How to multiply mixed numbers

There are four steps, and only the first one is new. Once both numbers are improper fractions, the rest is ordinary fraction multiplication that most students have already met.

Step 1: Convert each mixed number to an improper fraction. Multiply the whole number by the denominator, add the numerator, and write the total over the same denominator. For 2½: 2 × 2 = 4, plus 1 = 5, giving 5/2. This works because a whole contains exactly as many fraction pieces as the denominator names: one whole is two halves, three thirds, four quarters. You are trading the wholes in for pieces so that every part of the number is counted in the same unit.

Step 2: Multiply straight across. Numerator times numerator on top, denominator times denominator on the bottom. There is no common denominator to hunt for. That requirement belongs to addition and subtraction, and applying it here is wasted effort that often introduces mistakes.

Step 3: Simplify. Find the largest number that divides evenly into both the top and the bottom, and divide both by it. Doing this before the final conversion keeps the numbers small and the division easy.

Step 4: Convert back to a mixed number. Divide the numerator by the denominator. The quotient is the whole number, the remainder becomes the new numerator, and the denominator stays the same. If the remainder is zero, the answer is simply a whole number.

A worked example: 2½ × 1¾

A useful habit at the end is to estimate. The first number is a little more than 2, the second a little less than 2, so the answer should land somewhere near 4. It does. An estimate like this catches most careless errors before they reach the answer line.

Why you can't just multiply the wholes and the fractions separately

This is the single most common error, and it is worth ten minutes at the kitchen table. Faced with 2½ × 1¾, a lot of students multiply 2 × 1 = 2, then multiply ½ × ¾ = ⅜, and write 2⅜. The real answer is 4⅜, almost twice as much.

The reason is easiest to see as a rectangle. Draw one that is 2½ units wide and 1¾ units tall, then cut it along the whole-number marks. You do not get two pieces. You get four:

• 2 × 1: the big whole-by-whole block
• ½ × 1: a strip down the side
• 2 × ¾: a strip along the bottom
• ½ × ¾: the small corner square

Multiplying the wholes and the fractions separately counts only the first and last of these. The two strips in the middle, which make up most of the missing amount, get dropped entirely. The calculator above draws this rectangle for any problem with a whole number and a fraction on both sides, with the forgotten strips shaded blue, which is usually the moment the idea clicks.

Converting to improper fractions is not an arbitrary rule a textbook imposed. It is the shortcut that handles all four pieces automatically, so nobody has to remember to add them up.

Multiplying by a whole number or a plain fraction

Mixed number × whole number

Write the whole number as a fraction over 1, then follow the same four steps. For 3⅖ × 4, convert 3⅖ to 17/5 and 4 to 4/1, multiply across to get 68/5, and convert back to 13⅗. Set the fraction boxes in the calculator to 0 over 1 to run problems of this shape.

Mixed number × proper fraction

Convert only the mixed number; the proper fraction is already in the right form. For 2⅓ × ⅘, that is 7/3 × 4/5 = 28/15 = 1 13/15. Leave the whole-number box at 0 for the fraction side.

Common mistakes to watch for

• Multiplying the parts separately. The mistake described above. If the answer looks noticeably too small, this is almost always why.
• Looking for a common denominator. Needed for adding and subtracting, never for multiplying.
• Adding instead of multiplying when converting. The conversion is whole × denominator, then add the numerator. Reversing the order gives a different, wrong fraction.
• Stopping at the improper fraction. 35/8 is a correct value, but most worksheets want 4⅜. Read the instructions at the top of the page.
• Forgetting to simplify. 12/16 and 3/4 are the same amount, but only one of them will be marked right.