Fraction multiplication: worked examples (2/3 × 2/3 and 23 more)

The fraction multiplication question people ask more than any other is “what is 2/3 times 2/3 in fraction form?” — so let me answer it up front: 4/9. You multiply straight across, 2 × 2 = 4 on top and 3 × 3 = 9 on the bottom, and 4/9 is already in lowest terms. That single rule — numerators across, denominators across, then simplify — solves every problem on this page. The runners-up in the most-asked list — 2/3 x 3, 6×1/3, 1/2 times 2, 1/3 x 6, 3/4 times 2, 2/3 x 1/16 and 1/2 times 1/2 — are all in the table too. I have worked the 24 most-searched multiplication problems below, each in the same three steps, so you can check your own homework against them or lift the pattern for any problem of your own.

Why the rule works (and why answers shrink)

Fraction multiplication confuses people because the answer is often smaller than both numbers you started with. The reason is hiding in plain sight: multiplying by a fraction means taking a part of something. 2/3 × 2/3 asks for two-thirds of two-thirds. Picture a bar split into thirds; take 2 of those thirds; now split what you took into thirds again and keep 2 of them. You are holding 4 little pieces, each one-ninth of the original bar: 4/9. Of course it is smaller — you took a part of a part. Once I started reading “×” as “of,” every answer in the table below became something I could sanity-check before computing it.

The mechanics for whole numbers are one extra step: a whole number becomes a fraction by giving it a denominator of 1. 2/3 × 3 is 2/3 × 3/1 = 6/3 = 2. Nothing else changes — the rule is still “across, across, simplify.”

Two problems worked in full

2/3 × 2/3, step by step

1. Multiply the numerators: 2 × 2 = 4.
2. Multiply the denominators: 3 × 3 = 9.
3. Simplify if you can: 4 and 9 share no common factor, so 4/9 is final. (As a decimal check: 4 ÷ 9 = 0.444…, and 2/3 × 2/3 = 0.667 × 0.667 ≈ 0.444 ✓.)

3/4 × 2, with cancelling

Write the 2 as 2/1: 3/4 × 2/1. Before multiplying, notice the 2 on top and the 4 below share a factor of 2 — cancel them first (4 becomes 2, the 2 becomes 1): 3/2 × 1/1 = 3/2 = 1 1/2. Cancelling early keeps the numbers small; multiplying first (6/4) and simplifying afterward (÷ 2 → 3/2) reaches the same answer by a longer road. I cancel early whenever I can spot the shared factor, and the table below notes the problems where cancelling does the heavy lifting.

Every worked example

The full set, grouped the way the questions are usually asked. “Multiply across” shows the unsimplified product; the answer column is reduced, as a mixed number or whole number whenever the result is improper.

All 24 problems recomputed exactly. Note the phrasing twins — “2/3 × 3” and “2/3 times 3” are the same problem, as are “1/3 × 2″/”1/3 times 2” and “6 × 1/3″/”6 x 1/3” — each pair shares a row’s working but is listed so you can find your exact question.

A pattern worth one minute of your attention: look at how many answers come out as whole numbers. Any time the whole number is a multiple of the denominator — 3 with thirds, 6 with thirds, 2 with halves, 4 with halves — the denominator divides away completely and the answer is a whole number. And any fraction multiplied by its own denominator’s number (1/3 × 3, 1/2 × 2, 3 × 1/3) gives exactly 1. Those two patterns let me predict most of the table before touching it, which is the real skill: the multiplication itself is the easy part. The remaining searches are these same rows in the other notation — 1/4 times 2 = 1/2, 1/2 times 4 = 2, and 3 times 1/3 = 1 — worth one line of their own, since “times” and “×” and “x” are three spellings of one instruction.

The two-thirds ladder: 2/3 x 2, 2/3 x 3, 2/3 x 6, 2/3 x 9

Four of the most-searched problems are one family — two-thirds times a growing whole number — and lining them up shows the machinery better than any single problem. 2/3 x 2 = 4/3 = 1 1/3. 2/3 x 3 = 6/3 = 2 exactly. 2/3 x 6 = 12/3 = 4. 2/3 x 9 = 18/3 = 6. Read the answers: every time the whole number grows by 3, the answer grows by 2 — because 2/3 × 3 = 2 is the engine of the whole ladder, and 6 and 9 are just 3 twice and three times. Once you spot that, the reversed searches fall for free: multiplication does not care about order, so 3 x 2/3 = 2 and 6 x 2/3 = 4 are the same rungs climbed from the other side. I use this ladder as my mental benchmark: any “fraction × whole number” problem that surprises me gets checked against it.

Fraction times fraction: eight more, with cancelling

The main table leans on whole numbers; here is the pure fraction-times-fraction set, chosen because every row rewards cancelling before you multiply. Watch how often the answer lands on 1/2 — cross-cancellation is quietly doing that.

Eight fraction-times-fraction products, recomputed exactly, with the pre-cancellation that makes each one small before it is multiplied.

When the answer is bigger than both numbers

Everything so far multiplied proper fractions, so answers shrank. Feed the rule an improper fraction and it grows instead: 5/4 × 2 = 10/4 = 2 1/2 — bigger than the 2 you started with, because 5/4 is more than 1, and taking “five-quarters of” something means taking all of it plus a quarter more. The rule never changes; only the size of the multiplier decides the direction. That is the complete picture: multiply by less than 1, shrink; by exactly 1, stay; by more than 1, grow.

Multiplying three fractions

Three fractions are no harder — multiply all the numerators together and all the denominators together. The showpiece is 1/2 × 2/3 × 3/4: across the top, 1 × 2 × 3 = 6; across the bottom, 2 × 3 × 4 = 24; 6/24 simplifies to 1/4. Look closer and there is a telescope hiding in it: each numerator cancels the next denominator (the 2 kills the 2, the 3 kills the 3), leaving 1/4 standing almost immediately. The same trick runs 2/3 × 3/4 × 4/5 = 24/60 = 2/5: cancel the 3s and the 4s first and the problem is 2/5 before you have multiplied anything at all. Chains of fractions with consecutive numbers are built to telescope — when you see one, cancel diagonally before reaching for the big multiplication.

“Of” problems: where these answers live

Nearly every real-life multiplication of fractions arrives wearing the word “of,” and translating is the whole skill. Four translations, all answered by rows you have already seen:

• 2/3 of 9 students in a group of 9 are picked: 2/3 × 9 = 6 students.
• 3/4 of 2 hours of practice: 3/4 × 2 = 1 1/2 hours — the 3/4 times 2 row, in a tracksuit.
• 1/2 of 1/2 a cup of flour when halving a recipe: 1/2 × 1/2 = 1/4 cup — the most-used multiplication in home baking.
• 5/6 of 12 months of a year: 5/6 × 12 = 60/6 = 10 months.
• 1/3 of 6 slices of a six-slice pie: 1/3 × 6 = 2 slices.

When a word problem says “of,” I write “×” over it without a moment’s hesitation, and the problem becomes a table row. That single translation habit is worth more than any shortcut on this page.

Squaring a fraction: the headline, revisited

One special case deserves its own moment, because it is secretly the most-searched problem on this page in a second costume: multiplying a fraction by itself. 2/3 × 2/3 is “(2/3) squared,” and the area picture explains its answer — a square that is 2/3 of a unit on each side covers 2/3 × 2/3 = 4/9 of the unit square, which you can literally count on a 3-by-3 grid: 4 of the 9 little squares. The family: (1/2)² = 1/4 (a half-by-half square is a quarter of the whole), (3/4)² = 9/16, (5/6)² = 25/36. Squaring a proper fraction always shrinks it — you are taking a part of a part — and the grid picture is the fastest way I know to make that feel inevitable rather than merely true.

Method: standard fraction multiplication (a/b × c/d = ac/bd, then reduce by the GCD) per standard mathematics textbooks. Every product in the table was recomputed exactly for this article, and spot-checked in decimal form.