Knowing WHEN to add or subtract Adding and Subtracting All Sorts of Numbers We use addition when we know the parts and want to find the total. We use subtraction when we know the total and want to take some away, partition into two different groups, find out how much more or less, or find the difference between two numbers. Key words and phrases that indicate addition: sum, total, altogether, increased, plus, more than, added to, perimeter (perimeter is the distance around a shape and is found by adding the length of all the sides.) Key words and phrases that indicate SUBTRACTION: difference, how much more?, what is left?, take away, less than, reduced by, decreased by, minus. We can only add and subtract like things, such as oranges and oranges, or feet and feet. We do not add and subtract unlike things, like apples and oranges, or feet and meters. With fractions, this also means we can only add or subtract fractions with like denominators. This is because we need to add like things, halves and halves, thirds and thirds, fourths and fourths, and so on. We do not combine halves and fifths, just like we do not combine apples and oranges. Process for Solving Word Problems Write down the procedure. Even if you use your calculator to solve the problem, write down the numbers you are using, and show whether you used addition or subtraction. Include proper units. A number without units is not a complete answer. Answer using a complete sentence. This is a good way to make sure you include all the relevant information in your answer. Check that your answer makes sense. Example: Linda buys supplies for the math teachers at the bookstore. Yesterday she bought boxes of blue pens, boxes of red pens, boxes of green pens, and boxes of black pens. How many boxes of pens did she buy altogether? In this case, we want to find the total, so we should use addition. Show the procedure: + + + = Answer with a sentence that includes proper units: Linda bought boxes of pens. Example: If you remove cubic yards of sand from a pile containing 6 cubic yards, how much will be left? We will subtract to find out what is left. The total is 6 cubic yards, and we want to remove cubic yards. Show the procedure: 6 = 7 Answer with a sentence that includes proper units: There are 7 cubic yards left.
Example: At the bookstore, Jeff purchased some pens for $, a book for $, and a calculator for $07. His financial aid will cover $. How much additional money will he need to pay? In order to find out how much additional money Jeff will need, we must first know how much he will spend in total. Because we want the total we use addition. Show the procedure: + + 07 =. Jeff will spend a total of $. Jeff s financial aid will cover $ of the $ total. We know the total, and want to know what remains after financial aid, so we will use subtraction. Show the procedure: = 0 Answer with a sentence that includes proper units, in this case the units are dollars. Jeff will need an additional $0 to pay at the bookstore. Example: Anita drove to the store to buy two ingredients for dinner. She bought onions for $. and butter for $.9. At the register she paid with a $0 bill. How much change does she get? This is a two-step problem. First we must find the total of Anita s purchases (addition), then make change for the $0 bill (subtraction). Procedure: Anita s total purchases: $. + $.9 = $7.7 Anita s change: $0 $7.7 = $.6 Answer with a sentence and units: Anita should get $.6 in change. Check that your answer makes sense: Anita paid with a $0 bill, so she should get less than $0 back in change. This answer seems reasonable. Example: Jesse has three one-gallon containers. The first one has of a gallon of juice, the second has gallons of juice and the third has gallons of juice. How many total gallons of juice does Jesse have? We want to find the total amount of juice, so we will add. Show the procedure: + + = 6 reduce the fraction: 6 = Answer with a sentence that includes proper units: Jesse has of a gallon of juice.
Knowing HOW to add and subtract Note: Below are directions for doing addition and subtraction by hand, with whole numbers, decimals, and fractions. In this class you will always be allowed to use a calculator, but this might not be the case in future classes, or on some professional tests you may wish to take. Practicing addition and subtraction by hand will also help you understand the process more deeply, and be able to recognize when an answer makes sense, or when it does not. Adding and Subtracting Whole Numbers Adding by Grouping Example: Add + 79 We can break this apart: Add the hundreds: 00+00 = 00 Add the tens: 0 + 70 = 0 Add the ones: + 9 = Add the resulting pieces = Adding by Rearranging The idea that + is the same as + is called the commutative property for addition. Example: Add 7 + + We can rearrange the order to be 7 + + Since 7+ = 0, this makes things a bit easier 7 + + = 0 + = Subtracting using Expanded form Example: Subtract - 69 We can write this as: 00 + 0 + - 00 + 60 + 9 We can t take 9 from, so we borrow 0 from the 0 00 + 0 + - 00 + 60 + 9 Likewise we can t take 60 from 0, so we borrow 00 from the 00 00 + 0 + - 00 + 60 + 9 -------------------------- 00 + 0 + = Subtracting by Adjusting Values Example: Subtract 6 - If we add to both numbers, the difference will be the same, but easier to compute 6 + = 6 + = 0 ------------------------
Adding and Subtracting Decimals To add and subtract decimals, stack the numbers, aligning the place values and the decimal point. Add like place values, carrying as needed. The decimal point in the sum will be aligned with the decimal point in the numbers being added Example: Add. and.. +.. If one decimal has more decimal places than another, you can optionally write additional zeros on the number with less decimal places, since that doesn t change the value of the number. Example: Add.0 and..0.0 +. Writing additional zeros: +.00 7.70 Subtraction works the same way, but it is more important here to write the additional zeros if the top number has less decimal places than the bottom number. Example: Subtract. -.7..00.9 0 7. 9 0 -.7 -.7 -.7 -. 7. 7 9 Example: Ariel has a balance of $0. in her checking account. After paying an electric bill for $7.0 and a cell phone bill for $., how much will she have left in her account? We might start by adding the two bills: 7.0 +. 0.7 Now, subtracting this from $0.: 0. 0. 9. -0.7-0.7-0. 7 09. 9 Ariel will have $09.9 remaining in her account.
Adding and Subtracting Fractions Adding and Subtraction Fractions with Like Denominators To add or subtract fractions with the same denominator, we add or subtract the number of pieces of the whole. a b a b a b a b The denominator remains the same: and c c c c c c Example: Add and simplify Example: Subtract and simplify To add mixed numbers, add the whole parts and add the fractional parts. If the sum of the fractional parts is greater than, combine it with the whole part 7 Example: Add and simplify 9 9 Adding the whole parts. Adding the fractional parts, Now we combine these: 6 7 9 7. 9 9 9 9 To subtract mixed numbers, subtract the whole parts and subtract the fractional parts. You may need to borrow a whole to subtract the fractions Example: Subtract and simplify Since is larger than, we don t need to borrow., and, so Example: Subtract and simplify Since is smaller than, we need to borrow. We can say. Now we can subtract: and, so Alternatively, you can add or subtract mixed numbers my converting to improper fractions first: 6
Adding and Subtracting with Unlike Denominators Since can only add or subtract fractions with like denominators, if we need to add or subtract fractions with unlike denominators, we first need to give them a common denominator. Example: Add and simplify Since these don t have the same denominator, we identify the least common multiple of the two denominators,, and give both fractions that denominator. Then we add and simplify. 7 Example: Subtract and simplify The least common multiple of and is. We give both fractions this denominator and subtract. 7 Example: Add and simplify 7 7 9 7 9 7 6 We give these a common denominator of and add: 6 This can be reduced and written as a mixed number: To add and subtract mixed numbers with unlike denominators, give the fractional parts like denominators, then proceed as we did before. Example: Add and simplify Rewriting the fractional parts with a common denominator of : Adding the whole parts 7. Adding the fractional parts, Now we combine these: 7 Example: Subtract and simplify 6 Rewriting the fractional parts with a common denominator of 6: Since 0 6 is smaller than, we borrow: 6 6 0 6 0 =, and, so 6 = 6 9 9 7. 6 0