this with an odd $k$ and an $a$ and $b$ that aren't very close to $\frac{1}{2}k$. Let's try the more difficult:$$(109)(31)+(92)(14)$$
Example II: In this example, we can work out that $k=123$. First, we work out:$$\frac{1}{2}k=61.5 \\ 61.5 \approx 62$$Next, square:$$62^{2}=3,844$$Double that for the final step:$$2(3,844) = 7,688$$This is our starting point.
How much do we adjust this?$$109-62=47 \\ 31-62=-31$$If you're good with mentally multiplying numbers like this together, do it! Let's assume you're not, however:$$47 \approx 50 \\ -31 \approx -30$$
Multiplying those numbers together is much easier:$$50(-30) = -1,500$$Double that:$$2(-1,500) = -3,000$$
So, our final estimate would be:$$7,688 + (-3,000) \\ 7,688 - 3,000 \\ 4,688$$
The actual answer to the problem is $(109)(31)+(92)(14) = 4,667$, so our estimate of $4,688$ is quite good! How good?$$\frac{4,688}{4,667} \approx 1.0045$$It's not only within 5%, it's within 0.5%!
