Estimate, Find Missing Numbers and Solve Word Problems | Grade 5 Math | LevelKite

You can already add and subtract large numbers. The answer to an addition is called the sum. The answer to a subtraction is called the difference. But how do you know that your answer makes sense?

One way is to estimate. An estimate is a number that is close to the exact answer and quick to find in your head. It does not replace the exact answer. It tells you about how big the answer should be. If your exact work gives a number of a very different size, you know to look for a mistake.

To estimate, you round each number. A digit's place is its position in a number, such as the hundreds place or the thousands place. Rounding to the nearest thousand means replacing a number with the closest thousand. $6{,}348$ is between $6{,}000$ and $7{,}000$ . It is closer to $6{,}000$ , so it rounds to $6{,}000$ . A number exactly halfway, such as $6{,}500$ , rounds up to $7{,}000$ . A quick way to choose: look at the digit just to the right of the thousands place. $0$ to $4$ means the lower thousand. $5$ to $9$ means the higher thousand. Rounding to the nearest hundred works the same way, with the digit to the right of the hundreds place.

Round every number in a problem to the same place: all to the nearest thousand, or all to the nearest hundred. Then the rounded numbers are easy to add or subtract. $6{,}000 + 3{,}000$ is just $6$ thousands plus $3$ thousands. That makes $9$ thousands, or $9{,}000$ .

Two problems, each estimated two ways

Exact problem	Each number rounded to the nearest	Estimate	Exact answer	How far apart
$6{,}348 + 2{,}871$	thousand	$6{,}000 + 3{,}000 = 9{,}000$	$9{,}219$	$219$
$6{,}348 + 2{,}871$	hundred	$6{,}300 + 2{,}900 = 9{,}200$	$9{,}219$	$19$
$9{,}214 - 2{,}875$	thousand	$9{,}000 - 3{,}000 = 6{,}000$	$6{,}339$	$339$
$9{,}214 - 2{,}875$	hundred	$9{,}200 - 2{,}900 = 6{,}300$	$6{,}339$	$39$

How close will an estimate be? The nearest thousand is never more than $500$ away from a number. So rounding to the nearest thousand moves each number by $500$ at most. With two rounded numbers, the estimate is at most $1{,}000$ away from the exact sum or difference. In the table, those estimates are $219$ and $339$ away.

Rounding to the nearest hundred moves each number by $50$ at most. With two numbers, the estimate is at most $100$ away. In the table, those estimates are $19$ and $39$ away. So a smaller place gives a stricter check, and a larger place gives easier numbers.

Use this to judge an exact answer. Say you rounded two numbers to the nearest thousand:

If the answer is more than $1{,}000$ away from the estimate, there is a mistake. Recheck the rounding and the exact work.
If the answer is $1{,}000$ away or less, it is reasonable. That means it is the right size. It could still hold a small mistake, because an estimate checks the size of an answer, not every digit.

Worked example

Judge two answers with an estimate

Maya and Leo both subtract $8{,}105 - 3{,}692$ . Maya gets $5{,}413$ . Leo gets $4{,}413$ . Use an estimate to judge each answer.

Round each number to the same place. Use the nearest thousand. 8,105 is closer to 8,000 than to 9,000. 3,692 is closer to 4,000 than to 3,000.
$8{,}105 \rightarrow 8{,}000 \qquad 3{,}692 \rightarrow 4{,}000$
Subtract the rounded numbers to get the estimate. 8 thousands minus 4 thousands is 4 thousands.
$8{,}000 - 4{,}000 = 4{,}000$
Find how far Maya's answer is from the estimate. Two numbers were rounded to the nearest thousand, so a correct answer is at most 1,000 away. Her answer is 1,413 away. It is not reasonable, so it holds a mistake.
$5{,}413 - 4{,}000 = 1{,}413$
Do the same for Leo's answer. It is 413 away from the estimate. That is less than 1,000, so his answer is reasonable. It is the right size.
$4{,}413 - 4{,}000 = 413$
Reasonable means the right size, not that every digit is right. So subtract exactly to be sure. The exact difference matches Leo's answer.
$8{,}105 - 3{,}692 = 4{,}413$

Answer

Maya's answer, $5{,}413$ , is not reasonable because it is more than $1{,}000$ away from the estimate of $4{,}000$ . Leo's answer, $4{,}413$ , is reasonable, and the exact subtraction shows that it is correct.

A number sentence uses numbers, a sign such as $+$ or $-$ , and an equals sign to say that two amounts are equal, as in $650 + 250 = 900$ . Sometimes one number is missing, as in $? + 250 = 900$ . The $?$ stands for the number you need to find.

Think of the three numbers as one whole and two parts. The whole is the total. The parts are the two amounts that make up the total. An addition sentence joins the parts to make the whole, so the whole is the sum. A subtraction sentence starts with the whole, takes one part away and leaves the other part, so the whole is the number you subtract from.

Addition and subtraction are inverse operations. That means each one undoes the other. Start at $650$ and add $250$ , and you reach $900$ . Subtract $250$ from $900$ , and you are back at $650$ . This is why the same three numbers fit four sentences:

$650 + 250 = 900$ , $250 + 650 = 900$ , $900 - 250 = 650$ and $900 - 650 = 250$ .

Key concept

Two parts and one whole

Addition and subtraction tell about the same three numbers: two parts and their whole. Add the parts and you get the whole. Subtract one part from the whole and you get the other part.

So when one number is missing, in a number sentence or in a story, you can find it from the other two. Add to find a missing whole. Subtract to find a missing part. Then estimate with rounded numbers to check that your answer is the right size.

\text{part} + \text{part} = \text{whole} \qquad \text{whole} - \text{part} = \text{other part}

The same three numbers, with a different one missing each time

Number sentence	The missing number is	How to find it	Missing number
$? + 250 = 900$	a part	Subtract the known part from the whole: $900 - 250$	$650$
$250 + ? = 900$	a part	Subtract the known part from the whole: $900 - 250$	$650$
$? - 250 = 650$	the whole	Add the two parts: $650 + 250$	$900$
$900 - ? = 650$	a part	Subtract the known part from the whole: $900 - 650$	$250$

A word problem tells a short story with numbers and asks a question. You can turn the story into a number sentence with one number missing. Here is a plan:

Read the story. Ask which amount is the whole and which amounts are the parts.
Write a number sentence with $?$ for the number you need.
Find the missing number.
Check your answer against an estimate.

In this lesson, a one-step problem needs one addition or one subtraction. A two-step problem needs two of these, one after the other. The story leaves out a number you need, so you find that number first. Then you use it to answer the question.

For the estimate, round every number in the story to the same place and do the same steps with the rounded numbers. Each number rounded to the nearest thousand moves by $500$ at most, so three rounded numbers can put the estimate up to $1{,}500$ away from the exact answer.

Worked example

A one-step problem with a missing start

A library gave away $1{,}875$ old books. After that, it had $4{,}260$ books left. How many books did the library have before it gave the books away?

Write the story as a number sentence. Use ? for the number you do not know. The library started with ? books, took away 1,875, and had 4,260 left.
$? - 1{,}875 = 4{,}260$
Decide: whole or part? The missing number is the one you subtract from, so it is the whole. The books given away and the books left are its two parts. To find a missing whole, add the parts. Adding undoes the taking away.
$? = 4{,}260 + 1{,}875$
Add the two parts to get the whole.
$4{,}260 + 1{,}875 = 6{,}135$
Check the size with an estimate. Round both parts to the nearest thousand: 4,260 rounds to 4,000 and 1,875 rounds to 2,000. The estimate is 6,000. The exact answer is 135 away. That is less than 1,000, so it is reasonable.
$4{,}000 + 2{,}000 = 6{,}000$
Put your answer back into the number sentence to be sure. Taking 1,875 away from 6,135 leaves 4,260, which matches the story.
$6{,}135 - 1{,}875 = 4{,}260$

Answer

The library had $6{,}135$ books before it gave the old books away.

Worked example

A two-step problem

Two classes count their steps for a walking challenge. Class A walks $18{,}640$ steps. Class B walks $23{,}275$ steps. Their goal is $50{,}000$ steps in all. How many more steps do the classes need to reach the goal?

Make a plan. The question asks how far the classes are from 50,000 steps. First you need the number of steps walked so far. The story does not give it, so you add to find it. Then you compare it with the goal.
Step one: add the steps of the two classes. They are the two parts, and the steps so far are their whole.
$18{,}640 + 23{,}275 = 41{,}915$
Step two: the goal is the whole. The steps so far and the steps still needed are its two parts. The missing number is a part, so subtract the known part from the whole.
$41{,}915 + ? = 50{,}000 \quad \text{so} \quad ? = 50{,}000 - 41{,}915 = 8{,}085$
Estimate with every number rounded to the nearest thousand. 18,640 rounds to 19,000 and 23,275 rounds to 23,000. The goal of 50,000 is already an exact number of thousands, so it stays the same. Do the same two steps.
$19{,}000 + 23{,}000 = 42{,}000 \qquad 50{,}000 - 42{,}000 = 8{,}000$
Compare. The exact answer is 85 away from the estimate. With three numbers it could be up to 1,500 away, so 8,085 is reasonable.
$8{,}085 - 8{,}000 = 85$

Answer

The classes need $8{,}085$ more steps to reach their goal of $50{,}000$ steps.

Here is what to keep. Addition and subtraction link two parts and one whole. When a number is missing, first decide whether it is the whole or a part. Add the parts to find a missing whole. Subtract the known part from the whole to find a missing part. Word problems work the same way, in one step or in two. Then check: round every number to the same place and repeat the steps with the rounded numbers. If your exact answer is farther from that estimate than rounding can explain, look for a mistake.