ࡱ> >@= )bjbjWW 4h55!^^^^^rrrrtr>r)3)3)3>>>>>>>$?9B4>^)3)/)3)3)34>^^ I>999)3>^^>9)3>99j<<̈́Gg5<=_>0><,Bg5JB<B^<D)3)39)3)3)3)3)34>4>7>)3)3)3>)3)3)3)3B)3)3)3)3)3)3)3)3)3 $: Multiplying and Dividing Decimals Multiplying Decimals: Multiplying decimals is much like multiplying whole numbers, except a decimal must be inserted into the final answer. We must decide where to place the decimal in the answer. To decide where to place the decimal in the answer, we must count the number of total places behind the decimals in the multiplicands (the numbers being multiplied). The answer will equal the total number of places behind the answer in the product (answer). Examples: 1.) Multiply: 12.4 1.5 12.4 1.5 Multiply as if the numbers were whole numbers 124 15 620 1240 1860  12.4 x 1.5 620 1240 1860 There are two total numbers behind the decimal points (two in the first number + one in the second number). So, there will be three total decimal places in the final answer. 12.4 x 1.5 620 1240 18.60 Count from right to left. or 18.6 2.) Find the product of: 9.73 6.4 9.73 Multiply as if the numbers are whole numbers. 6.4 3892 58380 62272 9.73 6.4 3892 58380 62.272 (There are three total places behind the decimal). Using Order of Operations with Decimals Simplify: 429 32.6 3.2 = 429 104.32 The multiplication must be done before the subtraction. Now subtract.  EMBED Equation.DSMT4  Dividing Decimals: I. Changing a fraction to a decimal. To change a fraction to a decimal, we must use long division. The denominator is divided into the numerator. You must add a decimal and zeros to the dividend. The decimal point is placed in the quotient (answer) directly above the decimal point in the dividend. If the remainder is not zero, keep adding zeros until the division comes out with a remainder of zero OR round as indicated in the instructions. Examples: 1.) Change  EMBED Equation.DSMT4  to a decimal. Divisor Dividend Quotient   EMBED Equation.DSMT4  Put the decimal in the answer above the decimal in the divisor.  EMBED Equation.DSMT4  Ask: How many times does 20 go into 110? The answer is 5. Place the 8 behind the decimal, above the zero.  EMBED Equation.DSMT4  Now multiply back out. 5 20 = 100. Then subtract.  EMBED Equation.DSMT4  Add another zero.  EMBED Equation.DSMT4  The remainder is zero, so the problem is completed. Final Answer: .55 2.) Change  EMBED Equation.DSMT4  to a decimal.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Since the division does not come out evenly, add a decimal point and zero after the 25 and continue dividing.  EMBED Equation.DSMT4  Since the remainder is not zero, add another zero in the dividend.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The remainder is now zero, so the problem is completed. The final answer is: 6.25 3.) Change  EMBED Equation.DSMT4  to a decimal.  EMBED Equation.DSMT4  Now divide as in example 3 above.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The remainder is not zero, so add another zero to the dividend.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The remainder is not zero, so add another zero to the dividend.  EMBED Equation.DSMT4  The remainder is now zero so the problem is completed. Answer: 7.25 Sometimes the remainder will never come out to be zero. In these cases, you will be told to round to a particular place value. When this occurs, you should carry the division out one more place than what you will be rounding to. For example, if you are to round to the hundredths place, divide until you have a digit in the thousandths place. Examples: 1.) Convert  EMBED Equation.DSMT4  to a decimal. Round to the nearest thousandths place. (Note: Since we must round to the hundredths place, we will carry the answer out 1 place beyond the hundredths place.)  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The remainder is not zero, so add another zero to the dividend.  EMBED Equation.DSMT4  The remainder is not zero, so add another zero to the dividend.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The remainder is still not zero. The remainder continues to be 8. This is a repeating decimal. We need our answer rounded to the nearest thousandths place, so we are ready to round. 0.4166 rounded to the nearest thousandths place is 0.417 Final answer: 0.417. II. Dividing Decimals when the Divisor Contains a Decimal Point. To Divide when the Divisor Contains a Decimal Point: 1.) Move the decimal point in the divisor to the right end of the number. 2.) Move the decimal point in the dividend the same number of places to the right as the decimal point was moved in the divisor. 3.) Insert a decimal point in the answer (quotient) directly above the decimal point in the dividend. 4.) Divide similar to how whole numbers are divided. Add zeros if necessary in the dividend. Examples: 1.) Divide  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The decimal point must be moved to the end of the divisor.  EMBED Equation.DSMT4  Move the decimal two places to the right in the divisor and the dividend.  EMBED Equation.DSMT4  Place a decimal point in the answer (quotient) directly above the decimal point in the dividend.  EMBED Equation.DSMT4  Divide 68 into 54.4. The remainder is zero, so the problem is completed. Answer: 0.8 or .8 2.) Find the quotient:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The decimal point must be moved to the end of the divisor.   EMBED Equation.DSMT4  Move the decimal three places to the right in the divisor and the dividend. Note: If there were not three decimal places in the dividend, we would add zeros as place holders.  EMBED Equation.DSMT4  The decimal has now been moved. Now, divide just like whole numbers.  EMBED Equation.DSMT4  Place the decimal in the quotient (answer) directly above the decimal in the dividend.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The final answer is 9.3. 3.) Divide:  EMBED Equation.DSMT4   EMBED Equation.DSMT4  The decimal point must be moved to the end of the divisor.  EMBED Equation.DSMT4  Move the decimal two places to the right in the divisor and the dividend. Since there are not two decimal places in the dividend, we must add zeros as place holders.  EMBED Equation.DSMT4  The decimal has now been moved. Now divide. Just like whole numbers.  EMBED Equation.DSMT4   EMBED Equation.DSMT4   EMBED Equation.DSMT4  The remainder is zero, so the problem is completed. The final answer is 400 III. Notes for Solving Word Problems-Decimals Step 1: Read the problem through completely. Look for key words. Step 2: Write the number sentence from the problem. Step 3: Solve the problem. Elliot Jones has 103.4 pounds of candy to place in boxes. Each box holds 4.7 pounds of candy. How many boxes will Elliot need to hold the candy? Here we need to divide. Elliot is dividing the candy into boxes. Numeric Problem: 103.4 4.7 Answer: 22 A bookstore ordered 175 copies of a textbook at a cost of $58.25 each. What is the total cost of the textbooks? Numeric Problem: 175 $58.25 Answer: $10,193.75 The total ticket receipts for a play were $2362.50. If the tickets were $4.50 each, how many tickets were sold? Numeric Problem: $2362.50 $4.50 Answer: 525 Elliot Jones has 245 pounds of candy to place in boxes. Each box holds 17.5 pounds of candy. How many boxes will Elliot need to hold the candy? Here we need to divide. Elliot is dividing up the candy into boxes.  EMBED Equation.DSMT4   EMBED Equation.DSMT4  Move the decimal one place to the right in the divisor and the dividend.  EMBED Equation.DSMT4  - 175 7 00  EMBED Equation.DSMT4  - 175 700 - 700 0 The remainder is now zero, so the problem is completed. 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