Welcome to a lesson on subtracting fractions.
To subtract fractions with the same or like denominators,
we're going to keep the denominator the same,
and subtract the numerators,
and then we'll simply the fraction, if possible.
So for 3/5 minus 1/5, since both denominators are five,
it's going to stay five,
and now, we'll subtract the numerators.
Since three minus one is equal to two,
this difference is 2/5.
Let's take a look at the model for this.
Here's the model for 3/5, here's the model for 1/5,
notice if we have three pieces and we take away one piece,
we'll be left with two pieces,
where each piece represents 1/5, so the difference is 2/5.
So it's important to remember
that in order to subtract fractions,
we must have a common denominator,
so that the pieces that we're subtracting are the same size.
So, if we have fractions
with different or unlike denominators,
we must first find the least common multiple
of the denominators.
This is called the least common denominator, or LCD.
Once we find the LCD,
we're going to create equivalent fractions
with the common denominator,
then, we'll keep the denominator the same
and subtract the numerators.
And then, last, we'll simplify, if possible.
So, for 3/4 minus 1/2, we first need to find
the least common multiple of four and two,
which would also be the smallest number divisible
by both four and two, and this would be four.
Another way to recognize this
is that if we list multiples of two,
and list multiples of four, four will be the smallest number
these two lists have in common.
So, if I count by twos, we'd have two, four, six, so on.
If we count by fours, we have four, eight, 12, and so on.
Notice how four is the least common multiple.
So we want both denominators to be equal to four.
Well, this denominator's already four,
and since two times two is equal to four,
we can multiply this denominator by two,
as long as you do the same to the numerator.
This is like multiplying by 2/2, which is equal to one.
So now we have 3/4 minus, this would be 2/4,
and now that we have a common denominator,
meaning we're subtracting pieces of the same size,
the denominator's going to stay the same,
and now we subtract the numerators,
so we have three minus two, which is equal to one,
so our difference is 1/4, as we see here modeled.
Now, let's take a look at some more examples.
Here, we have 5/6 minus 7/12.
Again, the first step is to find the least common multiple,
or LCD, of six and 12,
this is the smallest number divisible by both six and 12,
which would be 12, but if we need to,
we can always list multiples of these.
We list multiples of six,
we would have six, 12, 18, and so on.
If we list multiples of 12, we would have 12, 24, 36, so on.
And now we can see the least common multiple is 12.
So, we want both denominators to be 12,
again, so we're subtracting pieces of the same size.
And since six times two is equal to 12,
we can multiply this by two,
as long as we do the same to the numerator.
So now, we would have 10/12 minus 7/12,
so our denominator is 12,
and now we'll subtract the numerators,
we have 10 minus seven, that's going to be equal to three,
so our difference is 3/12, but this does simplify
because three and 12 share a common factor of three.
To show this, we could write three as one times three,
and we could write 12 as four times three.
Three over three simplifies to one,
so our simplified difference is 1/4.
Next, we have 7/10 minus 3/8.
So, the first step is to determine the LCD,
or least common multiple, of 10 and eight.
The smallest number divisible by both 10 and eight,
and that would 40, but again, if we need to,
we can list multiples of 10 and eight,
so that'd be 10, 20, 30, 40, and so on.
And for eight, we'd have eight, 16, 24, 32, 40, and so on.
And here is our least common multiple.
So now, we want both denominators to be 40.
So we'll have 7/10 minus 3/8,
and since 10 times four is equal to 40,
we're going to multiply both the numerator
and the denominator by four.
Remember, this is like multiplying by one.
And since eight times five is equal to 40,
we're going to multiply both the top and bottom by five here,
so now we have 28/40
minus, this would be, 15/40.
So now, we have a common denominator of 40, stays the same,
and now we subtract the numerators,
28 minus 15 is equal to 13.
And this fraction does not simplify because 13 and 40
don't have any common factors other than one.
Let's take a look at two more examples.
Here, we have 5/12 minus 1/15.
So the first step here is to identify the LCD,
or least common multiple of 12 and 15.
If we can look at these two numbers
and recognize that 60 would be the LCD, that's great.
If not, we can always list multiples of both 12 and 15.
So for 12, we'd have 12, 24, 36, 48, 60 and so on.
For 15, we'd have 15, 30, 45, 60 and so on.
Here's our least common multiple of 60.
And now, we'll write equivalent fractions
with the denominator of 60 for both fractions.
Since 12 times five is equal to 60,
we're going to multiply both top and bottom by 5,
and since 15 times four is equal to 60,
we will multiply the top and bottom by four.
Here, we have 25/60 minus 4/60.
Denominator's the same, so we keep the same denominator
and subtract the numerators.
25 minus four is equal to 21,
but 21 and 60 do share a common factor of three.
To show this, we can write 21 as three times seven,
and 60 as three times 20,
so here's the common factor of three,
that simplifies to one,
and so our simplified difference is 7/20.
Now, for our last example, we have two minus 5/14.
So, to make two into fraction form, we'll write 2/1,
and now, the LCD, or least common multiple,
of one and 14 is just going to be 14,
so we'll have 2/1 minus 5/14.
We can drop these parentheses.
And now we can multiple 2/1 by 14/14,
so this is going to give us 28/14 minus 5/14,
denominator stays 14, and we subtract the numerators,
and 28 minus five is equal to 23.
And this fraction does not simplify,
so we have our difference.
I hope you found this helpful.