Nested Fractions Nicely Done

A nested fraction is a fraction where the numerator and denominators are fractions themselves. It’s like one of those composite photos that’s made up of tiny photographs. Those don’t often spark horror in someone’s heart the way a nested fraction does. Don’t despair! With a few tips, you can nicely handle nested fractions.

Step 1: Ignore the assumptions listed in the problem. They are they to ensure mathematical laws are followed – no one is going to divide by zero on their watch! But you don’t need them to solve the problem.

Step 2: Find the longest line between numerator and denominator (the longest “fraction bar” in math terms). Right now think of this problem like a fraction sandwich. We need to crack it open and make an open-faced sandwich! Where should we crack it open? At the longest line. I circled all the fraction bars below, which one is the longest?

Examining all of these lines, the one in purple is the longest.

Step 3: Now you know where to crack open this fraction sandwich! Take what’s on top, keep it as a fraction, and divide it by the fraction on the bottom

Tackling what’s inside the parentheses first would tidy things up, so let’s go for it. Right now those fractions are being subtracted, so to handle the subtraction, what do we need? Yes, we need a common denominator.

Now we need to divide. How do we divide fractions? Keep, change, flip. Keep the first fraction, change the division to a multiplication, and flip the last fraction.

Now we need to simplify as much as possible.

What do the top and bottom have in common? xy, so we can cross it out top and bottom

That is a simple as it can get. Our answer is b!