Section R.2. Fractions

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1 Section R.2 Fractions

2 Learning objectives Fraction properties of 0 and 1 Writing equivalent fractions Writing fractions in simplest form Multiplying and dividing fractions Adding and subtracting fractions Arithmetic operations on mixed numbers Vocabulary: numerator; denominator; whole number; least common denominator (LCD)

3 Fraction vocabulary 3/5, pronounced three-fifths, is a fraction The top number (3) is called the numerator The bottom number (5) is called the denominator A proper fraction is a fraction where the numerator is smaller than the denominator. 3/5 is a proper fraction. An improper fraction is a fraction where the numerator is larger than the denominator. 5/3 is an improper fraction. 5/3 can also be written as 1 2/3, which has a whole number part and a fraction part. This kind of number is called a mixed number.

4 Fraction properties of 0 and 1 6/6 = 6 divided by 6 = 1 Anything divided by itself = 1 6/1 = 6 divided by 1 = 6 Anything divided by 1 = itself 0/6 = 0 divided by 6 = 0 0 divided by anything = 0 6/0 is undefined; anything divided by 0 is undefined

5 Equivalent fractions Remember that 6/6 = 1; also, 6 x 1 = 6 Anything multiplied by 1 is itself So, we can take a fraction like ¼, multiply it by 1 and it would be ¼ 1 = 6/6, so ¼ x 6/6 = (1 x 6)/(4 x 6) = 4/24 ¼ = 4/24 These two fractions are Equivalent; ¼ is the simplified equivalent of 4/24

6 Simplify these fractions 3/3 40/4 16/1 42/49 11/27 0/7

7 Write each fraction as an equivalent 3/10, with a denominator of 40 4/7, with a denominator of 56

8 Simplifying fractions A fraction is simplified when the numerator and denominator have no common factors besides 1 5/11 is a simple fraction: 5 s factors are {1,5} and 11 s factors are {1,11} 6/12 is not a simple fraction: 6 s factors are {1,2,3,6} and 12 s factors are {1,2,3,4,6,12}. They have {2,3,6} in common The Greatest Common Factor of 6 & 12 is? We can simplify 6/12 by dividing both the numerator and the denominator by the greatest common factor.

9 Simplify the following fractions Hint: factor the numerator & the denominator; find the greatest common factor; then divide both 5/10 9/15 88/66 300/550

10 Multiplying fractions To multiply two fractions, we multiply numerator times numerator and denominator times denominator. Then, if possible, we try to simplify the result. 2/15 x 5/13:

11 Dividing fractions Dividing two fractions is the same as multiplying the first fraction with the reciprocal of the second. The reciprocal of a fraction is the fraction, flipped over. The reciprocal of 2/3 is 3/2. Reciprocal of 2/25 is 25/2 Divide ½ by 5/7:

12 Multiply or divide the following: 1/3 x 5/7 28/6 x 8/21 6/19 divided by 9/13 9/14 divided by 3/10

13 Adding and Subtracting Fractions Actually, adding and subtracting fractions are more involved than multiplying and dividing To add or subtract fractions, both must have the same denominator 3/5 + 1/5 = (3 + 1)/5 = 4/5 3/5 1/5 = (3 1)/5 = 2/5 What about 2/5 + ¼? Is it 3/9?

14 Adding & subtracting fractions This is why we learn the Least Common Denominator It is related to the Least Common Multiple (from R.1) When adding 2/5 and ¼, we need to find the LCD, which is the least common multiple of 5 and 4 The multiples of 5 are {5, 10, 15, 20, 25 } The multiples of 4 are {4,8,12,16,20,24 } The LCM of 5 & 4 is: So, then the LCD of 2/5 and ¼ is: Now, we write equivalent fractions for 2/5 and ¼ using the LCD: Then, finally, we can add them:

15 Add or subtract as indicated: 1/8 + 3/8 5/6 3/6 1/6 + 7/15 5/9 1/12 7/8 5/6

16 Performing operations on mixed numbers Divide 2 1/8 by 1 2/3? To perform on mixed numbers, first convert them to fractions 2 1/8 = 16/8 + 1/8 =? 1 2/3 = 3/3 + 2/3 =? 2 1/8 divided by 1 2/3 =? Divided by? =??

17 Adding mixed numbers Add: 2 1/ /3 First: write each number as a fraction: Next: find the LCD: Then: rewrite the two fractions using the LCD: Add the two fractions: Finally, simplify the result:

18 Perform the operations on mixed numbers 13 2/ / /7 2 1/3 x 6 3/7 3 4/5 divided by 1 2/5

19 Section R.3 Decimals and Percents

20 R.3 Decimals and percents Write decimals as fractions Add, subtract, multiply and divide decimals Round decimals to a given decimal place Write fractions as decimals Write percents as decimals and decimals as percents Vocabulary: decimal notation; place value; dividend; quotient; divisor; percent

21 Writing decimals as fractions Like fractions, decimals portray a part of a whole Decimal number: Each numeral in the number has a place value The 2 is in the place? The 8 is in the place? The 7 is in the place? The 6 is in the place? The 1 is in the place? Write as a mixed fraction:

22 Write the following decimals as fractions

23 Adding, subtracting, multiplying, and dividing decimals Add 3.25 and 11.6 Write the decimals so that the decimal points line up vertically Add or subtract as usual Make sure the decimal point remains in the right place in the answer

24 Add or subtract as indicated:

25 Multiplying decimals Multiply as if they were whole numbers The decimal point in the answer is placed so that the number of decimal places is equal to the sum of the number of decimal places in the numbers that were multiplied.: Multiply 0.03 x 0.6: Note this is the same as multiplying 3/100 x 6/10:

26 Dividing decimals 9.46 divided by 3.14 The first number (9.46) is the dividend The second number (3.14) is the divisor The answer is the quotient To divide decimals, move the decimal point all the way to the right in the divisor. Keep track of how many places right you moved it. Then move the decimal point in the dividend the same number of places to the right. Finally, divide. The decimal point in the quotient is right above the new decimal point in the dividend

27 Divide 9.46 by 3.14:

28 Multiply or divide as indicated: x divided by

29 Rounding decimals Rounding decimals to a place value to the right of the decimal point: Round to the nearest hundredth The hundredth digit is: The digit immediately to the right of it is: If this digit is 5 or greater, then add 1 to the hundredth digit, and drop all the digits to the right of it. Otherwise, just drop all the digits to the right of it rounded to the nearest hundredth is:

30 Round each decimal to the given place value 0.539, hundredth 2.35, tenth 27, , thousandth

31 Writing fractions as decimals Divide the numerator by the denominator, making sure you keep track of the decimal point. Write ¼ as a decimal: Write 2/3 as a decimal: Note that the 6 repeats. This is a repeating decimal. We write it as:

32 Write each fraction as a decimal: 2/5 2/3 3/8 19/6

33 Writing percents as decimals and decimals as percents Percent means per % means 53 per 100 To write 53% as a decimal, divide 53 by 100 = % is 7/100 as a fraction, or 0.07 as a decimal 63% is 63/100 as a fraction, or 0.63 as a decimal 109% is 109/100 as a fraction, or 1.09 as a decimal To write a decimal as a percent, move the decimal place two to the right, and add the percent sign.

34 Change the following percents to decimals 16% 5.1 % 300% 0.7%

35 Change the following decimals to percents

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