MAT00 Chapter - Decimals Decimal Fractions Section Using Decimal Notation Decimal fractions are fractions with 0, 00, 000, and so on in the denominator. 6 40,, and 0 00 000 Decimal fractions can be written in many ways. 0 (in fractional form) 0. (in decimal form) One-tenth (in words) 2 Hundreds Tens 4 Place Values Ones 9 Decimal Point. Tenths Hundredths Thousandths 6 Ten-Thousandths CQ-0 What decimal is indicated by the word five and four hundred sixtyseven ten-thousandths?.467.0467 00 0 and 0 00 000 0000,467,000 The decimal 49.6 can be written in words as three hundred forty-nine and three thousand eight hundred sixty-one ten-thousandths. 0.467 4 Fractions & Equivalent Decimals Changing from Fractions to Decimals Note the relationship between fractions and their equivalent numbers decimal form. 0 0. 49 000 0.049 6 Write the fraction 00 as a decimal. 6 0.6 00 one zero one decimal place three zeros three decimal places 27 Write the fraction as a decimal. 000 27.027 000 three zeros three decimal places 6 of
MAT00 Chapter - Decimals Changing from Decimals to Fractions Write the decimal 0.7 as a fraction. 0.7 7 000 Write the decimal 00 as a fraction. 00 4 0000 9 4 000 Simplify when possible. CQ-0 Write 0.0 in fractional form and reduce, if possible. 2000 200 2000 20 7 Section 2 Comparing, Ordering, and Rounding Decimals Comparing Decimals Comparing Two Numbers in Decimal Notation Start at the left and compare corresponding digits. If the digits are the same, move one place to the right. When two digits are different, the large number is the one with the larger digit. The numbers in the tenths place are both The numbers in the hundredths place are different. 4 Since 4 <, we know that 4 < 9 0 76.4 Ordering Decimals Place the following four decimals in order from smallest to largest. 76.00 76.40 76.44 Add zeros to make the comparison easier. Rearrange with the smallest first. 76.4, 76.44, 76., 76.4 CQ-0 Place the set of numbers 0.709, 0.7, 0.79, and 0.079 in the proper order from smallest to largest. 0.079,0.709,0.7,0.79 0.079,0.79,0.709,0.7 0.79,0.709,0.7,0.079 0.79,0.079,0.709,0.7 2 2 of
MAT00 Chapter - Decimals Rounding Decimals Rounding Decimals Find the decimal place (units, tenths, hundredths, and so on) for which rounding off is required. If the first digit to the right of the given place value is less than, drop it and all digits to the right of it. If the first digit to the right of the given place value is or greater, increase the number in the given place value by one. Drop all digits to the right of this place. 46.74 Round the 4 up to. 46.74 2 Do not round the 4 up. Rounding Decimals Round 9.4 to the nearest tenths. 9.4 9.4 Round 4.2902 to the nearest hundredths. 4.2902 4.0 hundredths place 46.7 46.74 hundredths place 4 CQ-0 Round 7.094 to the nearest thousandth. 7.09 7.09 7. Section Adding and Subtracting Decimals 7. 6 Adding Decimals The addition of decimals is related to the addition of fractions. 4 7 0 0 0 0. 0.4 0.7 Adding Decimals Write the numbers to be added vertically and line up the decimal points. Extra zeros may be added to the right of the decimal points if needed. Add all the digits with the same place value, starting with the right column and moving to the left. Place the decimal point of the sum in line with the decimal points of the numbers added. Adding Decimals Add 7.97 + 496.. 7.97 + 496.0 2.47 Write the numbers vertically and line up the decimal points. Add zeros where necessary for place holders. This value is the same as 47 2. 00 7 of
MAT00 Chapter - Decimals CQ-0. Add: 9 + 6.9 + 0. Subtracting Decimals 0.292 0.2 6.2 7.2 2 4 The subtraction of decimals is related to the subtraction of fractions. 4 2 0 0 0.4 Subtracting Decimals Write the numbers to be subtracted vertically and line up the decimal points. Extra zeros may be added to the right of the decimal points if needed. Subtract all the digits with the same place value, starting with the right column and moving to the left. Borrow when necessary. Place the decimal point of the difference in line with the decimal points of the two numbers being subtracted. 2 4 6 7 9 0 2 4 9 20 Subtracting Decimals Subtract 24967 24967 200 9.767 Write the numbers vertically and line up the decimal points Add zeros where necessary for place holders. Subtracting Decimals Subtract 62 9 9 0 7 0 0 0. 000 62 6.77 Write the numbers vertically and line up the decimal points. Add zeros where necessary for place holders. This value is the same as 767 9. 000 2 CQ-06. Subtract: 6.004 0.4 CQ-07. Simplify: 6 0. + 004 6.604.964 2 4 7.7 29 9.49 2 4 2 4 6 7 9 0 2 4 2 2 4 6 7 9 0 2 4 24 4 of
MAT00 Chapter - Decimals CQ-0. Mary Ann follows her investments in the stock market daily. Over a five-day period, she gained $9.7, gained $24, lost $., gained $244, and lost $49. How much did she lose overall? $20 $00.7 CQ-09. Boston had five snowstorms in one month, whose snowfall amounts were: 6. inches, 0.2 inches, 2 inches, 2 inches, and.9 inches. The normal amount of snow for the month was 2 inches. How many more inches of snow than normal fell that month? 29.6 inches 9.6 inches $0.6 49.6 inches $0.49 2 4 69 inches 2 4 2 4 6 7 9 0 2 4 2 2 4 6 7 9 0 2 4 26 Section 4 Multiplying Decimals 27 Multiplying Decimals The multiplication of decimals is related to the multiplication of fractions. 4 2 0. 0.04 0.02 0 00 000 one two three decimal place decimal places decimal places Multiplying Decimals Multiply the numbers just as you would multiply whole numbers. Find the sum of the decimal places in the two factors. Place the decimal point in the product so that the product has the same number of decimal places as the sum in step You may need to write zeros to the left of the number found in step 2 Multiplying Decimals Multiply 0.7 0. 0.7 0.4 0.06 2 decimal places decimal place decimal places in product (2 + = ) Multiplying Decimals Multiply 0.04 0.0 0.04 0.02 6 4 0.0006 decimal places decimal places decimal places in product ( + = 6) 29 Note the we need to insert three zeros before the 6. 0 of
MAT00 Chapter - Decimals CQ-0. Multiply: 7.2 0.027 CQ- Harold purchased 24 gallons of gasoline at a price of $ per gallon. How much did he pay? Round to the nearest cent. 97 $96.4 9.777 $9 0.97 0.9777 2 4 $99.9 2 4 $.00 2 4 6 7 9 0 2 4 2 4 6 7 9 0 2 4 2 Multiplying by a Power of 0 Observe the following pattern. one zero 0.02 0 = 0.02 0 = 0.2 Decimal point moved one place to the right. Multiplying by a Power of 0 Multiplication of a Decimal by a Power of 0 To multiply a decimal by a power of 0, move the decimal point to the right the same number of places as the number of zeros in the power of 0. two zeros 0.02 0 2 = 0.02 00 = three zeros 0.02 0 = 0.02 000 = 2 Decimal point moved two places to the right. Decimal point moved three places to the right. Multiply 9 0,000. 9 0,000 = 4,900. = 4,900 four zeros Decimal point moved four places to the right. 4 CQ- Multiply: 20 04 0 Section 0.00204 204. Dividing Decimals 0.0204 2 4 2 4 6 7 9 0 2 4 6 6 of
MAT00 Chapter - Decimals Dividing by a Whole Number Dividing by a Whole Number When dividing a decimal by a whole number, place the decimal point for the quotient directly above the decimal point in the dividend. 6 7. 2 6 2 4 4 0. The decimal points are aligned, one above the other. Divide as you normally divide whole numbers. 7 Divide 0.04 0.026 4 0.04 24 24 0 Note the zero after the decimal point. Dividing by a Whole Number Divide 262 Note that the remainder is not zero. 7.6 7 262 0 2 49 4 42 62 6 60 7 and round to the nearest hundredths. Carry out the division to the thousandths place, and then round. An extra zero was added to carry out the division to the required place. 7.6 is rounded to 7.69 (the nearest hundredths). 9 Dividing by a Decimal When dividing a decimal by another decimal, the division problem is converted into an equivalent problem that has a whole number as a divisor. Dividing by a Decimal Make the divisor a whole number by moving the decimal point to the right. Mark that position with a caret ( ). Count the number of places the decimal point moved. Move the decimal point in the dividend to the right the same number of places. Mark that position with a caret. Place the decimal point of your answer directly above the caret marking the point of the dividend. Divide as with whole numbers. 40 Dividing by a Decimal Divide 209 20. 9 Place the decimal point of the answer directly above the caret.. Divide as usual. 20 9 66 49 49 0 Move each decimal point to the right two places. Mark the new position by a caret ( ). CQ- Divide and round your answer to the nearest tenth: 9.67 0.4 2.4 2 2.7 2 4 4 2 4 6 7 9 0 2 4 42 7 of
MAT00 Chapter - Decimals CQ- Find the value of x: 0.x.2 CQ-. A butcher has 66 pounds of meat. He wants to pack the meat into packages weighing 0.7 pounds each. How many packages can he make? 49. packages 0. 0 packages 0.7 7 2 4 6 packages packages 2 4 2 4 6 7 9 0 2 4 4 2 4 6 7 9 0 2 4 44 Section 6 Converting Fractions to Decimals and the Order of Operations 4 Equivalent Fractions and Decimals A number can be expressed in two equivalent forms: as a fraction and as a decimal. 2 Fraction Decimal 4 2 three and one-fourth 0. 4 Same quantity, difference appearance 0.2 three and twenty-five hundredths Common equivalent fractions and decimals 0.2 0 0. 46 Converting a Fraction to a Decimal Terminating and Repeating Decimals Common equivalent fractions and decimals 2 0. 4 0.2 Converting a Fraction to an Equivalent Decimal Divide the denominator into the numerator until a) the remainder becomes zero, or b) the remainder repeats itself, or c) the desire number of decimal places is achieved. 0.2 0 0. 47 0.7 repeating digit 0. 0. 6 0.062 Terminating decimals (The remainder is zero when converting the fraction into a decimal.) repeating group of digits 0.9090 0. 90 Repeating decimals (When converting, the remainder is a digit or group of digits that repeats.) 4 of
MAT00 Chapter - Decimals Converting a Fraction to a Decimal Write as an equivalent decimal. 0.277.000 6 40 26 40 repeating remainders 0.27 CQ-6. Write 0.6 6 0.62 0.2 as a decimal. 2 4 49 2 4 6 7 9 0 2 4 0 CQ-7. Write 0.97 0.7 0.2 0666 6 as a decimal. 2 4 Ordering Fractions and Decimals Fill in the blank with one of the symbols <, =, or >. 0.7 4 Change the fraction into a decimal for easier comparison. 0.7 > 0.7 2 4 6 7 9 0 2 4 2 CQ-. Select the correct statement. 7 7 7 7 4 4 4 2 4 6 7 9 0 2 4 2 4 Order of Operations Order of Operations Do first Do last Perform operations inside any parentheses. Simplify any expressions with exponents. Multiply or divide from left to right. Add or subtract from left to right. Evaluate 9.6 + 6 (0.4) = 9.6 + 6 0.6 = 2 0.6 Exponents Addition = 04 Subtraction 4 9 of
MAT00 Chapter - Decimals Order of Operations Evaluate (4) 2 + 6 (2 0.7). (4) 2 + 6 (2 0.7) = (4) 2 + 6 0. Parentheses =.76 + 6 0. Exponents =.76 + 7.2 Division = 96 Addition CQ-9. Perform the operations in the proper order: 2 (0.24) 0.6 0.2 0.06 0.02 0. 0.92 2 4 0. 2 4 6 7 9 0 2 4 6 Section 7 Solving Applied Problems Using Decimals 7 Problem Solving Steps Understand the problem. a) Read the problem carefully. b) Draw a picture if this is helpful. c) Fill in the Mathematics Blueprint so that you have the facts and a method of proceeding in this situation. Solve and state the answer. a) Perform the calculations. b) State the answer, including the unit of measure. Check. a) Estimate the answer. b) Compare the exact answer with the estimate to see if your answer is reasonable. Mathematics Blueprint The Mathematical Blueprint is simply a sheet of paper with four columns. Each column tells you something to do. Mathematics Blueprint for Problem Solving Gather the Facts What Am I Asked to Do? How Do I Proceed? Key Points to Remember 9 Mathematics Blueprint # Kelsey is making holiday cookies, which she will give as gifts to her friends and co-workers. She has made pounds of cookies, which need to be divided up into equal packages containing 0.7 pound of cookies. How may packages can Kelsey make? Mathematics Blueprint for Problem Solving Gather the Facts There are lbs of cookies and each package is 0.7 lb. What Am I Asked to Do? Find the number of packages Kelsey can make. How Do I Proceed? Divide by 0.7. Key Points to Remember Move the decimal point over two places before dividing. Example continues. 60 0 of
MAT00 Chapter - Decimals Mathematics Blueprint # Kelsey is making holiday cookies, which she will give as gifts to her friends and co-workers. She has made pounds of cookies, which need to be divided up into equal packages containing 0.7 pound of cookies. How may packages can Kelsey make? 0.7 the number of packages 46 0.7 0.7 0 00 Kelsey can make 40 40 46 packages. 0 6 Mathematics Blueprint #2 David bought apples and pears at the grocery store for a fruit salad. At the checkout counter, the apples weighed 7 pounds and the pears weighed pounds. If the apples cost $29 per pound and the pears cost $49 per pound, how much did David spend on fruit? (Round your answer to the nearest cent.) Mathematics Blueprint for Problem Solving Gather the Facts 7 lbs of apples costing $29/lb.; lbs of pears costing $49/lb. What Am I Asked to Do? Find the total cost David spent on the fruit. How Do I Proceed? Apples: multiply 7 by 29; Pears: multiply by 49; Add the two products. Key Points to Remember Multiplication comes before addition. Example continues. 62 Mathematics Blueprint #2 David bought apples and pears at the grocery store for a fruit salad. At the checkout counter, the apples weighed 7 pounds and the pears weighed pounds. If the apples cost $29 per pound and the pears cost $49 per pound, how much did David spend on fruit? (Round your answer to the nearest cent.) Apples: 29 7 90 20 4 Pears: 49 92 490 62 4 62 6.6 David spent $6.7 on the fruit. 6 CQ-20. Dennis and Rob rented a van and drove from Boulder, CO to Palo Alto, CA. The cost was $00 for the first two weeks and then $0 per day after that. Also, they were not charged a mileage fee for the first 2,000 miles, but had to pay 2 cents a mile for every mile after that. They took their trip, and it took 9 days. Also, they drove 2,0 miles total. What was the total cost of the van rental? $92 $2 $0 $42 2 4 6 7 9 0 2 4 2 4 64 of