How to Multiply Fractions Step by Step

Suppose you want to multiply 2/3 by 4/5. Multiply the numerators: 2 times 4 equals 8. Then multiply the denominators: 3 times 5 equals 15. The answer is 8/15. Because 8 and 15 have no common factor greater than 1, the fraction is already in simplest form.

The rule works because multiplication of fractions can represent taking a fraction of another fraction. If you take one half of three quarters, you are finding 1/2 times 3/4. Multiplying gives 3/8. A diagram split into equal parts would show the same result, which is useful if the symbolic rule feels too automatic.

This is the central process for multiplying fractions. Numerator times numerator, denominator times denominator, then simplify. Do not add the denominators, and do not look for a common denominator unless a different operation requires one. Addition and subtraction are different because the pieces must refer to equal-sized parts before their numerators can be combined. Multiplication does not combine two sets of parts in that way, so the denominators can be multiplied directly.

You can often cancel common factors before multiplying. Suppose you have 3/10 times 5/9. The 3 and 9 share a factor of 3, so they reduce to 1 and 3. The 5 and 10 share a factor of 5, so they reduce to 1 and 2. You are left with 1/2 times 1/3, which gives 1/6.

This technique is sometimes called cross-cancelling. It does not change the value of the expression because you are dividing a numerator and a denominator by the same nonzero factor. It simply keeps the numbers smaller and reduces the chance of arithmetic mistakes.

When multiplying fractions, simplifying first is optional but often helpful. You can always multiply first and simplify the final fraction afterward. Both routes should lead to the same result. If the final numbers are large, look for common factors again. Simplifying is not a separate trick attached to the end of the method. It is simply a way to write the same fraction in a cleaner form.

A whole number can be written as a fraction with denominator 1. For example, 3 times 2/5 becomes 3/1 times 2/5, which equals 6/5. You can leave that as an improper fraction or write it as 1 1/5 if the question expects a mixed number.

Mixed numbers should usually be changed into improper fractions before multiplying. For example, 1 1/2 becomes 3/2. If you multiply 1 1/2 by 2/3, you get 3/2 times 2/3. The 3s and 2s cancel, leaving 1.

A quick how to multiply fractions checklist is useful: convert mixed numbers, simplify common factors if convenient, multiply across, and simplify the final answer. Also estimate the size. If you multiply two positive fractions smaller than 1, the result should be smaller than either starting fraction. You can also compare with decimals when a fraction is familiar. One half times one half should be one quarter, not one whole, because you are taking only half of an amount that was already half.

An area model makes the rule visible. Draw a square and split it into three equal columns, shading two of them to show 2/3. Then split the same square into five equal rows and shade four of them in a different colour to show 4/5. The square now contains fifteen small rectangles, which is where the denominator comes from, and the overlap where both shadings meet covers eight of them, which is the numerator. The picture shows why denominators multiply: cutting in both directions creates smaller pieces, and their number is the product of the two cuts.

In word problems, the word 'of' usually signals multiplication. If 3/4 of a class of 32 students walk to school, you calculate 3/4 times 32, which is 24. If 2/3 of those walkers live north of the school, then 2/3 of 3/4 of the class is 1/2 of the class, or 16 students. Working with the fractions first and applying the whole number last often keeps the numbers small. A quick sense check helps too: taking part of a group can never give you more people than the group started with.

Once you can multiply fractions, dividing them follows quickly. The reciprocal of a fraction is that fraction turned upside down, so the reciprocal of 2/3 is 3/2. A fraction multiplied by its reciprocal always gives 1, because 2/3 times 3/2 is 6/6. To divide by a fraction, multiply by its reciprocal instead. For example, 3/4 divided by 1/8 becomes 3/4 times 8/1, which is 24/4, or 6. A sense check confirms it: three quarters really does contain six eighths.

Why does flipping work? Dividing asks how many of one amount fit into another. Dividing by 1/2 asks how many halves fit, and every whole contains two of them, so dividing by 1/2 gives the same result as multiplying by 2. The same reasoning holds for any fraction: dividing by 2/3 means counting how many two-thirds pieces fit, which is one and a half times as many as the number of wholes. This also explains a result that surprises many learners, that dividing by a fraction smaller than 1 makes the answer larger.

Multiplying fractions is the everyday tool of probability. When two events are independent, meaning one does not affect the other, the chance of both happening is found by multiplying their probabilities. The chance of rolling a six on an ordinary dice is 1/6 and the chance of a coin landing heads is 1/2, so the chance of getting both together is 1/6 times 1/2, which is 1/12. This fits the meaning of 'of': you are finding one half of the one sixth of occasions when a six turns up.

The method still works when the first event changes the second, as long as you use the updated fraction. Suppose you draw two cards from a shuffled pack of 52 without putting the first back. The chance that the first is an ace is 4/52. If it is, 3 aces remain among 51 cards, so the chance of a second ace is 3/51. Cancelling first, 4/52 becomes 1/13 and 3/51 becomes 1/17, so the chance of two aces is 1/13 times 1/17, which is 1/221. Without cancelling you would face 12/2652, which simplifies to the same answer.

Successive discounts are a classic place where multiplying fractions beats intuition. A shop offers 20 percent off, then an extra 10 percent off at the till. It is tempting to add these to make 30 percent, but the second discount applies to a price that has already been reduced. Paying 20 percent less means paying 4/5 of the price, and the extra discount means paying 9/10 of that, so you pay 4/5 times 9/10, which is 36/50, or 18/25. That is 72 percent of the original price, a total saving of 28 percent rather than 30, whichever discount comes first.

Recipes work the same way. If a recipe for 6 people needs 3/4 of a cup of milk and you are cooking for 4, you need 4/6 of the recipe, which simplifies to 2/3. Then 2/3 times 3/4 gives 6/12, or 1/2 a cup. Cancelling first makes it quicker still: the 3s cancel, leaving 2/4, which is 1/2. Scaling every ingredient by the same fraction keeps all the proportions intact, so the dish tastes the same. Keeping the scale factor as a fraction rather than a rounded decimal such as 0.67 also stops small errors building up across a long list of ingredients.

Multiplying fractions comes down to multiplying numerators, multiplying denominators, and simplifying. Common denominators are not needed. Convert whole and mixed numbers into fractions first, cancel common factors when it helps, and use a rough size estimate to catch answers that do not make sense.