- Equivalent Fractions
- Reducing Fractions to Lowest Terms
- Raising Fractions to Higher Terms
Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses equivalent fractions, reducing fractions to lowest terms, and raising fractions to higher terms. By the end of the module students should be able to recognize equivalent fractions, reduce a fraction to lowest terms and be able to raise a fraction to higher terms.
Let's examine the following two diagrams.
Notice that both
Fractions that have the same value are called equivalent fractions. Equivalent fractions may look different, but they are still the same point on the number line.
There is an interesting property that equivalent fractions satisfy.
These pairs of products are called cross products.
If the cross products are equal, the fractions are equivalent. If the cross products are not equal, the fractions are not equivalent.
Thus,
Determine if the following pairs of fractions are equivalent.
The cross products are equals.
The fractions
The cross products are not equal.
The fractions
Determine if the pairs of fractions are equivalent.
, yes
, yes
, yes
, yes
It is often very useful to convert one fraction to an equivalent fraction that has reduced values in the numerator and denominator. We can suggest a method for doing so by considering the equivalent fractions
(Can you prove this?) So,
The fraction
A natural question is "Why did we choose to divide by 3?" Notice that
We can see that the factor 3 is common to both the numerator and denominator.
From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has reduced values in the numerator and denominator. The method is called reducing a fraction.
A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.
Consider the collection of equivalent fractions
Notice that each of the first four fractions can be reduced to the last fraction,
Observe a very important property of a fraction that has been reduced to lowest terms. The only whole number that divides both the numerator and denominator without a remainder is the number 1. When 1 is the only whole number that divides two whole numbers, the two whole numbers are said to be relatively prime.
A fraction is reduced to lowest terms if its numerator and denominator are relatively prime.
Reduce each fraction to lowest terms.
The fraction
Reduce each fraction to lowest terms.
Reduce each fraction to lowest terms.
Reduce each fraction to lowest terms.
Equally as important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator than the original fraction.
The fractions
Notice that
From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has higher values in the numerator and denominator. This method is called raising a fraction to higher terms.
A fraction can be raised to an equivalent fraction that has higher terms in the numerator and denominator by multiplying both the numerator and denominator by the same nonzero whole number.
The fraction
Most often, we will want to convert a given fraction to an equivalent fraction with a higher specified denominator. For example, we may wish to convert
This is possible to do because we know the process. We must multiply both the numerator and denominator of
We have some information. The denominator 8 was raised to 32 by multiplying it by some nonzero whole number. Division will give us the proper factor. Divide the original denominator into the new denominator.
Now, multiply the numerator 5 by 4.
Thus,
So,
Determine the missing numerator or denominator.
Determine the missing numerator or denominator.
32
12
4
150
88
For the following problems, determine if the pairs of fractions are equivalent.
equivalent
equivalent
not equivalent
not equivalent
not equivalent
equivalent
not equivalent
not equivalent
For the following problems, determine the missing numerator or denominator.
6
12
12
20
75
48
80
18
154
1,472
1,850
For the following problems, reduce, if possible, each of the fractions to lowest terms.
3
2
already reduced
A ream of paper contains 500 sheets. What fraction of a ream of paper is 200 sheets? Be sure to reduce.
There are 24 hours in a day. What fraction of a day is 14 hours?
A full box contains 80 calculators. How many calculators are in
There are 48 plants per flat. How many plants are there in
16
A person making $18,000 per year must pay $3,960 in income tax. What fraction of this person's yearly salary goes to the IRS?
For the following problems, find the mistake.
Should be
Cancel factors only, not addends;
Same as Exercise 115; answer is
((Reference)) Round 816 to the nearest thousand.
((Reference)) Perform the division:
0
((Reference)) Find all the factors of 24.
((Reference)) Find the greatest common factor of 12 and 18.
6
((Reference)) Convert
"Used as supplemental materials for developmental math courses."