Mr. Sharp here. In class recently we’ve been studying fractions. Here I have created a quick summary of what we’ve learned so far:
Remember the numerator is on top of a fraction: numerator/2.
The denominator is on the bottom of a fraction: 2/denominator.
Addition: Fractions with the same denominator are simple: 1/6 + 4/6 = 5/6
If denominators are different, we need to change one or both.
Change one: 1/3 + 3/6 =
2/6 + 3/6 = 5/6
3 times 2 is 6 (3 x 2 = 6), so that makes the denominators equal. We multiply the numerator by the same amount, in this case 2. So 1 times 2 is 2 (1 x 2 =2), leaving us with 2/6. Then adding the numerators is 2 + 3 = 5, giving us 5/6.
Change both: 1/3 + 1/5 =
5/15 + 3/15 = 8/15
3 times 5, giving us our first denominator of 15. 3 x 5 = 15. We take that same 5 and apply it to the numerator. 1 times 5 is five (1 x 5 = 15), leaving us with 5/15. The next number is 5 times three for 15 for our second denominator .5 x 3 = 15. That second numerator was a 1, so we do 1 times 3, the same as the second denominator. 1 x 3 = 3. The second fraction then becomes 3/15. Now they’re easy to add together, 5 + 3 is 8, giving us 8/15.
Subtraction: Much like addition, fractions with the same denominator are simple for subtraction.
4/6 – 1/6 = 3/6
(and yes, we can reduce this to ½, we’ll talk about that further down.)
If denominators are different, we need to change one or both.
Change one: 3/6 – 1/3 =
3/6 – 2/6 = 1/6
3 times 2 is 6 (3 x 2 = 6), so that makes the denominators equal. We multiply the second numerator by this same amount, so for this one, its times 2. Our operation is 1 times 2 is 2 (1 x 2 =2), leaving us with 2/6. Then we subtract our numerators, 3 – 2 = 1, so our final answer is 1/6.
Change both: 1/3 – 1/5 =
5/15 – 3/15 = 2/15
3 times 5, giving us our first denominator of 15. 3 x 5 = 15. The first numerator is 1, and we treat that the same as the first denominator, multiplying by 5. 1 times 5 is five (1 x 5 = 15), leaving us with 5/15. For the second denominator, 5 times three for 15. 5 x 3 = 15. We treat the second numerator the same as the second denominator, which means multiplying one by three. 1 x 3 = 3. The second fraction then becomes 3/15. Now we subtract the numerators, 5 – 3 = 2, so our answer is 2/15.
Multiplication: For multiplication, we multiply the numerators together, and the denominators together.
I want to start with something simple. Let’s do 2 x ¼. We can rewrite the problem as 2/1 x ¼. 2 x 1 = 2, and 1 x 4 = 4. So we’re left with 2/4, the same as ½.
2 x ¼ = ½.
Why is that? Think if we have two pizzas, and four people. Sharing equally, each person gets half a pizza.
Also, multiplying anything by a number smaller than one will make a number smaller. ½ is definitely smaller than 2.
Now let’s move onto something a bit more difficult.
2/7 x 3/5 = 6/35
So 2 x 3 = 6 for our numerator. 7 x 5 is 35 for our denominator. Our final answer is 6/35.
If this is a little confusing, just think of it in terms of decimal points. 2/7 is 0.28571428571, and 3/5 is 0.6. This is 0.28571428571 x 0.6. Both numbers are less than one, so our answer will be smaller than the numbers we started with. Think of it as having 2/7 only 0.6 of the time (or 0.6 times), so our answer will be smaller than both. 0.28571428571 x 0.6 = 0.17142857142, which is the same number you’d get if you tried 6 divided by 35 (6/35).
Division: The question with division is, “how many times does one fraction fit into another?” For example, with ½ ÷ ¼ = 2, we can fit ¼ into ½ two times. Remember, we can also think of this as 0.5 ÷ 0.25 =2.
½ ÷ ¼ = 2
To find our answer, we take the second fraction, and flip the numerator and denominator. So ¼ becomes 4/1. We then flip division to multiplication (÷ to x). Our new question then becomes ½ x 4/1. Remember our process for multiplication, for the numerators we have 1 x 4 = 4. Denominators 2 x 1 =2. This gives us 4/2. 4/2 is the same as 4 ÷ 2, which is 2.
The logic of this method has caused some confusion for our class, so here is an abstract demonstration of what is happening:
(a / b) ÷ (c / d) =
(a ÷ b) ÷ (c x 1/d) =
(a ÷ b) ÷ c ÷ (1/d) =
(a ÷ b) ÷ c x d =
(a ÷ b) x d ÷ c =
(a / b) x (d / c)
And here are a few links to useful websites on the subject:
http://www.mathnstuff.com/math/spoken/here/2class/70/frwhy.htm
http://www.nctm.org/publications/mathematics-teaching-in-middle-school/blog/reason-why-when-you-invert-and-multiply/
http://mathforum.org/dr.math/faq/faq.divide.fractions.html
Simplifications: Oftentimes with fractions, we will end up with an answer like 4/6. This is not the simplest way to write that number. Both 4 and 6 can be divided by 2, so let’s try that: 4 ÷ 2 = 2, while 6 ÷ 2 = 3. Our simplified fraction is 2/3.
This isn’t just true for dividing by two. Here are other examples:
3/9 simplifies to 1/3 (divide top and bottom by 3)
8/20 simplifies to 2/5 (divide top and bottom by 4)
35/56 simplifies to 5/8 (divide top and bottom by 7)
fiveast 