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Multiplying fractions by fractions
Free lesson with a video
In the video below, I start out by explaining that (1/2) x (1/3) means 1/2 of 1/3, and we find that visually. Similarly, (1/3) x (1/4) means 1/3 of 1/4. From this we get a shortcut that (1/m) x (1/n) = (1/mn).
Next, we find what is 2/3 of 1/4. First, we find 1/3 of 1/4 as being 1/12. Therefore, 2/3 has to be double that much, or 2/12.
After introducing the shortcut for fraction multiplication (multiply the numerators, multiply the denominators), I solve a few simple multiplication problems and a word problem.
Lastly, I justify the common rule for fraction division (sort of a "proof" for fifth grade level).
Most textbooks
simply “announce” the rule for multiplying fractions by
fractions. This lesson and the exercises in it will let you think and discover WHY the rule works. So please follow all
of the exercises.
| We have studied how to find |
1
2 |
of a whole number. For example |
1
2 |
of 24 is |
1
2 |
× 24 = 12. |
-
NOTE: The
word OF translates
into MULTIPLICATION.
| Finding |
1
2 |
of any fraction also means multiplying |
1
2 |
times that fraction! |
|
1
2 |
× |
1
3 |
means |
1
2 |
of |
1
3 |
. |
| Half of the 1/3-piece |
 |
is |
 |
. |
|
1
4 |
× |
1
3 |
means |
1
4 |
of |
1
3 |
. |
| A fourth part of the 1/3-piece |
|
is |
 |
. |
|
1. The pictures show how much
pizza is left, and you share it equally with one, two, or three other
people. Divide the pizza. What kind of part do you get? Write a multiplication sentence.
| a. Find |
1
2 |
of |
 |
1
2 |
× |
1
4 |
= |
|
|
|
|
[available in the book] |
|
|
|
|
[available in the book] |
| l. Find |
1
4 |
of |
 |
|
|
Shortcut - multiply fractions of the type 1/n You might have noticed that in the
above exercises, all of our fractions |
| were of the form |
1
n |
(where n is a whole number), |
and that we could have just
multiplied the
denominators to get the new
denominator. |
1
4 |
× |
1
5 |
= |
1
20 |
or |
1
2 |
× |
1
6 |
= |
1
12 |
|
|
|
We have now studied how to find 1/2 or 1/3 or 1/5
of some fractions. But how about finding some
other kind of fractional part?
Let's again compare this to finding fractional parts of whole numbers.
| Remember? To find |
3
4 |
of 16, or in other words |
3
4 |
× 16, you can first find |
1
4 |
of 16, which is 4. |
|
Then just take
that three times, which is 12. In other words, |
3
4 |
× 16 = 12. |
We can use the same exact idea when finding a fractional part of another fraction. |
| Example. Find |
2
3 |
of |
1
4 |
. First we find |
1
3 |
of |
1
4 |
, which is |
1
12 |
. |
| Then, |
2
3 |
of |
1
4 |
is double that much, or |
2
12 |
. |
|
|
|
| Example. Find |
4
5 |
of |
1
7 |
| First we find |
1
5 |
of |
1
7 |
, which is |
1
35 |
. Then, |
4
5 |
of |
1
7 |
is four times that much, or |
4
35 |
. |
|
Multiplying a fraction by a fraction means taking that fractional
part of the fraction.
It is just like taking a certain part of what is left over, when that leftover is already a fraction. |
2. The pictures show how much pizza is left, and you get a
certain part of the leftovers. How much will
you get? Write a multiplication sentence. Color in an answer picture.
|
(First find
1/4 of 1/2, then multiply the result by 3.)
| |
1
4 |
× |
1
2 |
= |
1
8 |
and |
1
8 |
× 3
= |
 |
|
|
[available in the book] |
|
|
[available in the book] |
|
|
[available in the book] |
|
Shortcut for multiplying fractions. Multiply the numerators to get the
numerator for the answer.
Multiply the denominators to get the denominator for the answer.
|
Let's compare.
|
The roundabout way |
Shortcut |
5
6 |
× |
1
2 |
= ? |
(First find
1/6 of 1/2, then multiply the result by 5.)
1
6 |
× |
1
2 |
= |
1
12 |
and |
1
12 |
× 5 = |
5
12 |
|
|
|
5
6 |
× |
1
2 |
= |
5 × 1
6 × 2 |
= |
5
12 |
|
|
|
2
8 |
× |
3
5 |
= ? |
(First find
1/8 of 3/5, then multiply the result by 2.
And, to find 1/8 of 3/5, first find 1/8 of 1/5, and
multiply that by 3.)
1
8 |
× |
1
5 |
= |
1
40 |
That by 3 is |
1
40 |
× 3 = |
3
40 |
|
|
| Then, that by 2 is |
3
40 |
× 2 = |
6
40 |
= |
3
20 |
(simplified) |
|
2
8 |
× |
3
5 |
= |
2 × 3
8 × 5 |
= |
6
40 |
= |
3
20 |
|
|
|
With the “roundabout way” we do multiplications separately.
With the
shortcut, we can
just do them all in “one go”. |
|
Study the examples on the right. Remember to always give your
final
answer as a mixed number and in
a simplified form.
|
|
|
 |
3
7 |
× |
4
9 |
= |
3 × 4
7 × 9 |
= |
12
63 |
= |
4
21 |
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
12
5 |
× |
9
8 |
= |
12 × 9
5 ×
8 |
= |
108
40 |
= |
27
10 |
= 2 |
7
10 |
|
3. Multiply. Give your answers in the lowest terms
(simplified) and as a mixed number, if possible.
| a. |
3
9 |
× |
2
9 |
|
|
[available in the book] |
|
[available in the book] |
|
| f. |
10 |
× |
5
7 |
|
4. Multiply. Give your answers in the lowest terms (simplified) and as a
mixed number, if possible.
| a. |
3
4 |
× |
7
8 |
|
|
|
|
|
[available in the book] |
|
|
[available in the book] |
|
| i. |
3
5 |
× |
1
10 |
= |
|
5. There was 1/4 of the pizza left. Marie ate 2/3 of that.
What part of the original pizza did she eat?
What part of the original pizza is left now?
6. [available in the book]
| 7. [available in the book] |
|
Brownies
3 cups sweetened carob chips
8 tablespoons Extra virgin olive oil
2 eggs
1/2 cup honey
1 teaspoon vanilla
3/4 cup whole wheat flour
3/4 teaspoon baking powder
1 cup walnuts or other nuts
|
|
|
[available in the book] |
|
The ideas in this fraction lesson are taken from the Math Mammoth Fractions 2 book. Only a few examples of each problem type are shown; you should make more problems of each kind for the student.
Next lesson: Fraction Multiplication and Area
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New! Times Tales is now on DVD!
The fast, FUN, and easy way to learn multiplication. Learn the upper times tales in two sittings using mnemonic stories.
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