Chapter 6 · Decimal operations
Check Decimal Answers and Fix Mistakes
Learn to check whether a decimal answer makes sense by estimating and thinking about the story, then find and fix mistakes in multi-step solutions.
You already know how to add, subtract, multiply and divide decimals, and how to solve word problems that take more than one step. Even careful work can go wrong, though. You might put a decimal point in the wrong place or leave out a step. So after you solve a problem, ask one more question: does my answer make sense?
An answer is reasonable when its size makes sense for the numbers in the problem and for the story the problem tells. An answer that is not reasonable is your signal to go back and look for a mistake. In this lesson you will learn two checks for size. Then you will use them to find and fix mistakes.
Check 1: Estimate with whole numbers. An estimate is a quick answer that is close to the exact answer but easier to find. To make one, round each decimal: swap it for the nearest whole number (a number with no decimal part, such as or ). Then work the problem with the rounded numbers. This check works best when rounding changes each number only a little, as it does in this lesson. A number such as would round to , and then the estimate would tell you very little.
Take . The number is close to , and is close to . So the exact answer should be close to , which is . The exact answer is . It sits right next to the estimate, so it passes this check.
Check 2: Ask whether the size fits the story. A number in a word problem stands for something real, such as a total cost or an amount of change. Some sizes cannot be true for that thing. The table shows three facts you can use in money problems. Each fact holds whenever every price is more than dollars.
| In the story | What must be true | Why |
|---|---|---|
You buy two or more of the same item. | The total is more than the price of one item. | Every extra item adds to the total. |
You pay and get change back. | The change is less than the amount you paid. | You get back only part of your money. |
Two or more people share a cost equally. | Each share is less than the whole cost. | The whole cost is split into smaller parts. |
Worked example
Which total makes sense?
Maya buys markers. Each marker costs dollars. Three classmates work out the total. They all get the digits , and , but they put the decimal point in different places: dollars, dollars and dollars. Which total is reasonable?
Estimate with whole numbers. The price 2.95 is closer to 3 than to 2, so round it to 3. Eight markers at about 3 dollars each cost about 24 dollars.
Compare each answer with the estimate of 24. The answer 2.36 is about ten times too small. The answer 236 is about ten times too big. Only 23.60 is close to 24.
Ask whether each size fits the story. Eight markers must cost more than one marker, but 2.36 dollars is less than the price of one. Each marker costs less than 3 dollars, so eight of them cost less than 24 dollars, but 236 dollars is far more than that.
Confirm with the exact total. Multiply each part of 2.95 by 8: 8 times 2 is 16, and 8 times 0.95 is 7.60. Add the two parts.
Answer
Only dollars is reasonable. It is close to the estimate of dollars and it fits the story. The exact total confirms it.
Key concept
Check the size, then check the steps
An answer is reasonable when it passes two checks. First, it is close to an estimate you make with whole numbers. Second, in a word problem, its size fits the story. If an answer fails either check, read the work one step at a time, find the mistake and fix it. Passing both checks tells you the size is about right. It does not tell you that every digit is right.
When an answer is not reasonable, you do not need to start over. Read the work one step at a time and ask three questions. Is the decimal point in the right place? Is this the right operation (adding, subtracting, multiplying or dividing) for this part of the story? Does the last step answer the question the problem asks? Each question matches a kind of mistake that is common in problems with more than one step. The table shows the clue to look for with each kind.
| Kind of mistake | What went wrong | Clue to look for |
|---|---|---|
Misplaced decimal point | The digits are right, but the point is one or two places away from where it belongs. | That step's result is or times too big or too small, like in place of . |
Wrong operation | A step adds, subtracts, multiplies or divides when the story calls for a different one. | The size often goes the wrong way, like a share that is bigger than the whole cost. |
Skipped step | The work stops before it answers the question. | The number comes from an earlier step and answers a different question, like the cost instead of the change. |
Worked example
Find the mistake: Leo's change
Leo buys sandwiches for dollars each. He pays with a dollar bill and wants to know his change. He writes and says his change is dollars. Find and fix his mistake.
Estimate the whole problem with whole numbers. Round 5.10 to 5. Three sandwiches cost about 15 dollars, so the change from 20 dollars is about 5 dollars.
Compare Leo's answer with the estimate. His 15.30 is less than the 20 dollars he paid, so the size fact about change does not catch the mistake. But 15.30 is about three times the estimate of 5, so it is not reasonable. Read his work one step at a time.
Test his multiplication. 3 times 5 is 15, and 3 times 0.10 is 0.30, so 15.30 is right. But 15.30 dollars is the cost of the sandwiches, and the question asks for the change. Leo skipped the last step.
Do the skipped step. The change is the part of the 20 dollars that is left after Leo pays the cost, so subtract the cost from 20.
Check the new answer. 4.70 is close to the estimate of 5. The cost and the change together make the 20 dollars Leo paid.
Answer
Leo skipped the subtraction step. The dollars is the cost of the sandwiches, not the change. His change is dollars.
Worked example
Find the mistake: sharing a cost
Four friends buy a pizza for dollars and drinks for dollars. A coupon takes dollars off. They share the cost equally. Ana writes , then , then . She says each friend pays dollars. Find and fix her mistake.
Estimate with whole numbers. Round 16.90 to 17 and 9.30 to 9. The food costs about 26 dollars, or about 24 dollars after the coupon. Split 4 ways, that is about 6 dollars each.
Compare. Ana's 96.80 is far more than the estimate of 6. It also fails the story check: one friend's share cannot be more than the whole cost, which is about 24 dollars.
Test each step. The addition is right: 16 plus 9 is 25, and 0.90 plus 0.30 is 1.20, which makes 26.20. The subtraction is right too, because a coupon lowers the cost. The last step is wrong. Sharing equally among 4 friends means splitting the cost into 4 equal parts, and that is division. Ana multiplied.
Redo the last step with division. Split 24.20 into 24 and 0.20. 24 divided by 4 is 6, and 0.20 divided by 4 is 0.05. Add the two parts.
Check the new answer. 6.05 is close to the estimate of 6. Four shares of 6.05 dollars make the whole cost again.
Answer
Ana used the wrong operation in her last step. She multiplied when the story called for division. Each friend pays dollars.
A decimal answer is not finished until you check it. Estimate with whole numbers, and ask whether the size fits the story. If the answer is far from your estimate or cannot be true in the story, read the work one step at a time. Look for a decimal point in the wrong place, a wrong operation or a skipped step. Then fix the mistake and check the new answer the same way.