If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Grade 6 (Virginia)>Unit 3

Lesson 4: Multiplying mixed numbers

# Multiplying mixed numbers

Multiplying mixed numbers is similar to multiplying whole numbers, except that you have to account for the fractional parts as well. By converting mixed numbers into improper fractions, you can multiply the two numbers together in a straightforward way. Once you have the product as an improper fraction, you can convert it back into a mixed number. Created by Sal Khan and Monterey Institute for Technology and Education.

## Want to join the conversation?

• When we have one mixed number and one whole number, why do we only multiply the numerator; for example; 9 x 1 1/12 = 9x13 /12, why can't we do 9 x 13/ 9x12?
• 1st question:
'When we have one mixed number and one whole number, why do we only multiply the numerators?'

•When calculating a Whole Number × a Fraction it can appear like only the numerators are multiplied, (but the denominators are too).

The unseen denominator math is:
(1 × other denominator), because all whole numbers have a denominator of one
,
so the calculation always equals the other denominator
.

So even without knowing why, by default we still get the correct denominator.

2nd question:
'9 x 1 1/12 = 9x13/12
Why can't we do 9x13/9x12?'

We don't multiply the Whole Number to both the numerator and denominator, because it mimics a Multiplicative Identity Fraction 9/9 = 1, (so ×1, no longer ×9).
So it doesn't answer to 9 × 1 1/12, it results in a wrong value.

First we transform the Mixed Number value into an Improper Fraction, (denominator × whole number + numerator, keep denominator), ex…

Nine times one and one twelfths.
=
9 × 1 1/12
=
9 × (12 × 1 +1)/12 ←transforming
=
9 × 13/12
= …
To multiply fractions:
(numerator × numerator), and
(denominator × denominator
).

A Whole Number's denominator always equals one, so that makes the multiplication always:
(1 × other denominator).

Therefore the whole number 9 has a denominator of one!

So the calculation is always the same, it's considered 'understood', so the following denominator math often isn't shown, except when learning it:

9 × 13/12
=
9/1 × 13/12 ←showing denominators
=
(9 × 13)/(1 × 12) ←often not shown
=
(9 × 13)/12

So mathematically the denominators are multiplied too, it's presumed 'known' to have occurred, we just don't bother writing it out because it always results in the denominator not equal to 1, the 'other' denominator.

Question 2
'9 x 1 1/12 = 9x13/12
Why can't we do 9x13/9x12?'

We can't do: 9×13/9×12,
9 ×numerator/9 ×denominator,
because it would be a miscalculation, and equivalent to: 9/9 × 13/12
.

9/9 is a Multiplicative Identity Fraction: the same numerator and denominator is equal to 1.

so it won't solve: 9 × 13/12

(9 × 13)/(9 × 12)
=
117/108
=
Simplify with GCF: 9
=
13/12 ←wrong value
It's multiplying by a fraction that equals one, so after we simplify, we're back to 13/12 again.

Complete calculations for:
nine times thirteen twelfths
=
9 × 13/12

=
(9 × 13)/(1 × 12) ←often unseen
=
(9 × 13)/12
=
117/12

simplify with GCF 3
=
39/4 ←correct value 🥳
=
9 3/4 ←mixed number form
=
9.75 ←decimal form

(≧▽≦) Hope this helps someone!
• When simplifying the fraction prior to multiplying it, why is it that you can change the numerator and denominator of opposite fractions? My understanding of simplification is that the purpose is to create a more wieldy, but equivalent, number, and my confusion is that when we do this simplification we get 7/1 (= to 7) and 9/5 (= to 1 4/5) which are not equivalent at all to 1 3/4 (7/4) and 7 1/5 (36/5) respectively. How does this give us the answer? Haven't we just arbitrarily created a different number, that will not give us the same answer as the numbers we've started with?
• Great Question!

How and Why is it the same answer?
(since the individual fractions before and after are not equal)

7/4 • 36/5 ←original values
=
7/1 • 9/5 ←cross simplified

first fractions:
7/4 ≠ 7/1
and second fractions:
36/5 ≠ 9/5

So…
Cross Cancellation simplifies before the fraction multiplication at a easier time, by the same GCF if used after multiplication.

Given:
One and three fourths × Seven and one fifths
1 3/4 • 7 1/5
=
Transform Mixed Numbers to Fractions: (denominator × whole number + numerator, keep denominator)
(4 · 1 + 3)/4 • (5 · 7 + 1)/5
=
7/436/5
=
Ok! this is where it diverges, with and without cross cancel simplifying…

Original Fractions
(not cross-cancelled)
=
7/436/5
=
252/20
=
Simply, GCF 4
(252 ÷4)/(20 ÷4)
=
63/5
=
12.6

Cross Cancel/Simplify
(In fraction multiplication, a numerator and denominator of opposite fractions divided by a common factor.)
=
7/4 • 36/5
1st denominator to 2nd numerator
GCF 4
=
7/(4 ÷ 4) • (36 ÷4)/5
=
7/1 • 9/5
=
63/5←same value 🥳
=
12.6

It works because…
Cross Cancelling is a short cut for longer Arithmetic processes

•In division: If we write out each step,the same simplification chance is available through Division:

7/4 • 36/5
=
(7 · 36)/(4 · 5) ←division chance
=
(7 · 9)/(1 · 5) ←becomes
=
63/5
←🥳

and the division chance works because…
★In multiplication: The original fractions are rewritten, and rearranged

1st fraction:
7/4 = 7 • 1/4
2nd fraction:
36/5 = 36 • 1/5

7/4 • 36/5
=
7 • 1/4 • 36 • 1/5
=
Multiplication is Commutative, (interchangeable, rearrangeable), we can swap the order of factors…
swap 7 with 36
=
36 • 1/4 • 7 • 1/5
=
36/4 • 7/5
=
9 • 7/5
=
63/5 ←🥳
=
12.6 ←decimal
=
12 3/5 ←mixed number

so…
Cross Cancelation is a shortcut to less arithmetic steps, and simpler values, each step of overall expression is equal, so we still get the same answer.

(≧▽≦) I hope this helps someone!
• What if you can't divide any of the Numerator's or Denominator's by anything?
• Good question Jennifer,

if you cant divide either the numerator or the denominator it will stay the same number,
10/10 = 10(/)2/10(/) = 5/5- you cannot
simplify 5/5 so it will stay 5/5.

((/) is division)
• idk if someone asked but When you multiply a whole number by a fraction, you only multiply and whole number by the numerator. It's because a whole number is a whole, (or wholes), which makes it unnecessary to multiply it with the denominator. Like for example, 9 = 9/1, correct? So when we do 9/1 * 1 1/2, the denominator is not effected.
• But if you multiply a whole number by a mixed number as your example, you have to be careful, it is not as simple as multiplying numerators and denominators.
• what is the opposite of 0.93
• To get an opposite of a number, just change the sign. If it is a negative, make it positive. If it is positive, make it a negative
• What if we don't have a remainder in the progress of making improper fraction to mixed number?
• If you have zero for a remainder, the fractional part of the mixed number has a zero in the numerator. So if you mixed number is something like 5 0/7 that is equal to simply 5. So, No Remainder = No Fractional part of a Mixed Number.
• i get this but he didnt explain what to do with a whole number?
• When dealing with whole numbers in an equation, you can easily convert them into fractions by placing them over 1. For instance, if you have the equation 1/3 x 5, you can convert 5 into a fraction by writing it as 5/1. This does not change the value of 5 but makes it easier to multiply. We can now proceed to multiply 1/3 by 5/1, which equals 5/3. Simplifying 5/3 gives us 1 2/3. I hope this explanation has been helpful in answering your question.

1/3 x 5 = ?
5 = 5/1
1/3 x 5/1 = 5/3
5/3 = 1 2/3
1/3 x 5 = 1 2/3
• what do we do if were multiplying a mixed number like 1 3/4 by a basic fraction like 5/6?
• Convert 1 3/4 into an improper fraction (7/4) then continue the usuals from there.
7x5=35 (n)
4x6=24 (d)
=35/24
Mixed: 1 11/24
• Can you use the same method to multiply 5×6 1/3?
• how do you multiply whole numbers by mixed numbers
• you convert the mixed number into a uneven whatever you call it like (20/10) then multuply like normal

## Video transcript

Multiply 1 and 3/4 times 7 and 1/5. Simplify your answer and write it as a mixed fraction. So the first thing we want to do is rewrite each of these mixed numbers as improper fractions. It's very difficult, or at least it's not easy for me, to directly multiply mixed numbers. One can do it, but it's much easier if you just make them improper fractions. So let's convert each of them. So 1 and 3/4 is equal to-- it's still going to be over 4, so you're still going to have the same denominator, but your numerator as an improper fraction is going to be 4 times 1 plus 3. And the reason why this makes sense is 1 is 4/4, or 1 is 4 times 1 fourths, right? 1 is the same thing as 4/4, and then you have three more fourths, so 4/4 plus 3/4 will give you 7/4. So that's the same thing as 1 and 3/4. Now, let's do 7 and 1/5. Same exact process. We're going to still be talking in terms of fifths. That's going to be the denominator. You take 5 times 7, because think about it. 7 is the same thing as 35/5. So you take 5 times 7 plus this numerator right here. So 7 is 35/5, then you have one more fifth, so you're going to have 35 plus 1, which is equal to 36/5. So this product is the exact same thing as taking the product of 7/4 times 36/5. And we could multiply it out right now. Take the 7 times 36 as our new numerator, 4 times 5 as our new denominator, but that'll give us large numbers. I can't multiply 7 and 36 in my head, or I can't do it too easily. So let's see if we can simplify this first. Both our numerator and our denominator have numbers that are divisible by 4, so let's divide both the numerator and the denominator by 4. So in the numerator, we can divide the 36 by 4 and get 9. If you divide something in the numerator by 4, you need to divide something in the denominator by 4, and the 4 is the obvious guy, so 4 divided by 4 is 1. So now this becomes 7 times 9, and what's the 7 times 9? It's 63, over 1 times 5. So now we have our answer as an improper fraction, but they want it as a mixed number or as a mixed fraction. So what are 63/5? So to figure that out-- let me pick a nice color here-- we take 5 into 63. 5 goes into 6 one time. 1 times 5 is 5. You subtract. 6 minus 5 is 1. Bring down the 3. 5 goes into 13 two times. And you could have immediately said 5 goes into 63 twelve times, but this way, at least to me, it's a little bit more obvious. And then 2 times 5 is 12, and then we have sorry! 2 times 5 is 10. That tells you not to switch gears in the middle of a math problem. 2 times 5 is 10, and then you subtract, and you have a remainder of 3. So 63/5 is the same thing as 12 wholes and 3 left over, or 3/5 left over. And if you wanted to go back from this to that, just think: 12 is the same thing as 60 fifths, or 60/5. 60/5 plus 3/5 is 63/5, so these two things are the same thing. These two things are equivalent. This is as an improper fraction. This is as a mixed number or a mixed fraction. But this is our answer right there: 12 and 3/5.