Real Numbers and Their Properties (Section 1.2: Fractions) Week 1 Real Numbers and Their Properties (Section 1.2: Fractions)

Week 1 Objectives This week students will: Utilize proper math terminology in written explanations. Compute basic operations with signed numbers. Exemplify properties of real numbers. Execute simplification techniques on expressions and equations.

Fraction Fractions are part of a whole number. A pizza sliced into four pieces. You ate one piece. Thus, you have eaten ¼ of the pizza, that is, one piece out of the four pieces. Here ¼ is a fraction. Just like we have done addition, subtraction, multiplication and division with whole numbers, similarly we can do all such operations with fractions also.

Vocabulary Numerator: the top number in a fraction is known as numerator Denominator: the bottom number in a fraction is known as denominator Numerator and denominators together are known as terms of fraction. Numerator Denominator

Vocabulary continues Proper Fraction: A fraction in which the value of the numerator is less than that of the denominator is known as a proper fraction Example: 2/3, ¾, 7/8 Improper Fraction: A fraction in which the value of the numerator is greater than or equal to that of the denominator is known as an improper fraction. Example 9/8, 17/13, 5/4

Vocabulary continues Equivalent fractions: Fractions that look different but represent the same numbers are known as equivalent fractions. Example ½ and 2/4 are equivalent fractions because they look different but represent the same number ½ or 0.5. Example: 1/3 or 3/9 or 4/12 all are equivalent fractions representing 1/3 or 0.333

Vocabulary continues Lowest Term: a fraction is said to be in lowest term if the numerator and denominator have no common factors other than the number 1. Example 2/5, 3/7, 8/9, 17/21 are all fractions in lowest term because 2 and 5 have no common factors; 3 and 7 have no common factors; similarly, 8 and 9, and 17 and 21.

Addition of fraction when denominator is same To add two fractions that have the same denominator, we add their numerators to get the numerator of the answer. The denominator in the answer is the same denominator as in the original fractions. Example: Add Step 1: Denominators are same. Both the fractions have denominator 8 Step 2: Add the numerators: 3 + 6 = 9 Step 3: Write down the solution:

Subtraction of fraction when denominator is same To subtract two fractions that have the same denominator, we subtract their numerators to get the numerator of the answer. The denominator in the answer is the same denominator as in the original fraction. Example: Step 1: Notice that the denominators are same Step 2: Subtract the numerators: 13 – 10 = 3 Step 3: Write down the solution:

Least common denominator (LCD) LCD for a set of denominators is the smallest number that is exactly divisible by each denominator. LCD is also known as least common multiple (LCM). Example: Find the LCD/LCM of 8 and 12. Step 1a: Find all the factors of 8 = 2*2*2. To find the factors, start with the smallest number, other than 1, like 2, 3, 4, etc which divides 8 exactly. 2 is that number. So, divide 8 by 2. You will be left with 4 (8/2 = 4). Again find the smallest number that divides 4 exactly. It is 2 (4/2 = 2), and when you divide 4 by 2, you are left with 2. Continue the process 2/2 = 1 and we stop here as we have got 1. Thus, the factors of 8 are 2*2*2. Step 1b: Find all the factors of 12 = 2*2*3 Step 1c: Form the LCD – take both the 2’s and 3 from the list of factors for 12. The list of factors of 8 are the three 2’s. But, already two 2’s are taken. Only one 2 is left. So, take that to get the LCD as: 2*2*2*3 = 24

Example of finding LCD Find the LCD of 24, 32 Step 1a: Factors of 24 = 2*2*2*3 Step 1b: Factors of 32 = 2*2*2*2*2 Step 2: Form the LCD Factors of 24 Factors of 32 2*2*2*3 2*2*2*2*2 2*2*2*3*2*2 = 96 Note: We have already taken three 2’s from factors of 24. Thus, we are not taking the remaining three 2’s from factors of 32

Quiz Find the LCD of 36 and 48 Solution is: Factors of 36 = 2*2*3*3

Addition with unlike denominators Add Step 1: Find the LCD of 4 and 6 Factors of 4 Factors of 6 2 *2 2*3 The LCD = 2*2*3 = 12 Step 2: LCD 12 divided by 4 is 3. So, multiply top and bottom of 5/4 with 3: Step 3: LCD 12 divided by 6 is 2. So, multiply top and bottom of 2/6 with 2: Step 4: Add the results from step 2 and 3:

Subtraction with unlike denominators Step 1: LCD of 4 and 6 is 12 as we found in the last slide Step 2: Multiply top and bottom of 5/4 with 3: Step 3: Multiply top and bottom of 2/6 with 2: Step 4: Do the subtraction:

More examples of addition 4 + 4 can be re-written as LCD of 1 and 3 is 3. 3 divided by 1 is 3 and thus multiply top and bottom of 4/1 with 3: 3 divided by 3 is 1 and thus if you multiply the top and bottom of 2/3 with 1, you will get 2/3 only. Add:

Comparison The inequality signs are <, >, ≥, ≤ “<“ is less than sign. Thus, 4 < 5 “>” is greater than sign. Thus, 4 > 3 “≤” is less than or equal to sign. Thus x ≤ 3 means the unknown x can have any values that are less than or equal to 3. “≥” is greater than or equal to sign. Thus, x ≥ 4 means the unknown x can have any values that are greater than or equal to 4.

Comparison of fractions Fractions can also be represented on the number line. Go to this website and move your mouse between 0 and 1 to see all the fractions, between 0 and 1, represented on the number line: http://www.mathsisfun.com/numbers/fraction-number-line.html To compare fractions, we need to have same denominator. Example: Compare 2/7 and 5/9 Step 1: LCD of 7 and 9 are 7*9 = 63. Step 2: Multiply top and bottom of 2/7 by 9: Step 3: Multiply top and bottom of 5/9 by 7: Step 4: Comparing 18/63 < 35/63  2/7 < 5/9

Mixed numbers Mixed numbers, as the name suggests, are numbers that are made up of whole numbers and fractions. Example: Additions and subtractions can also be done with mixed numbers. In order to do the operations, the mixed number must be changed into a fraction

Changing mixed number into fractions Example: Change into fraction. Step 1: Write the mixed number as a sum: 3 + Step 2: Rewrite the whole number as a fraction: Step 3: Find the LCD of 1 and 17; the LCD is 17. Step 4: Multiply the top and bottom of 3/1 by 17: Step 5: Multiply the top and bottom of 1/17 by 1, but we will get 1/17 back only. Step 6: Add: 51/17 + 1/17 = 52/17

Short cut method of changing mixed into fractions Example: Change into fraction. Step 1: Multiply the denominator with the whole number: 4*3 = 12 Step 2: Add the numerator to the number obtained in step 1: 12 + 2 = 14. Write the fraction with the number obtained in step 2 as the numerator; the denominator is the denominator of the original fraction: 14/3

Fractions turned into mixed numbers Covert the fraction 187/14 into mixed number: Step 1: Divide 187 by 14 to get the quotient = 13 and remainder = 5 Step 2: Then the mixed number is , that is the quotient becomes the whole number, the remainder becomes the numerator of the mixed number.

Multiplication of fractions The product of two fractions is a fraction whose numerator is the product of the two numerators; the denominator is the product of the two denominators. Thus Another example: Multiply 7 can be written as 7/1. Thus, we have

Multiplication of mixed numbers Multiplication of mixed numbers can also be done by first converting each mixed number into a fraction; and then, multiplying the fractions. Example: Multiply Step 1: Convert each mixed number into fractions: and Step 2: Multiply the fractions obtained in step 1: Step 3: The solution is 592/40 or in mixed number form as

Reducing fractions In the previous slide, we obtained the result The fraction 32/40 is not in the lowest term because 32 and 40 have factors in common. We need to reduce the fraction so that the numerator and denominator has no factors in common. We can do the reduction by following the steps: Step 1: Write out the factors of 32 and 40: Step 2: Cancel out the common factors from numerator and denominator: Thus, the reduced fraction of 32/40 is 4/5.

Percent During the recent election, you must have heard statements like “turnout in midterm elections is far lower, peaking at 48.7% .” 48.7% is read as 48 point seven percent Percent means per hundred. So, 48.7% means 48.7 out of 100

Converting percent to fractions To convert percent into fractions, write the percent out of 100 and then reduce the fraction into lowest form Example 1: 57% = Example 2: 25% =

Converting percent to decimal Step 1: Write the percent out of 100 Step 2: Reduce the fraction into lowest term Step 3: Write the lowest-term fraction as decimal Example: 30% Step 1: 30% is 30 out of 100 Step 2: Step 3: 3/10 = 0.3

Short-cut method for converting percent into decimals To change percent to a decimal, drop the % symbol and move the decimal point two places to the left. Example 1: 42% = .42 (note we have shifted two places to the left and put down the decimal point). Example 2: 201% = 2.01 Example 3: 80.1% = .801 Example 4. 0.5% = ._ _ 5(the decimal point was before 5. Shift it two places to the left; the two places are indicated by the blanks. In the blanks we put zeros. Thus, the final answer is 0.5% = 0.005 We have shifted two numbers to the left. The two numbers are 01. Thus put a decimal point before 0 to get 2.01

Changing decimal to percent There are two ways to convert a decimal to percent. Method 1 (short-cut method): move the decimal point two places to the right and use the % symbol. Method 2: Write the decimal as a fraction of lowest term and then multiply the top by 100; use the % symbol.

Example 1 Method 1: Convert 0.45 into percent Move the decimal two places to the right and use the % symbol. Thus, 0.45 = 45.% = 45% Method 2: Write 0.45 as a fraction: Multiply the top by 100: Cancel the 100 from top and bottom and use the % symbol to get

Example 2 Convert 0.09 into percent Step 1: Move the decimal point two places to the right: .09  09. Step 2: 09 is nothing but 9 Step 3: using % symbol we have 9% as the solution

Changing fractions to percent To change a fraction into percentage, we can follow either of the two methods: Method 1: Convert the fraction into a decimal and change the decimal into percent Method 2: Multiply the fraction by 100 and simplify (that is make sure everything is in lowest term).

Example 1 Convert ¾ into percent Method 1: Step 1: ¾ = 0.75 in terms of decimal. Step 2: Shift the decimal point two places to the right to get 75% Method 2: Multiply ¾ by 100 and simplify: