CHAPTER 7 Decimals, Percents, & Real Numbers

Transcription

1 MATH CHAPTER 7 Decimals, Percents, & Real Numbers Section 7. Decimals: Place Value, Estimation, and Mental Computation A decimal representation of a rational number is obtained by placing a dot, called a decimal point, between the whole part of the number and the fractional part of the number, expressed in place-value form. Ex. A) Write the following decimal fractions to show the place-value representation of the digits. Write each number in words, and express as rational numbers. (a) 9.43 (b) 2, PROPERTIES OF EXPONENTS If a and b are any numbers unequal to 0, and m and n are any exponents, then the following properties hold:. a m " a n = a m + n 4. a 0 = 2. (a m ) n = a mn 5. a "m = a m 3. (ab ) m = a m b m 6. a m a n = a m"n or a n"m A terminating decimal can be written as a fraction whose denominator is a power of ten. Thus, the only reduced fractions that can be written as terminating decimals are those whose denominators can be factored into powers of two and/or five only. Ex. B) If possible, change each of the following into a terminating decimal. If it is not possible, state so. (a) 3/250 (b) 7/30 (c) 5/6 (d) 42/70 A number N is written in scientific notation if N is written in the form: N = A 0 n where A < 0 and n is an integer. Ex. C) Write in scientific notation: (a) 63,245.6 (b) Ex. D) Write in standard notation. NOTE: when approximate numbers are added or subtracted, their sum or difference should be rounded to the same decimal place as the LEAST accurate term.

2 MENTAL ARITHMETIC: Knowing the fractional equivalents of decimals facilitates mental arithmetic: 20 = = = = 0. 2 = = = = = = Section 7.2 Decimal Arithmetic and Error Patterns Place Value Test for Inequality of Decimals: The greater of two positive decimals that are both less than will be the decimal with the greater digit in the tenths place. If these digits are equal, apply this test to the hundredths digit and so on. Ex. E) Order the following: 0.37, 0.4, 3 8, 0.373, 4 00, A repeating decimal is a non-terminating decimal in which (after a certain point) the same pattern of numbers repeats endlessly. The block of digits that is repeated is called the repetend. Every rational number can be represented by an integer or by a decimal that either terminates or eventually repeats. Also, every decimal that either terminates or eventually repeats can be represented by a rational number. Ex. F) Express 0 as a decimal fraction. Ex. G) Find a rational number representation of Ex. H) Find a rational number representation of 0.9. Ex. I) Demonstrate the addition of using (a) decimal squares and (b) the pencil-and-paper algorithm. Ex. J) Demonstrate the multiplication of the given decimal values. Check answers by estimation. (a) (b) Ex. K) Demonstrate the division of the given decimal values. Check answers by estimation. (a) (b) MATH CHAPTER 7 2

3 Practice problem over Section 7.2. Rewrite the numbers in order from least to greatest. If numbers have the same value, add an equals sign. Example: 0.4, 4, 0.04, 4 0 from least to greatest becomes 0.04, 4, 0.4 = 4 0. (a) 0.27, 2 7, 0.72, 7 2 (c).25, 2 5,.04, 4 (b) 3 0, 3.52, 55 6, 3.05, 3 2 (d).8, 8, 0.6, 0 8, 2 25 Section 7.3 Ratio and Proportion A ratio is an ordered pair of numbers, a/b or a : b, where b 0. A ratio expresses a relation in size between two entities. Ex. L) A baseball team played 20 games and won 5 of them. Find the ratio of: (a) games won to games played. (b) games lost to games played. (c) games won to games lost. Ex. M) Express in simplest form: 4 3 : The statement that the two ratios a b and c d are equal is called a proportion. Ex. N) A car travels 400 miles on 20 gallons of gasoline. How many gallons will be consumed on a 900-mile trip? Ex. O) An alloy is composed of three parts of tin and two parts of lead. How much tin is needed to make 60 kilograms of the alloy? Ex. P) If Carol can rake the leaves in 4 hours and Amy can rake the leaves in 5 hours, how long will it take the two of them to rake the leaves if they are working together? MATH CHAPTER 7 3

4 Practice problems over Section 7.3. Which dishwasher detergent is a better buy: the economy-sized 94-ounce box at $4.29, or the regular-sized 48-ounce box, on sale for $2.99? 2. On a map, a line segment 2 inches long represents 50 miles. How many miles long is a road that measures half an inch? 3. If it takes 0 minutes to saw a log into 3 pieces, how long does it take to saw it into 4 pieces? [Hint: draw a sketch.] 4. A flagpole casts a shadow 5 feet long. Sandra, whose height is 5 feet, 4 inches, is standing next to the flagpole. If she casts a shadow 6 inches long, what is the height of the flagpole? 5. If one man can paint an 8ʹ by 2ʹ wall in half an hour, how long would it take 3 men to paint a wall that was 8ʹ by 24ʹ? 6. Express the ratio of 20 minutes to 3 hours. Sections 7.4 & 7.5 Percents/Percents: Mental Computation, Estimation, and Change A percent is a ratio giving the number of parts per hundred. Ex. Q) Write as a percent: (a) 4/5 (b) (c) 2.3 Ex. R) Express as a decimal: (a) 2.6% (b) 2 % Ex. S) What percent of 90 is 5? Ex. T) The value 4 is 40% of what number? Ex. U) In a presidential election, only 54% of the registered voters went to the polls. If 42 million voted, how many registered voters were there at this time? Ex. V) Joan received a 2% raise on a salary of $2,500. What is her new salary? Ex. W) Rashad receives a 2% salary increase. If his new salary is $3,000 per month, what was his old salary? MATH CHAPTER 7 4

5 Practice problems over Sections 7.4 & 7.5. Of the 50,000 seats in a stadium, 35,650 were filled at the last game. What percent of the stadium was empty? 2. Thomas Sporting Goods buys a racket at a wholesale price of $7.40. The racket is marked up by 45%, and then during the March sale it is marked down by 30%. (a) What is the March sale price? (b) What is the markup rate on the sale price? 3. Jackie & Joan have gone out to dinner, and are going to split the check and tip. Jackie always gives 20% of the pre-tax amount of the check for a tip. Joan s rule of thumb is to give 5% of the total amount of the check for a tip. Which woman gives a higher tip (a) if tax is 8.25%? (b) If tax is 0%? (c) If tax is 20%? (d) How much would the tax rate need to be for Jackie & Joan to give the same tip? [Hint: One approach would be to choose a dollar value for the amount of the check.] Section 7.6 Rational, Irrational, and Real Numbers An irrational number is a number that is expressed as a non-terminating, nonrepeating decimal. Ex. X) Find an irrational number between /4 and /3. Square root: (i) b is the square root of a if b 2 = a (b 0) (ii) a is the square root of a with the property that a " a = a (a # 0) (iii) a can be written as a 2. (iv) (v) 0 = 0. -a where a is a positive number is not a real number. Ex. Y) Evaluate if possible: (a) 49 (b) " 8 (c) " 64 (d) 2 4 Fractional exponents: For any real number x and any natural number n, x n = n x whenever n x exists for real numbers. Also, x m n = n x m. Ex. Z) Find the value of the following: (a) 6 4 (b) 36 " 2 Ex. AA) Write as a radical. MATH CHAPTER 7 5

6 A radical is in simplest form if all possible whole number factors have been removed from the radicand. 3 Ex. AB) Write in simplest form: (a) 75 (b) 80 (c) 56 A real number is any number that is either rational or irrational. The real numbers are complete they account for every point on the number line. The sum of a rational number and an irrational number is an irrational number. The product of a rational number and an irrational number is an irrational number. Ex. AC) Are the irrational numbers closed for addition? For multiplication? Ex. AD) Are the following numbers rational or irrational? (a) 3 8 (b) 3 0 (c) 3 27 (d) 4 6 SUMMARY OF THE REAL NUMBER SYSTEM MATH CHAPTER 7 6