Reducing Fractions

Rule 10: Reducing Fractions

To reduce a simple fraction, follow the following three steps:

1.

Factor the numerator.

2.

Factor the denominator.

3.

Find the fraction mix that equals 1.

For example, reduce .

First: Rewrite the fraction with the numerator and the denominator factored.

Note all factors in the numerator and denominator are separated by multiplication signs.

Second: Find the fraction that equals 1. can be written which in turn can be written which in turn can be written .

Third: We have just illustrated that . Although the left side of the equal sign does not look identical to the right side of the equal sign, both fractions are equivalent because they have the same value. Check it with your calculator. and . This proves that the fraction can be reduced to the equivalent fraction .

Example 1: Reduce the fraction .

Answer. Factor the numerator and factor the denominator and look for the fractions in the mix that have a value of 1.

and

The fraction has been reduced into the equivalent fraction .
Now prove to yourself with your calculator that both fractions are equivalent. When you divide 120 by 180, you will get the same answer as when you divide 2 by 3.

Practice problems.

Problem 1: Reduce the fraction .

Problem 2: Reduce the fraction .

Problem 3: Reduce the fraction .

Problem 4: Reduce the fraction .

Problem 5: Reduce the fraction .

Factoring Integers

Rule 9: Factoring Integers

To factor an integer, simply break the integer down into a group of numbers whose product equals the original number. Factors are separated by multiplication signs. Note that the number 1 is the factor of every number. All factors of a number can be divided evenly into that number.

Example 1: Factor the number 3.

Answer:
Since 3 x 1 = 3, the factors of 3 are 3 and 1.

Example 2: Factor the number 10.

Answer:
Since 10 can be written as 5 x 2 x 1, the factors of 10 are 10, 5, 2, and 1. The number 10 can be divided by 10, 5, 2, and 1.

Example 3: Factor the number 18.

Answer:
The number 18 can be written as 18 x 1 or 9 x 2 or 6 x 3 or 3 x 3 x 2. Since 18 can be divided by 18, 9, 6, 3, 2, and 1, then 18, 9, 6, 3, 2, and 1 are factors of 18.

Example 4: Factor the number 24.

Answer:
The number 24 can be written as 24 x 1 or 12 x 2 or 8 x 3 or 4 x 6 or 2 x 2 x 2 x 3. Since 24 can be divided by 24, 12, 8, 6, 4, 3, 2, and 1, then 24, 12, 8, 6, 4, 3, 2, and 1 are factors of 24.

Example 5: Factor the number 105.

Answer:
The number 105 can be written as 105 x 1 or 21 x 5 or 3 x 7 x 5 or 15 x 7 or 35 x 3. Since 105 can be divided by 105, 35, 21, 15, 7, 5, 3, and 1, then 105, 35, 21, 15, 7, 5, 3, and 1 are factors of 105.

Example 6: Factor the number 1200 completely.

Answer:
This instruction means to factor 1200 into a set of prime factors (factors that cannot again be factored). The number 1200 can be written as 1200 x 1 or 100 x 12. Note the 100 can again be factored to 10 x 10 and the 12 can be factored to 6 x 2. So now you have 1200 = 100 x 12 = 10 x 10 x 6 x 2. This factored set can again be factored to (2 x 5) x (2 x 5) x (2 x 3) x 2 x 1. The number 1200 is factored completely as 5 x 5 x 3 x 2 x 2 x 2 x 2 x 1.

Work the following problems for practice with this skill.

Problem 1: Factor 15 completely.

Problem 2: Factor 300 completely.

Problem 3: Factor 4000 completely.

Problem 4: Factor -3 completely.

Problem 5: Is 3 a factor of 10? Why?

Problem 6: Is 7 a factor of 21? Why?

Problem 7: Is 4 a factor of 87? Why?