October 2026 - Multiplying & Dividing Fractions Study Guide

Multiplying

Three steps, and the middle one is optional but saves work.

1. Convert any mixed numbers
2. Cancel common factors
3. Multiply across
4. Simplify

Cancelling before you multiply is worth the habit. For 4/9 × 3/8, cancel the 4 with the 8 (leaving 1 and 2) and the 3 with the 9 (leaving 1 and 3), which reduces the whole problem to 1/3 × 1/2 = 1/6. Multiplying first gives 12/72, the same answer after a harder simplification.

One more thing worth knowing: multiplying by a fraction smaller than 1 makes a number smaller. If a question asks for a fraction *of* something, the answer must come out below the original, which is a useful sanity check on the answer choices.

Dividing: keep, change, flip

Division by a fraction is multiplication by its reciprocal. The rule is mechanical and it always works.

• Keep the first fraction exactly as it is.
• Change the division sign to multiplication.
• Flip the second fraction — swap its numerator and denominator.
• Then multiply as usual, cancelling first if you can.

A division that looks harder than it is

What each operation does to size

A quick plausibility check before you commit to an answer.

Terms to know

Reciprocal

A fraction turned upside down. The reciprocal of 2/5 is 5/2.

Cancelling

Dividing a numerator and a denominator by a common factor before multiplying.

Improper fraction

A fraction larger than 1, such as 9/4. Convert mixed numbers to this form before multiplying.

Of

In a word problem, "of" nearly always means multiply.

Per

In a word problem, "per" or "how many fit into" usually means divide.

What the TEAS asks most

The same core ideas appear in different wording. If you can answer these without stopping to think, you have what this section requires.

Key points

• Multiply straight across; no common denominator is needed.
• Divide by flipping the second fraction and multiplying.
• Cancel before multiplying — it is faster and avoids simplifying a large fraction later.
• Convert mixed numbers to improper fractions first.
• Multiplying by less than 1 shrinks a number; dividing by less than 1 grows it.