Subtracting Mixed Numbers with Renaming (Borrowing)

Recall from our discussion of subtracting whole numbers that, in problems in which a digit in the subtrahend is larger than the corresponding digit in the minuend, we need to borrow.

A similar situation can arise when we are subtracting mixed numbers. If the fraction on the bottom is larger than the one on top, we rename (or borrow from) the whole number on top.

To Add (or Subtract) Mixed Numbers with Different Denominators

• rewrite the fractions as equivalent fractions with a common denominator, usually the LCD;
• when subtracting, rename (or borrow from) the whole number on top if the fraction on the bottom is larger than the fraction on top;
• add (or subtract) the numerators, keeping the same denominator;
• add (or subtract) the whole numbers; and
• write the answer in simplest form.

EXAMPLE 1

Subtract: .

Solution

Let’s rewrite the problem vertically.

So

As in any subtraction problem, we can check our answer by addition.

TIP

Recall that and so on. That is, any fraction having the same numerator and denominator (both nonzero) equals 1.

EXAMPLE 2

Compute:

Solution

First, we write the problem vertically.

Because is larger than we need to rename .

Finally, we subtract and then write the answer in simplest form.

EXAMPLE 3

Find the difference between .

Solution

First, we write the equivalent fractions, using the LCD.

Then, we subtract by renaming.

Simplifying the answer, we get .

EXAMPLE 4

You hike from point A to point B along a trail. If the entire trail is miles long, will you have more or less than mile left to hike when you get to point B?

Solution

First we must find the difference between the length of the entire trail, miles, and the distance hiked, mile.

The distance left to hike is mile. Finally, we compare mile and mile.

Because . Therefore and you have less than mile left to hike when you get to point B.