Section 7-1 Introduction to Decimals. Decimals are an alternative was of representing fractions using our base-10 system.

Transcription

1 Section 7-1 Introduction to Decimals Decimals are an alternative was of representing fractions using our base-10 system. Connecting Decimals and Integers: Decimals and integers are both base-10 place value systems. The value of each digit depends on its place, and the value of each place (to the right) is one-tenth the value of the previous place. thousands hundreds tens ones tenths hundredths thousandths , Examples: Using the Base-10 blocks, if we let the Flat = 1, then how would you represent: a) 2.3 b) 1.25 c) 3.01 Examples: Use words to describe: a) 5.45 b) 4.06 c) Examples: Write the following decimals in expanded form with exponents: a) 2.41 b)

2 Connecting Decimals and Fractions By using the concept of place value, decimals can be translated into fractions. Examples: Convert the following decimals to fractions a) 0.35 b) 2.25 c) Examples: Convert the following fractions to decimals a) b) , 000 c) 7 20 d) e)

3 When fractions are converted to decimals, some will terminate and some will not. What causes a fraction to terminate? Theorem: A rational number a in simplest form can be written as a terminating decimal if and b only if the prime factorization of the denominator contains no primes other than or. Examples: Which of the following fractions can be written as terminating decimals? a) b) 7 34 c) d) 21/28 3

4 METHODS FOR ORDERING DECIMALS: Lining up the Decimals 1. Line up the numbers by place value. 2. Start at the left and find the first place where the place values are different 3. Compare these digits Example: Put in order from smallest to largest: 0.587, 0.059, 0.524,, Using equivalent fractions Example: Put in order from smallest to largest: 0.023, , 0.233,, 4

5 Section 7-2 Operations on Decimals Addition/Subtraction Line up the decimals or Convert to Fractions Examples: Add or Subtract as indicated using the standard algorithm and using Base-10 blocks. a) b)

6 Multiplication Example: Multiply using the standard algorithm taught in school Using Fractions: = This is the reason that there are 3 places behind the decimal place!!!! Note: Use your estimation skills to make sure the location of the decimal place seems reasonable! m n When multiplying decimals, if the denominators are 10 and 10, then the resulting product will contain 10 m n, therefore there will be m+n digits to the right of the decimal point. Using the Lattice Method: =

7 Division Examples: Divide using long division and the standard method of moving the decimal place of the divisor if necessary a) b) Why do we move the decimal place on the divisor?? = SCIENTIFIC NOTATION A number is in scientific notation if it is in the form a 10 b where 1 a 10 and b is an integer. Examples: Write each of the following in scientific notation a) 4,520,000,000 b)

8 Examples: Write each of the following as standard numerals a) b) Examples: Perform the indicated operation a) ( ) ( ) b) ( ) ( ) ROUNDING DECIMALS Example: Round to the nearest: a) Hundred b) Ten c) One d) Tenth e) Hundredth f) Thousandth

9 Section 7-3 Nonterminating Decimals Repeating Decimals Examples: Convert the following to decimals using long division a) 1 7 = b) 2 13 = When dividing by 7, the only non-zero remainders were 1, 2, 3, 4, 5, 6. As soon as a remainder re-occurs, you will see the repeating pattern. The repeating block of digits is called the repetend. Therefore, when dividing by 7, the repetend cannot contain more than 6 digits. When dividing by 13, the repetend cannot contain more than 12 digits. Example: Write 1 9 as a decimal. Then write 2 9, 3 9, and 7 9 as decimals. 9

10 Converting a Repeating Decimal into a Rational Number Examples: a) Write 0.3 as a fraction b) Write 0.36 as a fraction c) Write as a fraction 10

11 Ordering Repeating Decimals Example: Which is larger or ? = = Section 7-4 Percents n Percent means per-hundred n % 100 Examples: Write each of the following as percents a) 0.45 = 100 (0.45) = b) 1.26 = c) = d) 1 = e) 0.3 = 11

12 Fractions can be converted to percents by using the following proportion: Examples: Write each of the following as percents a b n 100 a) 2 5 = b) 9 40 = Examples: Convert the following percents to decimals: a) 5.8% = b) ¼ % = c) 150% Note: Estimation skills should tell you if your answer seems reasonable! 12

13 Applications Involving Percents: Example: What is 15% of 300? Example: Ricky took a 35-question test and he got 20 questions right. What percent of the problems did he get right? Example: A candy bar contains 12 grams of fat. If this is 19% of the maximum recommended daily intake of fat, what is the maximum recommended daily intake of fat? Example: A realtor earned $9975 from a home sale that involved a 7% commission. What was the selling price of the home? Example: Kirk made $45,000 last year and received a 2% raise. How much does he make now? 13

14 Example: A used car originally cost $1,700. One year later, it was worth $1,400. What is the percent of decrease in value? Example: Mark bought his house in 2000 for $59,000. It was recently appraised at $95,000. What is the percent of increase in value? Example: Amy bought a dress for $144 on a 20% off clearance rack. What was the original price of the dress? Mental Math with Percents It is relatively easy to find 50%, 25%, 10%, and 5% mentally Examples: Compute the following percents mentally: a) What is 50% of 250? b) What is 25% of 800? c) What is 10% of $35.00 d) What is 15% of $35.00 e) What is 1% of 160? f) What is 150% of 20? 14