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## Arithmetic (all content)

### Course: Arithmetic (all content)>Unit 5

Lesson 22: 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.

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• 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!
• 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)
• 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!
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• 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 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.
• 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
• Can you use the same method to multiply 5×6 1/3?
• Why does Sal write 4.1?