LESSON 3 : USING FACTORS AND MULTIPLES WITH FRACTIONS

Numbers

Todays Objectives

Students will be able to demonstrate an understanding of factors of whole numbers by determining the: prime factors, greatest common factor (GCF), least common multiple (LCM), square root, cube root, including:

Solving problems that involve perfect squares and cubes and square and cube roots

Using Factors and Multiples with Fractions

When simplifying a fraction (reducing to its lowest terms), the numerator and denominator are divided by common factors until the only common positive factor is the number 1.

This can be done by writing the numerator and denominator as a product of their prime factors, and cancelling.

Example

Simplify by factoring the numerator and denominator completely, and then dividing out common factors. Also, give the GCF of the numerator and denominator.Solution:

=

The GCF of 68 and 72 is 2 x 2 = 4

Example

Simplify by factoring the numerator and denominator completely, then cancelling out common factors (like the above example). Also, give the GCF.Solution:

GCF=4

Lowest Common Denominator (LCD)

Addition and subtraction of fractions requires that the fractions have a common denominator. The most desirable common denominator is the lowest common denominator (LCD), which is the smallest possible denominator.

Example

Write the fractions , , and as equivalent fractions in terms of their LCD by using factoring to determine the LCM of the denominators.

Solution:Prime Factorization of 8: 2 x 2 x 2 = 23

Prime Factorization of 12: 2 x 2 x 3 = 22 x 31

Prime Factorization of 18: 2 x 3 x 3 = 21 x 32

The LCM, which is also the LCD, is 23 x 32 = 72.

Example

Determine the sum of , , and in simplest form.

Solution:

The fraction 101/72 is in simplest form because 101 and 72 do not have any positive common factors other than 1.

Example

Write the fractions , , and as equivalent fractions with their LCD using factoring to determine the LCM of the denominators.

Solution: LCM of denominators = , , Determine the sum of , , and in simplest formSolution:

Solving Problems with Factors, Multiples, and Square or Cube Roots

A variety of problems involving factors, multiples, square and cube roots are possible. Look at the following examples:

In a store window display, one set of lights flashes every 12 seconds while another set of lights flashes every 14 seconds. If the two sets of lights are turned on at the same time, how often do they flash simultaneously?

Solution: The set of lights will flash simultaneously at a time interval that is the LCM of 12 and 14.

P.F. of 12: 2 x 2 x 3 = 22 x 31

P.F. of 14: 2 x 7 = 21 x 71

The LCM is 22 x 31 x 71 = 84The two sets of lights will flash simultaneously every 84 seconds.

Example

Tom and Jeremy have ordered 450 square patio stones with side lengths of 6in. What are the dimensions of the largest square patio they can build out of the patio stones, and how many stones will they have left over, if any?

Example

Solution: The patio will be built from the number of

stones that is the largest perfect square ≤ 450. Since 202 = 400 and 302 = 900, the perfect square must be between 202 and 302, and closer to 202. Try 212 and 222. Since 212 = 441 and 222 = 484, the required perfect square is 212 = 441.

Thus, the patio dimensions will be 21 stones per side, which is 21 x 6 = 126 inches/side. This is equivalent to 10 feet 6 inches per side. Since only 441 of the 450 patio stones would be used, there would be 9 stones left over.