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CalcMax

Multiplying Fractions Calculator

Range: -94,906,265 – 94,906,265

Range: 1 – 100,000

Range: -94,906,265 – 94,906,265

Range: 1 – 100,000

Result

1/2

Product (simplest form)

Before reducing
6/12
First fraction, cancelled
1
Second fraction, cancelled
1/2

A multiplying fractions calculator takes two fractions and gives you their product — and then shows you the shortcut that makes the multiplication easy. 2/3 times 3/4 is 1/2, and the reason it is worth showing is that the raw multiplication gives 6/12: correct, and a long way from the answer. Cross-cancelling gets there first. Before multiplying, it looks at each numerator against the denominator diagonally opposite it and divides both by whatever they have in common; here the 2 above the line cancels against the 4 below the other line, and the 3 cancels against the other 3. What is left is 1/1 times 1/2, which is the answer without a reducing step at the end. Both of those readings are on the panel. The four numbers this page prints are the answer in lowest terms, the same product before anything was cancelled, and the two factors as they look once the cancelling has been done — the last two being what makes the shortcut checkable rather than a trick to be memorised. Multiplying fractions is the one operation that needs no common denominator, and it is worth noticing why: adding quarters and thirds means finding twelfths first, while multiplying them multiplies the two denominators together and no such step exists. The one thing that trips people up is which pairs cancel, and the answer is the diagonal ones — numerator against the opposite denominator — never the two numerators against each other, because those are multiplied rather than cancelled.

Eight fraction products, cancellations shown

First fractionSecond fractionCancelled firstCancelled secondProduct
2/33/411/21/2
3/48/912/32/3
5/63/101/21/21/4
7/84/57/21/57/10
-2/33/4-11/2-1/2
-7/93/7-1/31-1/3
1/22111
99/100100/99111

Every row is the same two steps — cancel the diagonals, then multiply across — on a different pair, and the five columns are the two inputs, the two factors after cancelling, and the answer in lowest terms. Multiplying the two cancelled columns together always gives the product in the last column, which is the check that the cancelling was done against the right partners: if the two cancelled factors do not multiply to the product, a numerator was matched with the wrong denominator. The first three rows are pairs where both diagonals cancel and the factors come back simple; the third of them, 5/6 times 3/10, cancels in both directions and lands on 1/4. The fourth row is the other shape, where one diagonal is coprime and passes through untouched — 7 against 5 — so the first cancelled factor keeps a denominator of 2 while the other fully cancels. The fifth row is negative, and the minus sign survives the cancelling on the numerator. The sixth row has the two fractions cancelling fully, 3 against 9 and 7 against 7, so the second factor comes back as a bare 1. The seventh row multiplies a half by a whole number written as 2/1, which needs no special case. The eighth row is a fraction times its own reciprocal: everything cancels, both factors come back as 1, and the answer is 1 — the row where the shortcut has saved the most work, since multiplying straight across would have written down 9900 over 9900. Read the first two columns across and the answers down: the same four numbers rearranged give different products, which is why entering them in the wrong fields is worth checking against the unreduced line.

Formula

a/b × c/d = (a × c) / (b × d) Cross-cancelling: divide a and d by gcd(a, d), and c and b by gcd(c, b), before multiplying Then reduce the product

First numerator
The top of the first fraction: 2 in 2/3. It cancels against the denominator of the other fraction, not against its own, and not against the other numerator. It may be negative, and the sign travels with it through the cancelling and into the answer.
First denominator
The bottom of the first fraction: 3 in 2/3. It must be positive — a fraction carries its sign on the numerator, and that convention is what makes the cancelling unambiguous. It cancels against the second fraction's numerator.
Second numerator
The top of the second fraction: 3 in 3/4. It cancels against the first denominator, which in the default case is the same 3, so the two vanish together and both of the cancelled readings come out simpler.
Second denominator
The bottom of the second fraction: 4 in 3/4. It cancels against the first numerator — the 2 against the 4 in the default case — and it must be positive for the same reason the first denominator must be.
Cross-cancelling
Dividing one fraction's numerator and the other's denominator by their greatest common divisor before multiplying. Both diagonal pairs are cancelled, and the point of doing it first is that the product arrives already in lowest terms, so there is no long division at the end. It works because dividing one factor and multiplying by the same number cancel out.
Cancelled factors
The two fractions as they look once the diagonal pairs have been cancelled: 1/1 and 1/2 in the default case. They multiply to the answer's value, which is how the shortcut is checked — if the two readings do not multiply to the product, something was cancelled against the wrong partner. They are printed as they are, so a factor that cancels completely shows as a bare 1.
Product before reducing
What multiplying straight across gives, with no cancelling: 6/12 for the default case. It is the same number as the answer and a much larger pair of integers, and it is printed so that the difference between the two routes is visible rather than asserted.

The first use is the exercise: multiplying fractions is taught before dividing them, and it is the operation where the diagonal cancelling is introduced, which is why the panel prints both of the cancelled factors rather than just the answer. The second is checking work done by hand — enter the two fractions, read the cancelled factors against yours, and if yours differ the product will differ too. The third is a product that reduces a long way, such as 99/100 times 100/99, where multiplying straight across gives 9900/9900 and cross-cancelling gives 1 without any arithmetic at all. The fourth is a negative fraction, where the sign has to survive the cancelling: -2/3 times 3/4 is -1/2, and the sign belongs to the numerator that carries it. The fifth is a fraction multiplied by a whole number, which is just the second fraction written with a denominator of 1 — 1/2 times 2 is 1/2 times 2/1 — and the page takes it without any special case.

Worked examples

  1. 2/3 × 3/4

    1. Cancel the diagonals: 2 against 4 share a factor of 2, giving 1 and 2; 3 against 3 share 3, giving 1 and 1
    2. The two factors are now 1/1 and 1/2
    3. Multiply across: (1 × 1) ÷ (1 × 2) = 1/2
    4. Without cancelling it would have been (2 × 3) ÷ (3 × 4) = 6/12, which is the same number

    The default case, and the one that shows what cross-cancelling buys: the two routes produce 1/2 and 6/12, the same value, and only one of them is already in lowest terms. Note the first cancelled factor prints as a bare 1 rather than as 1/1 — every one of its numbers cancelled, and a denominator of 1 is written without one. The check is in the last two readings: multiply them and you get the answer, 1 × 1/2 = 1/2.

  2. 7/8 × 4/5

    1. Cancel the diagonals: 7 against 5 share nothing, so both stay; 4 against 8 share 4, giving 1 and 2
    2. The two factors are now 7/2 and 1/5
    3. Multiply across: (7 × 1) ÷ (2 × 5) = 7/10
    4. Without cancelling: (7 × 4) ÷ (8 × 5) = 28/40, the same number with bigger integers

    The case where only one of the two diagonals cancels: 7 and 5 are coprime, so that pair passes through untouched and the first factor keeps a denominator of 2, while the other diagonal cancels fully and the second factor drops to 1/5. Half the work of cross-cancelling is recognising when there is nothing to do, which is why a reading that comes back unchanged is information rather than a failure. The answer 7/10 is already in lowest terms, and the unreduced line 28/40 shows what the long route would have handed over.

  3. 99/100 × 100/99

    1. Cancel the diagonals: 99 against 99 share 99, giving 1 and 1; 100 against 100 share 100, giving 1 and 1
    2. Both factors are now 1/1
    3. Multiply across: 1 × 1 = 1
    4. Without cancelling: (99 × 100) ÷ (100 × 99) = 9900/9900, which is also 1

    A fraction multiplied by its own reciprocal, which is the case that shows how much the shortcut saves: the long route builds 9900 over 9900 and then has to reduce it back to 1, while cross-cancelling never writes either number down. The two readings come back as bare 1s because both factors cancelled completely, which is the shape of a reciprocal in this notation. It is also the clearest illustration of why the cancelling works — the 99 and the 100 are each being multiplied and divided by the same amount.

  4. -2/3 × 3/4

    1. Cancel the diagonals: -2 against 4 share a factor of 2, giving -1 and 2; 3 against 3 share 3, giving 1 and 1
    2. The two factors are now -1/1 and 1/2
    3. Multiply across: (-1 × 1) ÷ (1 × 2) = -1/2
    4. The sign belongs to the numerator throughout

    A negative fraction, where the sign rides along with the numerator through the cancelling: -2 and 4 share a factor of 2, and what is left is -1 and 2, so the minus is still there after the cancellation rather than being cancelled away with it. Note that the denominator stays positive on every line — a fraction carries its sign on top, which is exactly the convention that keeps this unambiguous. One negative factor makes a negative product; two would have made a positive one.

  5. 1/2 × 2

    1. Write the whole number as a fraction: 2 is 2/1
    2. Cancel the diagonals: 1 against 1 share nothing; 2 against 2 share 2, giving 1 and 1
    3. Multiply across: 1 × 1 = 1
    4. So a half of two is one

    A whole number, which needs no special case: 2 is 2/1, so the page treats it as any other fraction and the diagonal cancelling finds the 2 against the 2. Without that step the product would be 2/2 and would still have to be reduced, which is the difference the shortcut makes on an example this small. The answer coming out as a bare 1 rather than 1/1 is the same convention as everywhere else on the page: a denominator of one is not written.

Limitations

Both denominators must be positive, since a fraction carries its sign on the numerator — and that convention is doing real work here, because the cancelling matches numbers up diagonally and would be ambiguous if either denominator could be negative. The numerators may be negative, and the sign travels into the answer. All four numbers are limited in size: the numerators and the denominators are each capped at about ninety-four million, which is the largest value for which the two numerators multiplied together still land on an integer that can be represented exactly. A product past that would be quietly inexact, so the page refuses the entry rather than printing a number that looks right. Zero is accepted in a numerator — the product is then zero, and both cancelled factors come back as zero over something — but a denominator of zero is not, since the fraction would not exist. The page multiplies two fractions and nothing else: it does not add, subtract or divide, and it does not handle a mixed number written as a whole number beside a fraction, which has to be turned into an improper fraction first. Finally, the answer is in lowest terms while the factor readings are not necessarily: a cancelled factor can still print as something like 2/2 if its own numbers happen to share a factor, and that is the reading being honest about what the cancelling did rather than the answer being wrong.

Frequently asked questions

How do you multiply two fractions?
Multiply the numerators together and the denominators together. 2/3 times 3/4 is (2 × 3) over (3 × 4), which is 6/12, and 6/12 reduces to 1/2. No common denominator is needed, unlike adding — that is the one thing about multiplying fractions that makes it easier than adding them.
What is cross-cancelling?
Dividing a numerator and the denominator diagonally opposite it by their greatest common divisor before multiplying. In 2/3 times 3/4, the 2 cancels with the 4 and the 3 cancels with the other 3, leaving 1/1 times 1/2 — which is already the answer. It works because dividing one factor and multiplying by the same number cancel out.
Which numbers cancel — the ones across or the ones down?
Always across, diagonally: each numerator against the denominator of the other fraction. The two numerators are multiplied together, not cancelled against each other, and likewise the two denominators. Cancelling the wrong pair gives a wrong answer rather than no answer, which is why the two cancelled factors are printed on the panel.
Do I have to cross-cancel?
No, and the page shows both routes: 6/12 and 1/2 are the same number. What cross-cancelling buys is that the product arrives in lowest terms, so there is no reducing step at the end — on 99/100 times 100/99 the shortcut gives 1 while the long route writes down 9900 over 9900 first.
Can I multiply a fraction by a whole number?
Yes. Write the whole number over 1 — 2 becomes 2/1 — and then it is an ordinary multiplication. 1/2 times 2 is 1/2 times 2/1, the 2s cancel diagonally, and the answer is 1. There is no special rule for whole numbers.
What about negative fractions?
The sign belongs to the numerator and travels with it. -2/3 times 3/4 is -1/2: the -2 cancels with the 4 to leave -1 and 2, so the minus survives the cancelling. The denominators stay positive throughout, which is the convention that makes the diagonal matching unambiguous.
Why is one of the cancelled factors printed as a bare 1?
Because everything in it cancelled. When a numerator and the opposite denominator share all their factors, what is left is 1 over 1, and a denominator of 1 is written without one — the same convention the rest of the page uses. A bare 1 is a factor that has been reduced away rather than a missing reading.

References

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