Converting Fractions Study Guide (page 2)
Introduction to Converting Fractions
Do not worry about your difficulties in mathematics. I can assure you mine are still greater.
—Albert Einstein, theoretical physicist (1879–1955)
This lesson begins with another definition of a fraction. Then you'll see how to reduce fractions and how to raise them to higher terms—skills you'll need to do arithmetic with fractions. Before actually beginning fraction arithmetic (which is in the next lesson), you'll learn some clever shortcuts for comparing fractions.
A fraction is defined as a part of a whole. Here's a new definition, which you'll find useful as you move into solving arithmetic problems involving fractions.
A fraction means "divide."
The top number of the fraction is divided
by the bottom number.
Thus, means "3 divided by 4," which may also be written as 3 ÷ 4 or 4√3. The value of is the same as the quotient (result) you get when you do the division. Thus, , which is the decimal value of the fraction. Notice that of a dollar is the same thing as 75¢, which can also be written as $0.75, the decimal value of .
Example: Find the decimal value of .
Divide 9 into 1 (note that you have to add a decimal point and a series of zeros to the end of the 1 in order to divide 9 into 1):
The fraction is equivalent to the repeating decimal 0.1111 etc., which can be written as . (The little "hat" over the 1 indicates that it repeats indefinitely.)
The rules of arithmetic do not allow you to divide by zero. Thus, zero can never be the bottom number of a fraction.
It's helpful to remember the decimal equivalent of the following fractions:
Reducing a Fraction
Reducing a fraction means writing it in lowest terms, that is, with smaller numbers. For instance, 50¢ is of a dollar, or of a dollar. In fact, if you have 50¢ in your pocket, you say that you have half a dollar. We say that the fraction reduces to . Reducing a fraction does not change its value. When you do arithmetic with fractions, always reduce your answer to lowest terms. To reduce a fraction:
- Find a whole number that divides evenly into the top number and the bottom number.
- Divide that number into both the top and bottom numbers and replace them with the quotients (the division answers).
- Repeat the process until you can't find a number that divides evenly into the top and bottom numbers.
It's faster to reduce when you find the largest number that divides evenly into both numbers of the fraction.
Example: Reduce to lowest terms .
|Two steps:||One step:|
|1. Divide by 4:||1. Divide by 8:|
|2. Divide by 2:|
When the top and bottom numbers both end in zeros, cross out the same number of zeros in both numbers to begin the reducing process. (Crossing out zeros is the same as dividing by 10; 100; 1,000; etc., depending on the number of zeros you cross out.) For example, reduces to when you cross out two zeros in both numbers:
There are tricks to see if a number is divisible by 2, 3, 4, 5, and 6. Use the tricks in this table to find the best number to use when reducing fractions to lowest terms:
Raising a Fraction to Higher Terms
Before you can add and subtract fractions, you have to know how to raise a fraction to higher terms. This is actually the opposite of reducing a fraction. To raise a fraction to higher terms:
- Divide the original bottom number into the new bottom number.
- Multiply the quotient (the step 1 answer) by the original top number.
- Write the product (the step 2 answer) over the new bottom number.
Example: Raise to 12ths.
|1.||Divide the old bottom number (3) into the new one (12):||3√12=4|
|2..||Multiply the quotient (4) by the old top number (2):||4 ×2 = 8|
|3.||Write the product (8) over the new bottom number (12):|
|4.||Check: Reduce the new fraction to make sure you get back|
|the original fraction.|
A reverse Z pattern can help you remember how to raise a fraction to higher terms. Start with number 1 at the lower left, and then follow the arrows and numbers to the answer.
|Divide 3 into 12||Write the answer here|
|Multiply the result of by 2|
Which fraction is larger, or ? Don't be fooled into thinking that is larger just because it has the larger bottom number. There are several ways to compare two fractions, and they can be best explained by example.
||Use your intuition: "pizza" fractions. Visualize the fractions in terms of two pizzas, one cut into 8 slices and the other cut into 5 slices. The pizza that's cut into 5 slices has larger slices. If you eat 3 of them, you're eating more pizza than if you eat 3 slices from the other pizza. Thus, is larger than .|
||Compare the fractions to known fractions like . Both and are close to . However, is more than , while is less than . Therefore, is larger than . Comparing fractions to is actually quite simple. The fraction is a little less than , which is the same as ; in a similar fashion, is a little more than , which is the same as . ( may sound like a strange fraction, but you can easily see that it's the same as by considering a pizza cut into 5 slices. If you were to eat half the pizza, you'd eat slices.)|
||Change both fractions to decimals. Remember the fraction definition at the beginning of this lesson? A fraction means divide: Divide the top number by the bottom number. Changing to decimals is simply the application of this definition.|
|Because 0.6 is greater than 0.375, the corresponding fractions have the same relationship: is greater than .|
||Raise both fractions to higher terms. If both fractions have the same denominator, then you can compare their top numbers.|
|Because 24 is greater than 15, the corresponding fractions have the same relationship: is greater than .|
||Shortcut: cross multiply. "Cross multiply" the top number of one fraction with the bottom number of the other fraction, and write the result over the top number. Repeat the process using the other set of top and bottom numbers.|
|Since 24 is greater than 15, the fraction under it, , is greater than .|
It's time to take a look at your pocket change again! Only this time, you need less than a dollar. So if you found extra change in your pocket, now is the time to be generous and give it away. After you gather a pile of change that adds up to less than a dollar, write the amount of change you have in the form of a fraction. Then reduce the fraction to its lowest terms.
You can do the same thing with time intervals that are less than an hour. How long till you have to leave for work, go to lunch, or begin your next activity for the day? Express the time as a fraction, and then reduce to lowest terms.
Converting Fractions Sample Questions
- Reduce to lowest terms.
- Raise to 16ths.
Divide by 3:
|1.||Divide the old bottom number (8) into the new one (16):|
|2.||Multiply the quotient (2) by the old top number (3):||2 ×3=6|
|3.||Write the product (6) over the new bottom number (16):|
|4.||Check: Reduce the new fraction to make sure you get back the original.|
Find practice problems and solutions for these concepts at Converting Fractions Practice Questions.
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