1.2.1 Adding Whole Numbers And Applications

SummaryYou can add numbers with more than one digit using any method, including the partialsums method. Sometimes when adding, you may need to regroup to the next greaterplace value position. Regrouping involves grouping ones into groups of tens, groupingtens into groups of hundreds, and so on. The perimeter of a polygon is found by addingthe lengths of each of its sides.1.2.1 Self Check SolutionsSelf Check AA local company built a playground at a park. It took the company 124 hours to plan outthe playground, 243 hours to prepare the site, and 575 hours to build the playground.Find the total number of hours the company spent on the project.800 130 12 942 hoursSelf Check BFind the perimeter of the trapezoid in feet.300 500 500 900 2,200 ftSelf Check CA city was struck by an outbreak of a new flu strain in December. To prevent anotheroutbreak, 3,462 people were vaccinated against the new strain in January. In February,1,298 additional people were vaccinated. How many people in total receivedvaccinations over these two months?3462 1298 4,7601.28

1.2.2 Subtracting Whole Numbers and ApplicationsLearning Objective(s)1 Subtract whole numbers without regrouping.2 Subtract whole numbers with regrouping.3 Solve application problems using subtraction.IntroductionSubtracting involves finding the difference between two or more numbers. It is a methodthat can be used for a variety of applications, such as balancing a checkbook, planning aschedule, cooking, or travel. Suppose a government official is out of the U.S. onbusiness for 142 days a year, including travel time. The number of days per year she isin the U.S. is the difference of 365 days and 142 days. Subtraction is one way ofcalculating the number of days she would be in the U.S. during the year.When subtracting numbers, it is important to line up your numbers, just as with addition.The minuend is the greater number from which the lesser number is subtracted. Thesubtrahend is the number that is subtracted from the minuend. A good way to keepminuend and subtrahend straight is that since subtrahend has “subtra” in its beginning, itgoes next to the subtraction sign and is the number being subtracted. The difference isthe quantity that results from subtracting the subtrahend from the minuend. In 86 – 52 34, 86 is the minuend, 52 is the subtrahend, and 34 is the difference.Objective 1Subtracting Whole NumbersWhen writing a subtraction problem, the minuend is placed above the subtrahend. Thiscan be seen in the example below, where the minuend is 10 and the subtrahend is 7.Example10 – 7 ?Problem10– 73Answer10 – 7 3When both numbers have more than one digit, be sure to work with one place value at atime, as in the example below.1.29

ExampleProblem689 – 353 ?689–353First, set up the problemand align the numbers byplace value.689–3536Then, subtract the ones.689–35336Next, subtract the tens.Finally, subtract thehundreds.689–353336Answer 689 – 353 336Lining up numbers by place value becomes especially important when you are workingwith larger numbers that have more digits, as in the example below.ExampleProblem9864– 7439864– 7431–9864743219,864 – 743 ?First, set up the problem andalign the numbers by placevalue.Then, subtract the ones.Next, subtract the tens.1.30

–9864743121Now, subtract the hundreds.–98647439121There is no digit to subtract inthe thousands place, so keepthe 9.Answer9,864 – 743 9,121Self Check ASubtract: 2,489 – 345.Subtracting Whole Numbers, with RegroupingObjective 2You may need to regroup when you subtract. When you regroup, you rewrite thenumber so you can subtract a greater digit from a lesser one.When you’re subtracting, just regroup to the next greater place-value position in theminuend and add 10 to the digit you’re working with. As you regroup, cross out theregrouped digit in the minuend and place the new digit above it. This method isdemonstrated in the example below.ExampleProblem–32254763,225 – 476 ?First, set up the problem and align the digits by placevalue.1 153225– 47691 11 153225– 476492 11 11 153225– 476749Since you can’t subtract 6 from 5, regroup, so 2 tensand 5 ones become 1 ten and 15 ones. Now you cansubtract 6 from 15 to get 9.Next, you need to subtract 7 tens from 1 ten. Regroup2 hundreds as 1 hundred, 10 tens and add the 10tens to 1 ten to get 11 tens. Now you can subtract 7from 11 to get 4.To subtract the digits in the hundreds place, regroup3 thousands as 2 thousands, 10 hundreds and addthe 10 hundreds to the 1 hundred that is already inthe hundreds place. Now, subtract 4 from 11 to get 7.1.31

2 11 11 15Since there is no digit in the thousands place of thesubtrahend, bring down the 2 in the thousands placeinto the answer.3225– 4762749Answer3,225 – 476 2,749Self Check BSubtract: 1,610 – 880.Checking Your WorkYou can check subtraction by adding the difference and the subtrahend. The sum shouldbe the same as the minuend.ExampleProblemCheck to make sure that 7 subtracted from 12 isequal to 5.12 – 7 55 712AnswerHere, write out the original equation.The minuend is 12, the subtrahend is 7,and the difference is 5.Here, add the difference to thesubtrahend, which results in the number12. This confirms that your answer iscorrect.The answer of 5 is correct.Checking your work is very important and should always be performed when timepermits.Subtracting Numbers, Using the Expanded FormAn alternative method to subtract involves writing numbers in expanded form, as shownin the examples below. If you have 4 tens and want to subtract 1 ten, you can just think(4 – 1) tens and get 3 tens. Let’s see how that works.1.32

Example45 – 12 ?Problem45 40 512 10 2Let’s write the numbers in expandedform so you can see what theyreally mean.45 40 512 10 230Look at the tens. The minuend is40, or 4 tens. The subtrahend is 10,or 1 ten. Since 4 – 1 3, 4 tens – 1ten 3 tens, or 30.45 40 5–12 10 230 3Look at the ones. 5 – 2 3. So, 30 3 33.Answer 45 – 12 33Now let’s use this method in the example below, which asks for the difference of 467and 284. In the tens place of this problem, you need to subtract 8 from 6. What can youdo?Example467 – 284 ?ProblemStep 1: Separate by place valueWrite both the minuend and thesubtrahend in expanded form.4 hundreds 6 tens 7 ones2 hundreds 8 tens 4 onesHere, we identify differences that arenot whole numbers. Since 8 is greaterthan 6, you won’t get a whole numberdifference.Step 2: Identify impossible differences6–8 []Regroup one of the hundreds fromthe 4 hundreds into 10 tens and add itto the 6 tens. Now you have 16 tens.Subtracting 8 tens from 16 tens yieldsa difference of 8 tens.Step 3: Regroup3 hundreds 16 tens 7 ones– 2 hundreds 8 tens 4 ones1 hundred 8 tens 3 onesCombining the resulting differencesfor each place value yields a finalanswer of 183.Step 4: Combine the parts1 hundred 8 tens 3 ones 183Answer467 – 284 1831.33

Self Check CA woman who owns a music store starts her week with 965 CDs. She sells 452 by theend of the week. How many CDs does she have remaining?Example45 – 17 ?ProblemWhen you try to subtract 17 from45, you would first try to subtract 7from 5. But 5 is less than 7.45 40 517 10 7Let’s write the numbers in expandedform so you can see what theyreally mean.45 30 1517 10 7Now, regroup 4 tens as 3 tens and10 ones. Add the 10 ones to 5 onesto get 15 ones, which is greaterthan 7 ones, so you can subtract.45 30 15– 17 10 720 8Finally, subtract 7 from 15, and 10from 30 and add the results: 20 8 28.Answer45 – 17 28Solve Application Problems Using SubtractionObjective 3You are likely to run into subtraction problems in every day life, and it helps to identifykey phrases in a problem that indicate that subtraction is either used or required. Thefollowing phrases appear in problem situations that require subtraction.Phrase or wordLess thanTake awayDecreased byExample problemThe cost of gas is 42 cents per gallon less than it waslast month. The cost last month was 280 cents pergallon. How much is the cost of gas this month?Howard made 84 cupcakes for a neighborhood picnic.People took away 67 cupcakes. How many didHoward have left?The temperature was 84oF in the early evening. Itdecreased by 15o overnight. What was thetemperature in the morning?1.34

Subtracted fromThe differenceJeannie works in a specialty store on commission.When she sells something for 75, she subtracts 15from the 75 and gives the rest to the store. Howmuch of the sale goes to the store?What is the difference between this year’s rent of 1,530 and last year’s rent of 1,450?The number of pies sold at this year’s bake sale was15 fewer than the number sold at the same event lastyear. Last year, 32 pies were sold. How many pieswere sold this year?Fewer thanWhen translating a phrase such as “5 fewer than 39” into a mathematical expression, theorder in which the numbers appears is critical. Writing 5 – 39 would not be the correcttranslation. The correct way to write the expression is 39 – 5. This results in the number34, which is 5 fewer than 39. The chart below shows how phrases with the key wordsabove can be written as mathematical expressions.Phrasethree subtracted from sixthe difference of ten and eightNine fewer than 40Thirty-nine decreased by fourteenEighty-five take away twelveFour less than one hundred eightExpression6–310 – 840 – 939 – 1485 – 12108 – 4ExampleProblemEach year, John is out of the U.S. onbusiness for 142 days, including travel time.The number of days per year he is in the U.S.is the difference of 365 days and 142 days.How many days during the year is John in theU.S.?The words “the difference of”suggest that you need to subtractto answer the problem.365-142First, write out the problem basedon the information given and alignnumbers by place value.1.35

365-1423Then, subtract numbers in theones place.365-14223Subtract numbers in the tensplace.365-142223AnswerFinally, subtract numbers in thehundreds place.John is in the U.S. 223 days during the year.Self Check DTo make sure he was paid up for the month on his car insurance, Dave had to pay thedifference of the amount on his monthly bill, which was 289, and what he had paidearlier this month, which was 132. Write the difference of 289 and 132 as amathematical expression.ExampleProblem An African village is now getting cleaner water than itused to get. The number of cholera cases in the villagehas declined over the past five years. Using the graphbelow, determine the difference between the number ofcholera cases in 2005 and the number of cases in 2010.1.36

Number of cholera cases YearThe words “the difference” suggest that youneed to subtract to answer the problem.500-200300AnswerFirst, use the graph to find the number ofcholera cases per year for the two years: 500 in2005 and 200 in 2010.Then write the subtraction problem and alignnumbers by place value. Subtract the numbersas you usually would.500 – 200 300 casesSummarySubtraction is used in countless areas of life, such as finances, sports, statistics, andtravel. You can identify situations that require subtraction by looking for key phrases,such as difference and fewer than. Some subtraction problems require regrouping to thenext greater place value, so that the digit in the minuend becomes greater than thecorresponding digit in the subtrahend. Subtraction problems can be solved withoutregrouping, if each digit in the minuend is greater than the corresponding digit in thesubtrahend.In addition to subtracting using the standard algorithm, subtraction can also can beaccomplished by writing the numbers in expanded form so that both the minuend andthe subtrahend are written as the sums of their place values.1.37

1.2.2 Self Check SolutionsSelf Check ASubtract: 2,489 – 345.2,144Self Check BSubtract: 1,610 – 880.730Self Check CA woman who owns a music store starts her week with 965 CDs. She sells 452 by theend of the week. How many CDs does she have remaining?965 – 452 513Self Check DTo make sure he was paid up for the month on his car insurance, Dave had to pay thedifference of the amount on his monthly bill, which was 289, and what he had paidearlier this month, which was 132. Write the difference of 289 and 132 as amathematical expression.The difference of 289 and 132 can be written as 289 – 132.1.38

1.2.3 EstimationLearning Objective(s)1 Use rounding to estimate sums and differences.2 Use rounding to estimate the solutions for application problems.IntroductionAn estimate is an answer to a problem that is close to the solution, but not necessarilyexact. Estimating can come in handy in a variety of situations, such as buying acomputer. You may have to purchase numerous devices: a computer tower andkeyboard for 1,295, a monitor for 679, the printer for 486, the warranty for 196, andsoftware for 374. Estimating can help you know about how much you’ll spend withoutactually adding those numbers exactly.Estimation usually requires rounding. When you round a number, you find a newnumber that’s close to the original one. A rounded number uses zeros for some of theplace values. If you round to the nearest ten, you will have a zero in the ones place. Ifyou round to the nearest hundred, you will have zeros in the ones and tens places.Because these place values are zero, adding or subtracting is easier, so you can find anestimate to an exact answer quickly.It is often helpful to estimate answers before calculating them. Then if your answer is notclose to your estimate, you know something in your problem-solving process is wrong.Using Rounding to Estimate Sums and DifferencesObjective 1Suppose you must add a series of numbers. You can round each addend to the nearesthundred to estimate the sum.ExampleProblemEstimate the sum 1,472 398 772 164 by rounding each number to thenearest hundred.1,472 .1,500398 . 400First, round each numberto the nearest hundred.772 .800164 .2001.39

1,5 0 0400800 2002,9 0 0AnswerThen, add the roundednumbers together.The estimate is 2,900.In the example above, the exact sum is 2,806. Note how close this is to the estimate,which is 94 greater.In the example below, notice that rounding to the nearest ten produces a far moreaccurate estimate than rounding to the nearest hundred. In general, rounding to thelesser place value is more accurate, but it takes more steps.ExampleProblemEstimate the sum 1,472 398 772 164 byfirst rounding each number to the nearestten.1,472 .1,470398 . 400First, round each number to thenearest ten.772 .770164 .160121470400770 16000121470400770 160800121470400770 1602800Next, add the ones and then thetens. Here, the sum of 7, 7, and 6is 20. Regroup.Now, add the hundreds. The sumof the digits in the hundreds placeis 18. Regroup.Finally, add the thousands. Thesum in the thousands place is 2.Answer The estimate is 2,800.1.40

Note that the estimate is 2,800, which is only 6 less than the actual sum of 2,806.Self Check AIn three months, a freelance graphic artist earns 1,290 for illustrating comic books, 2,612 for designing logos, and 4,175 for designing web sites. Estimate how much sheearned in total by first rounding each number to the nearest hundred.You can also estimate when you subtract, as in the example below. Because you round,you do not need to subtract in the tens or hundreds places.ExampleProblemEstimate the difference of 5,876 and 4,792 by firstrounding each number to the nearest hundred.5,876 .5,9004,792 .4,8005,9 0 0– 4,8 0 01,1 0 0AnswerFirst, round each number to the nearesthundred.Subtract. No regrouping is needed sinceeach number in the minuend is greaterthan or equal to the correspondingnumber in the subtrahend.The estimate is 1,100.The estimate is 1,100, which is 16 greater than the actual difference of 1,084.Self Check BEstimate the difference of 474,128 and 262,767 by rounding to the nearest thousand.Solving Application Problems by EstimatingEstimating is handy when you want to be sure you have enough money to buy severalthings.ExampleProblemWhen buying a new computer, you find thatthe computer tower and keyboard cost 1,295,the monitor costs 679, the printer costs 486, the 2-year warranty costs 196, and asoftware package costs 374. Estimate thetotal cost by first rounding each number tothe nearest hundred.1.41Objective 2

1,295 .1,300679 . 700First, round each number to thenearest hundred.486 .500196 .200374 .40021300700500200 4003,1 0 0AnswerAdd.After adding all of the roundedvalues, the estimated answer is 3,100.The total cost is approximately 3,100.Estimating can also be useful when calculating the total distance one travels overseveral trips.ExampleProblemJames travels 3,247 m to the park, then 582 mto the store. He then travels 1,634 m back tohis house. Find the total distance traveled byfirst rounding each number to the nearest ten.3247 .3,250582 .580First, round each number to thenearest ten.1634 . 1,63013250580 163060Adding the numbers in the tensplace gives 16, so you need toregroup.113250580 1630460Adding the numbers in thehundreds place gives 14, soregroup.1.42

1132 50580 16305,4 6 0Adding the numbers in thethousands place gives 5.AnswerThe total distance traveled was approximately5,460 meters.In the example above, the final estimate is 5,460 meters, which is 3 less than the actualsum of 5,463 meters.Estimating is also effective when you are trying to find the difference between twonumbers. Problems dealing with mountains like the example below may be important toa meteorologist, a pilot, or someone who is creating a map of a given region. As in otherproblems, estimating beforehand can help you find an answer that is close to the exactvalue, preventing potential errors in your calculations.ExampleProblemOne mountain is 10,496 feet high and anothermountain is 7,421 feet high. Find thedifference in height by first rounding eachnumber to the nearest 100.10,496 .10,5007,421 .7,40010500– 74003,1 0 0First, round each number to thenearest hundred.Then, align the numbers andsubtract.The final estimate is 3,100, whichis 25 greater than the actual valueof 3,075.AnswerThe estimated difference in height between thetwo mountains is 3,100 feet.Self Check CA space shuttle traveling at 17,581 miles per hour decreases its speed by 7,412 milesper hour. Estimate the speed of the space shuttle after it has slowed down by roundingeach num

miles. The two numbers to be added, 1,240 and 530, are called the . addends .The total distance, 1,770 miles, is the sum. Adding Whole Numbers, without Regrouping Objective . Adding numbers with more than one digit requires an understanding of placevalue . The place value of a digit is the