Example
Reduce .
Let's see: 58 is divisible by 2, but not by 3, and 87 is divisible by 3, but not by 2. What can you do?
As you learned in Chapter 3, you can express whole numbers as the product of their prime factors. So 58 works out to 2 × 29 and 87 works out to 3 × 29. So your fraction is . Now you can divide both the numerator and the denominator by 29. (This is called "canceling" the 29.) The result is .
For any fraction you might encounter, expressed as a whole number divided by a whole number, you can write both whole numbers as the product of their prime factors, like with . Then, cancel, cancel, cancel until there are no more primes to cancel.
Augmenting a Fraction
Sometimes you need to do the opposite of reducing. Instead of converting to , you may need to convert to . You do this by multiplying the numerator and denominator by 2.
You can augment of a pizza by using a pizza cutter. Suppose the full pizza has been sliced into four equal slices, and three of the slices ( of the pizza) are topped with mushrooms. Cut each of the four slices in half. There are now 8 equal slices, so each slice is of the pizza. Six of those slices are topped with mushrooms. (The three slices of mushroom pizza became six slices when you did the cutting.) That means that of the pizza is topped with mushrooms—the same of the pizza that was topped with mushrooms before you used the pizza cutter.
Adding Fractions
Do you have any loose change? A friend would like to borrow a quarter. Do you happen to have another quarter he can borrow? Don't worry; it's just a loan. And while you're at it, let him borrow still another quarter. All right, how many quarters does he owe you?
If he borrowed one quarter from you, then another quarter, and then still another quarter, he borrowed three quarters from you. In other words, he borrowed , or a total of .
Here's another question: How much is ? It's . And how much is ? Go ahead and add them up. It's . When you add fractions with the same denominator, all you have to do is add the numerators.
Now, how much is ? It's . But you can reduce that to . What did you really do just then? You divided the numerator (3) by 3 and you divided the denominator (6) by 3. There's a law of arithmetic that says when you divide the top of a fraction by any number, you must also divide the bottom of that fraction by the same number.
Now add together . What did you come up with? Was it 2? You did this: .
In order to add fractions, they must have the same denominator:
Inside TrackShould you reduce your fractions to lowest possible terms? If you left as is (instead of making it 1), is it wrong? No, but by convention, we always reduce our fractions as much as possible. Indeed, there are mathematicians who can't go to sleep at night unless they're sure that every fraction has been reduced to its lowest possible terms. You probably wouldn't want to keep these poor people up all night, so always reduce your fractions. |
But with Unlike Denominators …
Have you ever heard the expression, "That's like adding apples to oranges?" You can add apples to apples; you can add oranges to oranges. But you can't add apples to oranges.
Can you add and ? Believe it or not, you can. The problem is that they don't have the same common denominator. You need to give them a common denominator before they can be added. Do you have any idea how to do this?
You need to convert into and into . Here's how to do it:
Once the fractions have a common denominator, you can add them:.
Caution!Remember the old arithmetic law: What you do to the bottom of a fraction (the denominator) you must also do to the top (the numerator). |
Example
What is the smallest denominator you can use? You need to augment the three fractions so that they all have the same denominator. In other words, you need to find the LCD (least common denominator) for 4, 6, and 8.
One way to find the LCD is to run the multiples of 4, 6, and 8, looking for the first number that appears in all three lists:
- Multiples of 4: 4, 8, 12, 16, 20, 24, …
- Multiples of 6: 6, 12, 18, 24, …
- Multiples of 8: 8, 16, 24, …
The first number that appears on all three lists is 24, so 24 is your least common denominator.
Another way of finding the LCD is to write each of the denominators, 4, 6, and 8, as the product of its prime factors:
- 4 = 2 × 2
- 6 = 2 × 3
- 8 = 2 × 2 × 2
The number you want needs to have three 2's (to accommodate the 8) and one 3 (to accommodate the 6). So, the LCD is 2 × 2 × 2 × 3, which is 24.
You can now augment each of the fractions to a fraction in which the denominator is 24:
How to Subtract Fractions
Subtracting fractions is not much different from adding them, except for a change of sign. If the denominators are the same, we subtract the numerators.
Example
The denominators are the same, so do the subtraction: 5 – 2 = 3. The answer is .
If the denominators are different, you augment the fractions, just like when you add fractions, so that they both have the same denominator. Then, subtract the numerators.
Example
Augment the . Your problem is now .
How to Multiply Fractions
You'll find that multiplying fractions is different from adding and subtracting them because you don't need to find a common denominator before you do the math operation. Actually, this makes multiplying fractions easier than adding or subtracting them.
Example
Multiply both the numerators and denominators straight across.
- , which can be reduced to .
Fuel for ThoughtPROBLEM: How much is one-third of one-eighth?
You can see by the way this problem is worded that of means "multiply," or "times." The question would be the same if it were, "How much is one-third times one-eighth?" |
Canceling Out
When you multiply fractions, you can often save time and mental energy by cancelling out. Here's how it works.
Example
- How much is ?
You divided the 6 in by 3 and you divided the 3 in by 3. In other words, the 3 in the 6 and the 3 in the 3 cancelled each other out.
Cancelling out helps you reduce fractions to their lowest possible terms. While there's no law of arithmetic that says you have to do this, it makes multiplication easier because it's much easier to work with smaller numbers.
Dividing Fractions
The division of fractions is like multiplication, but with a twist. You'll find the trick is to turn a division problem into a multiplication problem.
Let's get right into it. When a fraction problem says to divide, just say no. When a fraction problem asks you to divide the first fraction by a second fraction, you say "no" to the division. Instead, you take the first fraction and multiply it by the reciprocal of the second fraction.
The reciprocal of a fraction is found by turning the fraction upside down. The reciprocal of 4 is , and the reciprocal of is 4. In other words, 4 and are reciprocals of each other.
The reciprocal of –4 is – and the reciprocal of – is –4. In other words, 4 and – are reciprocals of each other.
Reciprocals come in pairs, and the numbers in a reciprocal pair are either both negative or both positive. This is because two negative reciprocals multiply to +1 and two positive reciprocals multiply to +1.
Fuel for ThoughtZero is the only number that doesn't have a reciprocal. This is because any number times zero is zero, so there is no number to multiply zero by that will give you 1. |
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