Simplifying Fractions
-
Writing a Number as a Product of Primes
We call a whole number greater than one prime if it cannot be divided evenly except by itself and one. For example the number 7 is prime but the number 6 is not, because
6 = 2 x 3
A number that is not prime is called composite. The first primes are
2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53 These are all prime
One of the most common uses of primes is to write a number as a product of primes.
Example
Write the number 140 as a product of prime numbers.
Solution
We write
140 = 10 x 14 = (2 x 5) x (2 x 7) = 2 x 2 x 5 x 7
We can also use a factor tree to write a number as a product of primes
Example
Write the number 882 as a product of primes using a factor tree.
So that
196 = 2 x 2 x 7 x 7
Exercise
Write each number as a product of prime numbers
-
48
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-
882
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Remark: As we have seen from the following examples the task of writing a number as a product of primes is always possible. In fact, the result will always be the same. This remark is so important that it is called
The Fundamental Theorem of Arithmetic
Every composite number can be written in exactly one way as a product of prime numbers
-
- Reducing Fractions
Consider a pizza that has been cut into four slices such that two of the slices are left. There
is
more than one way of writing this as a fraction. One way is
2
4
since that are two slices left out of four total. The other way is to notice that exactly one-half of the pizza is left. So we write
1
2
Notice that we can write the numerator and denominator of the first fraction as
2 2 x 1
4 2 x 2
2 1
=
2 2
1
= 1 x
2
1
=
2
We define a common factor of two number to be a number that is a divisor of both. As was shown in the example, we can always divide out a common factor.
Example
Simplify
18
24
Solution
We see that 6 is a common factor of 18 and 24, so
18 6 x 3 3
24 6 x 4 4
Exercises
Simplify the following
-
14
35
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-
24
33
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If we do not immediately see the common factor, we can factor the numerator and the denominator into their prime factorizations and then cancel all common factors
Example
Simplify
126
350
Solution
We write
126 = 2 x 63 = 2 x 3 x 21 = 2 x 3 x 3 x 7
and350 = 10 x 35 = 2 x 5 x 5 x 7
so that126 2 x 3 x 3 x 7
350 2 x 5 x 5 x 7
3 x 3 9
=
5 x 5 25
Exercises
-
90
165
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-
225
441
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-
14
- Testing for Equality
How can we tell if two fractions are equal? One way is to simplify and see if they simplify to the same fraction. An easier way is to take the cross products and test for equality.
Example
24 ? 8
27 9
We check
24 x 9 = 27 x 8
Since these are both equal to 216. We can conclude that the two fractions are equal.
Example
Show that the following two fractions are not equal
3 6
4 7
Solution
We have
3 x 7 = 21 and 6 x 4 = 24
Since
2124
We can conclude that the two fractions are not equal
Exercises
Determine which pairs of fraction are equal
- 3
? 5
11 14
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- 16
? 18
40 45
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- 3
? 5
- Applications
Example
Gasoline costs 144 cents per gallon at the pump. 54 cents of this goes to taxes. What fractional part of the cost goes to taxes?
Solution
We write
54 9 x 6 2 x 3 x 3 x 3
144 12 x 12 2 x 2 x 2 x 2 x 3 x 3
3 3
2 x 2 x 2 8
We can conclude that 3/8 of the total cost goes to taxes.
Exercise
You have found that at your restaurant out of the 96 patrons, 8 complained that food took too long to come. What fractional part of the patrons made this complaint?
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