SECTION 3. ROUNDING, ESTIMATING, AND USING A CALCULATOR Exact numbers are not always necessary or desirable. Sometimes it may be necessary to express the number which is a result of a calculation to a place value that is appropriate for the situation. For example, in a problem involving money, the answer is usually expressed to the nearest cent. Therefore, if a calculation produced an answer of $37.43792, the answer would be rounded to $37.44. How a measurement is rounded also conveys the precision of the instrument used to make the measurement. A length measurement using a metric ruler might be 54 mm, indicating that the smallest unit the ruler can measure is 1 mm. A micrometer reading would be given as 54.00 mm, indicating that this instrument can measure 0.01 mm as its smallest unit. The rounding of a number involves the determination of its significant place value. Any number may be rounded and, quite often, in several ways (i.e., thousands, ones, hundredths, etc.). To round a whole number or decimal number to any place value, we will be using the following steps: ROUNDING A NUMBER STEP 1 STEP 2 STEP 3 STEP 4 Identify the place value (precision) desired in the number to be rounded. Look at the first digit to the right of the selected place value to be rounded. If this digit is 4 or less, leave the selected place value digit the same. If this digit is greater than 5, increase the selected place value digit by 1. If the number is exactly 5 and more digits following the 5, increase the selected place value digit by 1. If the number is exactly 5 with no other digits following the 5, round the selected place value to the nearest even digit. Change all digits to the right of the rounding digit to zeros. Drop all trailing zeros to the right of the decimal point. Example 3.1 Solution: Round 7.32586 to the nearest thousandth. Since the digit 5 is in the thousandths place value position, we look to the digit to the right which is an 8 in this example. Since 8 is greater than 5, we round the 5 up to a 6. We now have 7.32600 following step 3. The last step is to drop the two trailing zeros, giving us a final answer of 7.326 rounded to the nearest thousandth.
Example 3.2 Solution: Round 126.9 to the nearest hundred. Since the digit 1 is in the hundreds place value position, we look to the digit to the right which is a 2 in this example. Since 2 is less than 5, we do not increase the 1 digit. Performing step 3 gives us 100.0. Dropping the trailing zero to the right of the decimal point gives us our final answer, which is 100 rounded to the nearest hundred. Example 3.3 Solution: Round 5802.95074 to the nearest tenth. Since the digit 9 is in the tenths place value position, we look to the digit to the right which is a 5 in this example. Since we round up when the digit is 5 or more, we round the 9 up to a 10. Since we can t write 10, we write the zero and carry the one to the next place value. We now have 5803.00000 following step 3. The last step is to drop the four trailing zeros, giving us a final answer of 5803.0 rounded to the nearest tenth. (Note how we left the zero in the tenth place value position. Without that zero, it would have appeared as though we rounded to the nearest whole number rather than the nearest tenth.) A practical use of rounding is the process of estimating answers to calculations. In order to get a rough idea of an answer, we can round each number to the first nonzero digit in the number and perform the indicated operation. For example, suppose we purchased three items costing $21.55, $6.75, and $12.03. To get an estimate of the total cost, we would round the costs to be $20, $7, and $10. Since 20 + 7 + 10 is easier to calculate in our head than the original numbers, we have an estimate of $37. The actual answer is $40.33 which is close to our estimate. The rest of the unit will ask us for an estimate and an actual answer to give us practice in finding an estimate before an actual answer is found. We are living in a world where electronic calculators are commonly used in finding answers to calculations. A long division problem that may take several minutes to perform by hand will be calculated in several seconds using a calculator. However, the calculator does have a drawback. It cannot think. Only the operator of the calculator has the ability to think. The abuse of calculator usage happens when the user applies the calculator without having any idea of what kind of an answer they would expect. For example, suppose the calculation was 84.9 17.3. An estimate for the difference involves rounding each number, namely 80 20, which gives us an approximate answer of 60. If we accidentally hit the division key rather than the subtraction key on our calculator, the calculator gives me an answer of 4.9075145. A person would have given an obvious wrong answer had he not estimated first and expected an answer close to 60. Some employers give pre-employment tests to prospective employees. Many of these tests contain basic math skills of adding, subtracting, multiplying, and dividing whole numbers, decimals, fractions, and percents without the use of a calculator.
So we encourage you to practice enough problems in the course to make sure you do not forget these skills, but we also realize that employers will expect that you know how to correctly use the calculator to increase productivity. And the correct use of the calculator involves estimating answers to avoid writing obvious wrong answers that arise because of data entry mistakes. PROBLEM SET 1.3 1. Round 307.9 to the nearest ten. 2. Round 0.4146 to the nearest thousandth. 3. Round 2,180.7 to the nearest hundred. 4. Round 7.480751 to the nearest thousandth. 5. Round 1.5008 to the nearest thousandth. 6. Round 327,291 to the nearest thousand. 7. Round 85.63970 to the nearest thousandth. 8. Round.05193049 to the nearest ten-thousandth. 9. Round 219.499 to the nearest whole number. 10. Round 57,295 to the nearest ten-thousand. 11. Round 50.17095 to the nearest ten-thousandth. 12. Round 2910.255027 to the nearest ten-thousandth. 13. Round 901.4992 to the nearest thousandth. 14. Round 46.8212098 to the nearest thousandth. 15. Round 308.09091 to the nearest whole number. 16. Round 1095.998910 to the nearest ten-thousandth. 17. Round 12.045678 to the nearest hundred-thousandth. 18. Round 740.306 to the nearest ten.
SECTION 4. MULTIPLYING WHOLE NUMBERS AND DECIMALS Multiplication is performed as repeated addition. For example, 268+268+268+268+268 is the same as 268 x 5. Since repeated addition is very time consuming, we choose to use multiplication instead of repeated addition whenever possible. The answer to a multiplication problem is called the product. The number that does the multiplying is called the multiplier. The number multiplied by the multiplier is called the multiplicand. The multiplicand and the multiplier are sometimes referred to as the factors of the product. How do we multiply two numbers together without the use of a calculator? Suppose we need to find the product of 652 and 34. First, we put the factors in columnar form and align the place values. 652 x 34 We start with the ones place value digit of the bottom number (multiplier) and multiply it by the entire top number (multiplicand). Carry the number for any product larger than 9. 2 652 x 34 2608 6x4=24, 24+2=26 Next, we multiply the tens place value digit of the multiplier by the entire multiplicand. Line the first number that we write down directly under the same column as the place value that we are multiplying by. 1 652 x 34 2608 1956 Our last step is to add the numbers that we have written down, column by column. 11 2608 + 1956 22168 This gives us our final answer. The product of 652 and 34 is 22,168.
In order to multiply two decimal numbers, we will be using the following procedure: MULTIPLYING TWO DECIMAL NUMBERS STEP 1 STEP 2 STEP 3 Multiply the two decimal numbers as if they were whole numbers. Find the total number of digits to the right of the decimal point in the multiplier and in the multiplicand. Using the total found in step 2, place the decimal point that number of digits from the right side of the answer. Example 4.1 Find the product of 4.701 and 0.3. 2 Solution: Step 1. 4701 x 3 14103 Step 2. The multiplicand (4.701) has three decimal digits and the multiplier (0.3) has one. The total is four, so the product should have 4 digits to the right of the decimal point. Step 3. 4.701 x 0.3 1.4103 Example 4.2 Find the product of 704.1 and 0.13 rounded to the nearest hundredth. Solution: Step 1. 7041 x 13 21123 7041 91533 Step 2. The multiplicand has one decimal digit and the multiplier has two decimal digits, so the total is three. Step 3. 704.1 x 0.13 91.533 Rounding 91.533 to the nearest hundredth gives us our final answer of 91.53.
Since we often use a calculator to find the product of two numbers, we need a quick and easy way to estimate the product to help avoid giving an answer that is obviously wrong. To estimate, we will round both factors to the left most nonzero digit and multiply. Using this technique on the example 4.2 to estimate the product of 704.1 and 0.13 would work like this: 704.1 rounds to 700 0.13 rounds to 0.1 700 x 0.1 = 70 So 70 is an estimate to the product of 704.1 and 0.13. The estimate tells me that a reasonable answer to the problem will be close to 70 and the actual answer of 91.533 is close to my estimate. When we repeat multiplication of the same number, we can shorten the way we write the problems by using an exponent. An exponent, or power, defines repeated multiplication. For example, 7 x 7 x 7 x 7 can be written as 7 4. The exponent tells us to multiply the number beneath it, called the base number, by itself as many times as the power indicates. The exponent 2 is read as squared, the exponent 3 is read as cubed, and any other exponent number is read as to the power. For example: 7 4 is read as seven to the fourth power 3 2 (three squared) = 3 x 3 = 9 5 3 (five cubed) = 5 x 5 x 5 = 125 10 6 (ten to the sixth power) = 10 x 10 x 10 x 10 x 10 x 10 = 1,000,000 Your calculator will perform the repeated multiplication called for when using exponents if it has an exponential or power key, which are shown below: x y or y x or To use the power key, we first enter the number that is being multiplied by itself (called the base). Then we touch the power key, followed by the exponent, followed by the = key. Most calculators have special keys for the exponents 2 and 3. The square key looks like this: x 2 The third power key, or the cubed key looks like this: x 3 When using the special power keys, we don t enter the exponent number after hitting the special power key.
PROBLEM SET 1.4 Find an estimate of the product, then find the actual product of each of the following: 1) 83 x 75 2) 628 x 57 3) 760 x 4.03 estimate estimate estimate actual actual actual 4) 0.395 x 0.201 5) 1.95 x 312 6) 0.32 x 0.4 estimate estimate estimate actual actual actual 7) 9012 x 47 8) 4.26 x 37 9) 225 x.0125 estimate estimate estimate actual actual actual Use your calculator s power key to find the following products. 10. 4 5 = 11. 12 3 = 12. (0.74) 2 =
13. (9.2) 3 = 14. 2 7 = 15. (1.25) 2 =