# Simplifying Fractions

**Writing a Number as a Product of Prime Factors**

A prime number is a whole number greater than 1 that cannot be evenly

divided except by itself and 1. If you examine all the whole numbers from 1 to

50, you will find 15 prime numbers.

**The First 15 Prime Numbers**

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47

**Composite Number
**A composite number is a whole number greater than 1 that can be divided by

whole numbers other than 1 and itself.

**Divisibility Tests**

1. A number is divisible by 2 if the last digit is 0, 2, 4, 6, or 8.

2. A number is divisible by 3 if the sum of the digits is divisible by 3.

3. A number is divisible by 5 if the last digit is 0 or 5.

**Writing Numbers as Products of Prime Factors**

**Determining If a Number is Prime**

Determine which of these whole numbers are prime. If a number is composite,

write it as the product of prime factors.

**Reducing a Fraction to Lowest Terms**

A fraction is called simplified , reduced, or in lowest terms if the numerator
and

denominator have only 1 as a common factor .

**Equality Test for Fractions **

After we simplify, how can we check that a reduced fraction is equivalent to the

original fraction? If two fractions are equal, their diagonal products or cross

products are equal. This is called the equality test for fractions. Equivalent

fractions are two fractions that are equal. For any two fractions where a, b, c,

and d are whole numbers and b ≠ 0, d ≠ 0, ifthen
a×d=b×c.

**Determining Whether Two Fractions are Equal**

** Real -World Applications**

1. Medical students frequently work long hours. Susan worked a 16-hour shift,

spending 12 hours in the emergency room and 4 hours in surgery. What

fractional part of her shift was she in the emergency room? What fraction

part of her shift was she in surgery?

2. William works for a wireless communications company
that makes beepers

and mobile phones. He inspected 315 beepers and found that 20 were

defective. What fractional part of the beepers were not defective?

**2.2 Homework**

Pages 124–127 #1, 3, 5, 11, 17, 21, 27, 35, 41, 47, 51, 59, 63, 69, 77, 81, 89

## 2.3 – Converting Between Improper Fractions and Mixed

Numbers

**Vocabulary**

• Proper Fraction - A fraction in which the numerator is less than the

denominator. The fractions are proper
fractions.

• Improper Fraction - A fraction in which the numerator is greater than or equal

to the denominator. The fractions are all
improper fractions.

• Mixed Number - A number created by the sum of a whole number greater

than 1 and a proper fraction. The numbers are
both mixed

numbers. Mixed numbers are sometimes referred to as mixed fractions.

** Changing a Mixed Number to an Improper Fraction**

1. Multiply the whole number by the denominator of the fraction.

2. Add the numerator of the fraction to the product found in step 1.

3. Write the sum found in step 2 over the denominator of the fraction.

**Examples of Changing Mixed Numbers to Improper
Fractions**

**Changing an Improper Fraction to a Mixed Number**

1. Divide the numerator by the denominator.

2. Write the quotient followed by the fraction with the remainder over the

denominator.

**Examples of Changing Improper Fractions to Mixed
Numbers**

**Reduce Each Mixed Number**

**Reduce Each Improper Fraction**

**Change to a Mixed Number and Reduce**

**Applications**

1. For the Northwestern University alumni homecoming, the students studying

sculpture have made a giant replica of the school using
pounds of

clay. Change this number to an improper fraction.

2. To paint the walls of the new gymnasium, the custodian
took measurements.

Based on his calculations , he determined he needed
gallons of paint.

Write this as a mixed number.

**2.3 Homework**

Pages 132–134 #1, 7, 13, 23, 27, 37, 41, 43, 47, 63, 77, 85

## 2.4 – Multiplying Fractions and Mixed Numbers

Multiplying Two Fractions That Are Proper or Improper

In general, for all positive whole numbers, a, b, c, and d,

**Multiplying a Whole Number by a Fraction**

**Multiplying Mixed Numbers**

Change any mixed number to an improper fraction before multiplying.

Applications

1. An area in the Midwest is a designated tornado danger zone. The land is

miles long and
miles wide. Find the area of the tornado
danger

zone. (Hint: The area of a rectangle is the product of the length times the

width.)

2. A jeep has gallons
of gas. The jeep averages 12 miles per gallon. How

far will the jeep be able to go on what is in the tank?

3. The dormitory rooms in Selkirk Hall are being carpeted.
Each room requires

square feet of carpet. If there are 30
rooms, how much carpet is

needed?

4. Russ purchased a new Buick LeSabre for $26,500. After
one year the car was

worth 4/5 of the purchase price. What was the car worth after one year?

5. There were 1340 students are the Beverly campus of
North Shore Community

College during the spring 2006 semester. The registrar discovered that 2/5 of

these students live in the city of Beverly. He further discovered that 1/4 of
the

students living in Beverly only attend classes on Monday, Wednesday and

Friday. How many students at the Beverly campus live in the city of Beverly

and attend classes only on Monday, Wednesday, and Friday?

**2.4 Homework**

Pages 139–141 #1, 13, 23, 29, 33, 41, 49, 53, 59, 61

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