Fractions and Mixed Numbers Study Guide (page 2)
Find practice problems and solutions for these concepts at Fractions and Mixed Numbers Practice Problems.
What's Around The Bend
- What's a Fraction?
- Changing a Fraction without Changing Its Value
- Addition and Subtraction with Fractions
- Multiplying and Dividing Fractions
- Reciprocals on the Number Line
- Introducing Mixed Numbers
- Converting to a Fraction
- Converting a Fraction to a Mixed Number
- Adding and Subtracting Mixed Numbers
- Multiplying and Dividing Mixed Numbers
Fractions are used to represent parts of a whole. You can think of the fraction bar as meaning "out of." You can also think of the fraction bar as meaning "divided by."
Although most people call the top part of the fraction the "top" and the bottom part of a fraction the "bottom," the technical names are numerator and denominator.
You are never allowed to have a zero in the denominator. A fraction whose denominator is zero is undefined.
A proper fraction has a numerator that is smaller than its denominator. Examples are , , and . Improper fractions have numerators that are bigger than their denominators. Examples include , , and .
What if the numerator and denominator are equal (making the fraction equal to 1), as is the case with , , and ? Are these proper or improper fractions? The rule is that these fractions must be called improper fractions.
A fraction that represents a particular part of the whole is sometimes referred to as a fractional part. For example, let's say that a family has 4 cats and 2 dogs. What fractional part of their pets are cats? Because 4 out of the total 6 animals are cats, the fractional part of their pets that are cats is equal to . You can reduce this fraction to .
Fuel for Thought
A proper fraction (with positive numerator and denominator) has a value that is less than 1. If the value is 1 or greater, the fraction is called an improper fraction.
You can identify an improper fraction easily: Its numerator is as big as or bigger than its denominator. Is that a no-no? No, it is not a no-no. There is really nothing improper about an improper fraction.
There's one more term you need to know, and then we can stop talking about fractions and start using them. That term is mixed number, which consists of a whole number paired with a proper fraction. Examples would include , , and .
Working with Fractions
What is it that doesn't get smaller when you reduce it? A fraction! There are two operations you can do to change the look and feel of a fraction without changing its value: You can reduce it (as in converting to ) or you can augment it (as in converting to ).
Reducing a Fraction
When you reduce to , you don't change the value of the fraction. How is it possible to change the fraction without changing its value?
Well, both and reside at the same point on the number line. That means is the same number as . A piece of lumber that is of an inch thick has the same thickness as a piece of lumber that is of an inch thick. The principle used to preserve the value of the fraction was to divide the numerator and the denominator by the same number (in this case, 2).
When you reduce a fraction, you preserve its value.
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.
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:
Should 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:.
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).
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.
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.
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.
Multiply both the numerators and denominators straight across.
- , which can be reduced to .
Fuel for Thought
PROBLEM: 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?"
When you multiply fractions, you can often save time and mental energy by cancelling out. Here's how it works.
- 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.
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 Thought
Zero 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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