Adding Fractions
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3
Here’s a fraction:
4
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3
This is the denominator: 4
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This is the numerator: 3
4
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3
So we’re working with fourths: 4
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And we have three of them: 3
4
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1 1
4 4
1
4
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1
Here’s another fraction:
4
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1
4
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How do we add these two fractions?
3 1
+ = ?
4 4
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Are the denominators the same?
3 1
+ = ?
4 4
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We use this “common” denominator in the answer.
3 1
+ =
4 4 4
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We add the numerators for the answer’s numerator.
3 1 4
+ =
4 4 4
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The answer is four fourths.
3 1 4
+ =
4 4 4
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1 1
4 4
1 1
4 4
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Four fourths and “the whole pie”
are the same thing!
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4
= 1
4
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This is something important to remember.
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Whenever the numerator and the denominator
of a fraction are the same number,
the fraction equals 1.
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4
= 1
4
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15
= 1
15
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123
= 1
123
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a zillion
= 1
a zillion
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Also, any number multiplied by 1
equals that number.
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15 x 1 = 15
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123 x 1 = 123
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a zillion x 1 = a zillion
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Now we’re ready to move on.
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How do we add these two fractions?
3 1
+ = ?
8 16
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Are the denominators the same?
3 1
+ = ?
8 16
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We need a “common” denominator
to add these fractions.
3 1
+ = ?
8 16
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We can use 16 as a common denominator.
3 1
+ = ?
8 16
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We won’t need to change the second fraction.
3 1
+ = ?
8 16
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The second fraction already has our
common denominator of 16.
3 1
+ = ?
8 16
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We’ll need to change the first fraction
to sixteenths.
3 1
+ = ?
8 16
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But we don’t want to change the value
of the first fraction.
3 1
+ = ?
8 16
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So just changing the denominator
from 8 to 16 won’t work.
3 3
≠
8 16
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Here’s where we use what we know
about fractions that equal 1.
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Remember that any number, multiplied by 1,
leaves the number’s value unchanged.
3 3
x 1 =
8 8
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And remember that any fraction with the
same numerator and denominator equals 1.
2
= 1
2
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We can multiply the first fraction by 1
(written as two over two), and we’ve
converted the fraction to the
common denominator of 16
without changing its value!
3 2 6
x
=
x
8 2 16
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Now we can add the numerators for our answer.
6 1 7
+ =
16 16 16
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How do we add these two fractions?
1 1
+ = ?
3 16
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This time 16 won’t work as the common denominator.
1 1
+ = ?
3 16
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Three is not a factor of 16.
1 1
+ = ?
3 16
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Three doesn’t “go into” 16 without a remainder.
1 1
+ = ?
3 16
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We need another strategy for getting
a common denominator.
1 1
+ = ?
3 16
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Here’s a strategy that always works.
1 1
+ = ?
3 16
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We can multiply the denominators.
3 x 16 =
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Forty-eight can be our common denominator.
3 x 16 = 48
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 48.
1 ?
x =
3 ? 48
1 ?
x =
16 ? 48
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 48.
1 16 16
x =
3 16 48
1 ? ?
x =
16 ? 48
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 48.
1 16 16
x =
3 16 48
1 3 3
x =
16 3 48
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Finally, we add the numerators
to arrive at the answer.
16 3 19
+ =
48 48 48
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How do we add these two fractions?
1 1
+ = ?
9 15
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We don’t have a common denominator.
1 1
+ = ?
9 15
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Nine won’t go into 15 without a remainder,
so 15 won’t work as a common denominator.
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We could multiply the denominators.
That would give us a common denominator of 135.
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Here’s another strategy for finding
a common denominator.
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We begin writing the multiples of the
two denominators.
9 x 1 = 9
9 x 2 = 18
9 x 3 = 27
9 x 4 = 36
9 x 5 = 45
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We begin writing the multiples of the
two denominators.
9 x 1 = 9 15 x 1 = 15
9 x 2 = 18 15 x 2 = 30
9 x 3 = 27 15 x 3 = 45
9 x 4 = 36
9 x 5 = 45
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Before long, we discover a common multiple
of the two denominators.
9 x 1 = 9 15 x 1 = 15
9 x 2 = 18 15 x 2 = 30
9 x 3 = 27 15 x 3 = 45
9 x 4 = 36
9 x 5 = 45
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In fact, we’ve identified the least common multiple
of the denominators, haven’t we?
9 x 1 = 9
15 x 1 = 15
9 x 2 = 18
15 x 2 = 30
9 x 3 = 27
15 x 3 = 45
9 x 4 = 36
9 x 5 = 45
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Forty-five can be our common denominator,
and it should be more manageable than 135.
9 x 1 = 9
15 x 1 = 15
9 x 2 = 18
15 x 2 = 30
9 x 3 = 27
15 x 3 = 45
9 x 4 = 36
9 x 5 = 45
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 45.
1 5 5
x =
9 5 45
1 ?
x =
15 ? 45
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 45.
1 5 5
x =
9 5 45
1 3 3
x =
15 3 45
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We add the numerators
to arrive at the answer.
5 3 8
+ =
45 45 45
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How do we add these two fractions?
1 1
+ = ?
40 72
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We lack a common denominator.
1 1
+ = ?
40 72
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Because 40 won’t “go into” 72 without a
remainder, 72 won’t work as a
common denominator.
1 1
+ = ?
40 72
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We could multiply the denominators.
That would give us a common denominator of 2880.
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Here’s yet another strategy for finding
a common denominator.
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The strategy uses prime factorization.
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We need lists of the prime factors
of the two denominators.
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We use a technique sometimes called
upside-down division to make our lists
of the prime factors.
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Divide each denominator by the lowest prime
that will go into it without a remainder. Do the
same to the answer until the answer is a prime.
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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The numbers along the left and at the bottom
are the prime factors of the denominators.
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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Now we can build our common denominator.
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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We use each prime factor the most times
it appears in any one of the lists.
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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The prime factor 2 appears three times in the left list
and three times in the right list. The most it appears,
then, is three times.
2x2x2
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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The prime factor 3 appears two times in the right list
and not at all in the left list. The most it appears,
then, is two times.
2x2x2x3x3
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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The prime factor 5 appears one time in the left list
and not at all in the right list. The most it appears,
then, is one time.
2x2x2x3x3x5
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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We multiply our collected prime factors
to arrive at the least common multiple of 40 and 72,
which also works as a common denominator.
2 x 2 x 2 x 3 x 3 x 5 = 360.
2 40 2 72
2 20 2 36
2 10 2 18
5 39
3
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 360.
1 9 9
x =
40 9 360
1 ?
x =
72 ? 360
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Now we multiply each fraction by the
version of 1 that gets us to the
common denominator of 360.
1 9 9
x =
40 9 360
1 5 5
x =
72 5 360
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We add the numerators
to arrive at the answer.
9 5 14
+ =
360 360 360
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