A trail starts feet above sea level and ends on a peak feet above sea level. How do you find the climb, place by place, when the top number keeps running short?

Grade 4 · 3.NBT.A.2 · 4.NBT.B.4

The standard subtraction algorithm

A trail starts 2{,}738 feet above sea level and ends on a peak 6{,}415 feet above sea level. How do you find the climb, place by place, when the top number keeps running short?

You need

Regrouping in subtractionGrade 2 Fact familiesGrade 1

Unlocks

Subtracting across zerosGrade 4

Where you are 7 in Multi-digit subtraction

Number9

Arithmetic5

Addition and subtraction12

Multi-digit subtraction7

▴ 2 above Regrouping in subtraction Subtracting three-digit numbers by place value, including across zero

The standard subtraction algorithm

Subtracting across zeros Counting up to find a difference

When you reach for it

A hiking trail starts at a trailhead 2{,}738 feet above sea level. It climbs to the top of a peak that is 6{,}415 feet above sea level. How many feet does the trail climb?

The climb is the difference between the two heights, 6{,}415 - 2{,}738. No basic fact gives that, and the numbers are too big to hold in your head comfortably.

The standard subtraction algorithm handles numbers of any size. You write one number above the other, lined up by place value. Then you subtract ones from ones, tens from tens, hundreds from hundreds, and so on, starting at the right. When a place on top has fewer than the place below it, you unbundle one unit from the place to its left: one ten becomes ten ones, one hundred becomes ten tens. That trade is regrouping in subtraction, where one ten is broken into ten ones because a place does not have enough.

Reach for it in two kinds of problem. Taking away asks how many are left: a hiker at 7{,}429 feet walks down 3{,}615 feet. Comparing asks how much more: how much taller is one peak than another. Both are answered by the same subtraction, as in taking away and comparing.

The words: in 6{,}415 - 2{,}738 = 3{,}677, the answer 3{,}677 is the difference.

Two facts carry the whole method. First, place value: each place is worth ten of the place on its right, the same columns you see in a The place value chart. So one unit of a place can always be traded for ten units of the next place to the right, and the number stays the same. Second, the subtraction facts up to 18 - 9: every column ends as one of them.

The procedure

Part 1: lining up places, and unbundling one ten

Start with a difference where every place on top is big enough: 4{,}869 - 2{,}345.

Write 2{,}345 under 4{,}869 so that ones sit under ones, tens under tens, and so on. The right ends line up. When the numbers have different lengths, this matters most: the ones digits always share a column.

\begin{array}{r} 4{,}869 \\ -\;2{,}345 \\ \hline 2{,}524 \end{array}

Now subtract one place at a time.

Ones. 9 - 5 = 4 ones.
Tens. 6 - 4 = 2 tens.
Hundreds. 8 - 3 = 5 hundreds.
Thousands. 4 - 2 = 2 thousands.

The difference is 2{,}524. Why may you do it this way? The number 2{,}345 is 2{,}000 + 300 + 40 + 5. Taking it away is the same as taking away each of those parts. And it is fair to take ones from ones and tens from tens, because 4{,}869 is 4 thousands, 8 hundreds, 6 tens and 9 ones, so each part has its own place to come from.

Now a difference where one place runs short: 5{,}382 - 1{,}247.

Ones. The top has 2 ones. You need to take away 7 ones, and 2 is not enough. Look one place to the left. The tens place holds 8 tens. Unbundle one of them: one ten becomes ten ones. Now the tens place holds 7 tens and the ones place holds 2 + 10 = 12 ones.

Figure 1One ten is unbundled into ten ones. Both pictures show 82, so the trade does not change the number.

The picture shows why this is allowed. Both sides are 82: 8 \text{ tens} + 2 \text{ ones} = 80 + 2 and 7 \text{ tens} + 12 \text{ ones} = 70 + 12. The number did not change. Only the way it is bundled changed.

Write the trade down. Cross out the 8 in the tens and write 7 above it. Cross out the 2 in the ones and write 12 above it. Then 12 - 7 = 5 ones.

Tens. The tens place now holds 7, not 8. So 7 - 4 = 3 tens.

Hundreds. 3 - 2 = 1 hundred.

Thousands. 5 - 1 = 4 thousands.

\begin{array}{r} 5{,}382 \\ -\;1{,}247 \\ \hline 4{,}135 \end{array}

The difference is 4{,}135.

The same trade works between any two neighboring places. One hundred is ten tens. One thousand is ten hundreds. One ten thousand is ten thousands. Every place is worth ten of the place on its right, so the move is always the same.

Some people call this trade “borrowing.” Nothing is borrowed and nothing is paid back: one unit is unbundled into ten smaller ones and stays in the number. This lesson calls it unbundling.

Part 2: unbundling in several places

Back to the trail: 6{,}415 - 2{,}738. This time the top runs short three times.

Ones. 5 ones, take away 8: not enough. Unbundle one ten. The tens go from 1 to 0, and the ones go from 5 to 15. Then 15 - 8 = 7 ones.

Tens. The tens place now holds 0, and you need to take away 3: not enough. Unbundle one hundred. The hundreds go from 4 to 3, and the tens go from 0 to 10. Then 10 - 3 = 7 tens.

Hundreds. The hundreds place now holds 3, and you need 7: not enough. Unbundle one thousand. The thousands go from 6 to 5, and the hundreds go from 3 to 13. Then 13 - 7 = 6 hundreds.

Thousands. The thousands place now holds 5. Then 5 - 2 = 3 thousands.

The difference is 3{,}677. The trail climbs 3{,}677 feet.

Figure 26{,}415 - 2{,}738 = 3{,}677 with its record of trades. Each crossed-out digit has its new value written just above it.

The record of trades in the figure is worth reading slowly. Each trade changes two places: it lowers the place on the left by 1 and raises the place on the right by 10. A place can be changed twice, once by the place on its right taking from it and once when it takes from the place on its left. That is what happened to the tens: 1, then 0, then 10.

This is why the algorithm starts at the right. A trade always lowers the place to the left. Working from the right, you settle every trade before you reach the place it changes, so no digit you have already written in the answer ever has to be changed.

The top number still has the same value after all the trades. It now reads 5 thousands, 13 hundreds, 10 tens and 15 ones:

5{,}000 + 1{,}300 + 100 + 15 = 6{,}415.

One trade is always enough for one place. After a trade the top digit is at least 10, and the digit below it is at most 9. So the column can always be subtracted.

Part 3: estimating first, and checking by adding back

Two habits go with the algorithm. One comes before it and one comes after.

Estimate first. Round each number to a friendly place and subtract in your head, as in estimating a difference. For the trail, 6{,}415 is about 6{,}000 and 2{,}738 is about 3{,}000, so the climb is about 6{,}000 - 3{,}000 = 3{,}000 feet. The answer 3{,}677 is near that, so it is in the right range. An estimate catches a big slip, such as numbers lined up on the left instead of the right.

Check by adding back. Every subtraction fact has a partner addition fact, the idea of Fact families. The facts 7 - 3 = 4 and 4 + 3 = 7 belong to one family. The same is true for big numbers. The difference is exactly the amount you must add to the smaller number to reach the larger one. So if 6{,}415 - 2{,}738 = 3{,}677, then

3{,}677 + 2{,}738 = 6{,}415.

Do the addition with the standard addition algorithm. Ones: 7 + 8 = 15, write 5, carry 1. Tens: 7 + 3 + 1 = 11, write 1, carry 1. Hundreds: 6 + 7 + 1 = 14, write 4, carry 1. Thousands: 3 + 2 + 1 = 6. The sum is 6{,}415, the number you started with, so the difference is right.

Adding back catches small slips that an estimate cannot see. If the sum comes out wrong, the place where it first goes wrong points to the column to redo.

Take away or compare. For a comparing problem, a bar picture helps you see that subtraction is the right move. Draw the two amounts as bars from the same starting edge. The piece of the longer bar that sticks out past the shorter one is the difference.

Figure 3Comparing two heights. The dashed piece is how much taller one peak is, and it is found by subtracting.

Worked examples

Example 1

Part 1: no unbundling at all

Mahina reads that one peak is 9{,}587 feet tall and another is 4{,}352 feet tall. How much taller is the first peak?

Solution

“How much taller” compares two heights, so subtract the smaller from the larger: 9{,}587 - 4{,}352. Line up the ones.

\begin{array}{r} 9{,}587 \\ -\;4{,}352 \\ \hline 5{,}235 \end{array}

Ones: 7 - 2 = 5. Tens: 8 - 5 = 3. Hundreds: 5 - 3 = 2. Thousands: 9 - 4 = 5. Each top digit is at least as big as the digit below it, so no place needs a trade.

The first peak is 5{,}235 feet taller. Check: 5{,}235 + 4{,}352 = 9{,}587.

Example 2

Part 1: unbundling a thousand

Dariush is at a lookout 7{,}429 feet above sea level. He hikes down 3{,}615 feet to a lake. How high is the lake?

Solution

He loses height, so take away: 7{,}429 - 3{,}615.

Ones: 9 - 5 = 4. Tens: 2 - 1 = 1. Hundreds: the top has 4 hundreds and you need 6. Not enough. Unbundle one thousand into ten hundreds: the thousands go from 7 to 6, and the hundreds go from 4 to 14. This is the same trade as one ten for ten ones, one place further left. Now 14 - 6 = 8 hundreds. Thousands: 6 - 3 = 3.

\begin{array}{r} 7{,}429 \\ -\;3{,}615 \\ \hline 3{,}814 \end{array}

The lake is 3{,}814 feet above sea level. Check: 3{,}814 + 3{,}615 = 7{,}429.

Example 3

Part 2: two trades that are not side by side

Find 7{,}264 - 3{,}518, and show that the top number keeps its value after the trades.

Solution

Ones: 4 ones, take away 8. Unbundle a ten: tens 6 \to 5, ones 4 \to 14. Then 14 - 8 = 6.

Tens: the tens place now holds 5. Then 5 - 1 = 4. No trade is needed here.

Hundreds: 2 hundreds, take away 5. Unbundle a thousand: thousands 7 \to 6, hundreds 2 \to 12. Then 12 - 5 = 7.

Thousands: 6 - 3 = 3.

\begin{array}{r} 7{,}264 \\ -\;3{,}518 \\ \hline 3{,}746 \end{array}

The difference is 3{,}746. Each place is decided on its own: a trade happens only where the top runs short, and the tens did not.

After the two trades the top reads 6 thousands, 12 hundreds, 5 tens and 14 ones, and

6{,}000 + 1{,}200 + 50 + 14 = 7{,}264.

So the trades only rebundled the number. Check by adding: 3{,}746 + 3{,}518 = 7{,}264.

Example 4

Part 2: a trade in every place, numbers of different lengths

Nkosana runs a postcard printer. It has 24{,}361 postcards in stock, and he ships 8{,}975 of them. How many are left?

Solution

Take away: 24{,}361 - 8{,}975. The second number has only four digits, so line up the right ends: the 5 goes under the 1, and nothing sits under the 2 ten thousands.

Figure 424{,}361 - 8{,}975 = 15{,}386. Each column from the ones to the thousands needs one trade, and the numbers line up on the right.

Ones: 1 - 5 runs short. Unbundle a ten: tens 6 \to 5, ones 1 \to 11. Then 11 - 5 = 6.

Tens: 5 - 7 runs short. Unbundle a hundred: hundreds 3 \to 2, tens 5 \to 15. Then 15 - 7 = 8.

Hundreds: 2 - 9 runs short. Unbundle a thousand: thousands 4 \to 3, hundreds 2 \to 12. Then 12 - 9 = 3.

Thousands: 3 - 8 runs short. Unbundle a ten thousand: ten thousands 2 \to 1, thousands 3 \to 13. Then 13 - 8 = 5.

Ten thousands: 1 - 0 = 1, since nothing is taken from that place.

There are 15{,}386 postcards left. Check: 15{,}386 + 8{,}975 = 24{,}361.

Example 5

Part 3: estimate, compare, check

One peak is 14{,}236 feet tall. A second is 9{,}568 feet tall. Estimate how much taller the first is, then find the exact difference and check it.

Solution

This compares two heights, as in the bar picture in Part 3: the difference is the dashed piece.

Estimate. Round to the nearest thousand: 14{,}000 - 10{,}000 = 4{,}000 feet.

Subtract. 14{,}236 - 9{,}568, right ends lined up.

Ones: 6 - 8 runs short. Tens 3 \to 2, ones 6 \to 16, and 16 - 8 = 8.
Tens: 2 - 6 runs short. Hundreds 2 \to 1, tens 2 \to 12, and 12 - 6 = 6.
Hundreds: 1 - 5 runs short. Thousands 4 \to 3, hundreds 1 \to 11, and 11 - 5 = 6.
Thousands: 3 - 9 runs short. Ten thousands 1 \to 0, thousands 3 \to 13, and 13 - 9 = 4.
Ten thousands: 0 left, and nothing below it.

The difference is 4{,}668 feet. It is close to the estimate of 4{,}000.

Check by adding. 4{,}668 + 9{,}568: ones 8 + 8 = 16, write 6, carry 1; tens 6 + 6 + 1 = 13, write 3, carry 1; hundreds 6 + 5 + 1 = 12, write 2, carry 1; thousands 4 + 9 + 1 = 14. The sum is 14{,}236, so the first peak is 4{,}668 feet taller.

Example 6

Part 3: adding back finds a slip

Mahina works out 8{,}342 - 4{,}567 and gets 3{,}875. Check her answer by adding back, and correct it if it is wrong.

Solution

If the difference is right, then 3{,}875 + 4{,}567 must be 8{,}342. Add: ones 5 + 7 = 12, write 2, carry 1; tens 7 + 6 + 1 = 14, write 4, carry 1; hundreds 8 + 5 + 1 = 14, write 4, carry 1; thousands 3 + 4 + 1 = 8. The sum is 8{,}442, not 8{,}342.

The sum is 100 too big, and the ones and tens match. So look at the hundreds column.

Redo the subtraction. Ones: 2 - 7 runs short; tens 4 \to 3, ones 12, and 12 - 7 = 5. Tens: 3 - 6 runs short; hundreds 3 \to 2, tens 13, and 13 - 6 = 7. Hundreds: the hundreds place now holds 2, not 3. Then 2 - 5 runs short; thousands 8 \to 7, hundreds 12, and 12 - 5 = 7. Thousands: 7 - 4 = 3.

The difference is 3{,}775. The hundreds digit is 7, not 8, because the hundreds place holds 2 after the tens trade. Check: 3{,}775 + 4{,}567 = 8{,}342.

Example 7

Part 3: a climb in two legs

Dariush starts at a trailhead 3{,}186 feet above sea level. He camps at 6{,}857 feet, and the next day he reaches the summit at 9{,}524 feet. How many feet does he climb each day? Show that the two climbs add up to the whole climb.

Solution

Day one takes the camp height minus the trailhead height: 6{,}857 - 3{,}186. Ones: 7 - 6 = 1. Tens: 5 - 8 runs short; hundreds 8 \to 7, tens 15, and 15 - 8 = 7. Hundreds: 7 - 1 = 6. Thousands: 6 - 3 = 3. He climbs 3{,}671 feet.

Day two is 9{,}524 - 6{,}857. Ones: 4 - 7 runs short; tens 2 \to 1, ones 14, and 14 - 7 = 7. Tens: 1 - 5 runs short; hundreds 5 \to 4, tens 11, and 11 - 5 = 6. Hundreds: 4 - 8 runs short; thousands 9 \to 8, hundreds 14, and 14 - 8 = 6. Thousands: 8 - 6 = 2. He climbs 2{,}667 feet.

The whole climb is 9{,}524 - 3{,}186. Ones: 4 - 6 runs short; tens 2 \to 1, ones 14, and 14 - 6 = 8. Tens: 1 - 8 runs short; hundreds 5 \to 4, tens 11, and 11 - 8 = 3. Hundreds: 4 - 1 = 3. Thousands: 9 - 3 = 6. That is 6{,}338 feet.

The two days add to 3{,}671 + 2{,}667 = 6{,}338, which matches. That is a check on all three subtractions at once.

Boundaries

Whole numbers, larger on top. The algorithm finds a whole-number difference only when the top number is at least as large as the bottom one. In 2{,}738 - 6{,}415 the thousands run short with no place to the left to unbundle from. The answer there is not a whole number; it is a negative integer, met later in adding and subtracting integers. “How much taller” always puts the taller height on top.
Places lined up on the right. The method rests on taking ones from ones and tens from tens. If the numbers are lined up on the left, the columns hold different places and the answer means nothing.
Unbundling needs something to unbundle. Every trade in this lesson takes one unit from a place that holds at least 1. When the place to the left holds 0, as in 5{,}000 - 1{,}846, the trade has to reach further left first; that case is subtracting across zeros.
One trade per place. A place receives at most one trade from its left. After it, the top digit is at least 10 and the bottom digit at most 9, so the column always works.

Where this goes next

The same column method subtracts money and decimals once the decimal points are lined up, and it sits inside every longer calculation, such as the long division that subtracts at each step. The habit of checking by adding back stays useful for every subtraction you will ever do, because addition and subtraction always come as a family.

Exercises

Part 1: lining up and one trade

Find 5{,}768 - 2{,}431.
Find 9{,}946 - 6{,}723.
Find 4{,}573 - 1{,}248.
Find 8{,}192 - 5{,}371.
Find 63{,}854 - 21{,}627.
One peak is 7{,}386 feet tall and a lower one is 4{,}152 feet tall. How much taller is the first peak?
For each difference, say which place needs a trade (if any), then find it: (a) 5{,}437 - 2{,}316, (b) 5{,}437 - 2{,}381, (c) 5{,}437 - 2{,}527.

Part 2: trades in several places

Find 5{,}231 - 2{,}874.
Find 8{,}523 - 4{,}756.
Find 6{,}342 - 3{,}587.
Find 9{,}253 - 2{,}968.
Find 41{,}726 - 18{,}359.
Find 52{,}624 - 7{,}839.
A ranger station sits 3{,}567 feet above sea level, below a peak that is 8{,}341 feet above sea level. How much higher is the peak than the station?
Nkosana’s printer has 7{,}352 postcards and 2{,}486 are mailed. Find how many are left, and list every trade in the record, place by place.

Part 3: estimating, checking, and word problems

Estimate 7{,}832 - 3{,}946 by rounding to the nearest thousand, then find it exactly.
Nkosana writes 6{,}253 - 3{,}478 = 2{,}785. Check by adding back. If it is wrong, find which column to redo and give the right difference.
Is 5{,}724 - 2{,}968 = 2{,}756? Decide by adding back.
One peak is 12{,}483 feet tall and another is 9{,}657 feet tall. Estimate how much taller the first is, then find the exact difference.
A printer made 6{,}521 postcards and 2{,}864 have been sold. How many are still unsold?
Mahina hikes from a trailhead at 2{,}947 feet to a lake at 5{,}163 feet, then on to a summit at 8{,}425 feet. How many feet does she climb on each leg? Check that the two legs add up to the whole climb.
Find the missing number: 4{,}386 + \square = 9{,}352.

Reasoning

Explain why the algorithm starts with the ones and moves left. Use 6{,}415 - 2{,}738 worked from the left to show what goes wrong.
Dariush says unbundling changes the top number, so the answer cannot be right. In 8{,}523 - 4{,}756 (Exercise 9), write what each place of the top number holds after all the trades, and show that it still adds up to 8{,}523.
Explain why, after one trade, a column always has enough on top to subtract the digit below it. Then explain why adding back is a fair check on any subtraction.

Answers

3{,}337. Ones 8 - 1 = 7, tens 6 - 3 = 3, hundreds 7 - 4 = 3, thousands 5 - 2 = 3; no trades.
3{,}223. Ones 6 - 3 = 3, tens 4 - 2 = 2, hundreds 9 - 7 = 2, thousands 9 - 6 = 3.
3{,}325. Ones 3 - 8 runs short: tens 7 \to 6, ones 13 - 8 = 5. Then tens 6 - 4 = 2, hundreds 5 - 2 = 3, thousands 4 - 1 = 3. Check: 3{,}325 + 1{,}248 = 4{,}573.
2{,}821. Ones 2 - 1 = 1, tens 9 - 7 = 2. Hundreds 1 - 3 runs short: thousands 8 \to 7, hundreds 11 - 3 = 8. Thousands 7 - 5 = 2.
42{,}227. Ones 4 - 7 runs short: tens 5 \to 4, ones 14 - 7 = 7. Then 4 - 2 = 2, 8 - 6 = 2, 3 - 1 = 2, 6 - 2 = 4.
3{,}234 feet taller. 7{,}386 - 4{,}152: 6 - 2 = 4, 8 - 5 = 3, 3 - 1 = 2, 7 - 4 = 3.
(a) No trade; 3{,}121. (b) The ones are fine; the tens need a trade (3 - 8 runs short, so a hundred is unbundled); 3{,}056. (c) Only the hundreds need a trade (4 - 5 runs short, so a thousand is unbundled); 2{,}910.
2{,}357. Ones: tens 3 \to 2, 11 - 4 = 7. Tens: hundreds 2 \to 1, 12 - 7 = 5. Hundreds: thousands 5 \to 4, 11 - 8 = 3. Thousands 4 - 2 = 2.
3{,}767. Ones: tens 2 \to 1, 13 - 6 = 7. Tens: hundreds 5 \to 4, 11 - 5 = 6. Hundreds: thousands 8 \to 7, 14 - 7 = 7. Thousands 7 - 4 = 3.
2{,}755. Ones: tens 4 \to 3, 12 - 7 = 5. Tens: hundreds 3 \to 2, 13 - 8 = 5. Hundreds: thousands 6 \to 5, 12 - 5 = 7. Thousands 5 - 3 = 2.
6{,}285. Ones: tens 5 \to 4, 13 - 8 = 5. Tens: hundreds 2 \to 1, 14 - 6 = 8. Hundreds: thousands 9 \to 8, 11 - 9 = 2. Thousands 8 - 2 = 6.
23{,}367. Ones: tens 2 \to 1, 16 - 9 = 7. Tens: hundreds 7 \to 6, 11 - 5 = 6. Hundreds 6 - 3 = 3. Thousands: ten thousands 4 \to 3, 11 - 8 = 3. Ten thousands 3 - 1 = 2.
44{,}785. Ones: tens 2 \to 1, 14 - 9 = 5. Tens: hundreds 6 \to 5, 11 - 3 = 8. Hundreds: thousands 2 \to 1, 15 - 8 = 7. Thousands: ten thousands 5 \to 4, 11 - 7 = 4. Ten thousands 4 - 0 = 4.
4{,}774 feet higher. 8{,}341 - 3{,}567: ones 11 - 7 = 4 (tens 4 \to 3), tens 13 - 6 = 7 (hundreds 3 \to 2), hundreds 12 - 5 = 7 (thousands 8 \to 7), thousands 7 - 3 = 4.
4{,}866 postcards left. Trades: ones 2 \to 12 with tens 5 \to 4; tens 4 \to 14 with hundreds 3 \to 2; hundreds 2 \to 12 with thousands 7 \to 6. The thousands need no trade: 6 - 2 = 4.
Estimate 8{,}000 - 4{,}000 = 4{,}000. Exact: 3{,}886. Ones: tens 3 \to 2, 12 - 6 = 6. Tens: hundreds 8 \to 7, 12 - 4 = 8. Hundreds: thousands 7 \to 6, 17 - 9 = 8. Thousands 6 - 3 = 3.
Wrong. 2{,}785 + 3{,}478 = 6{,}263, which is 10 too big, and the ones match, so redo the tens. After the ones trade the tens hold 4; 4 - 7 runs short, so hundreds 2 \to 1 and 14 - 7 = 7. The right difference is 2{,}775, and 2{,}775 + 3{,}478 = 6{,}253.
Yes. 2{,}756 + 2{,}968 = 5{,}724: ones 6 + 8 = 14, tens 5 + 6 + 1 = 12, hundreds 7 + 9 + 1 = 17, thousands 2 + 2 + 1 = 5.
Estimate 12{,}000 - 10{,}000 = 2{,}000 feet. Exact: 2{,}826 feet. Ones: tens 8 \to 7, 13 - 7 = 6. Tens 7 - 5 = 2. Hundreds: thousands 2 \to 1, 14 - 6 = 8. Thousands: ten thousands 1 \to 0, 11 - 9 = 2.
3{,}657 unsold. 6{,}521 - 2{,}864: ones 11 - 4 = 7 (tens 2 \to 1), tens 11 - 6 = 5 (hundreds 5 \to 4), hundreds 14 - 8 = 6 (thousands 6 \to 5), thousands 5 - 2 = 3. Check: 3{,}657 + 2{,}864 = 6{,}521.
First leg 5{,}163 - 2{,}947 = 2{,}216 feet. Second leg 8{,}425 - 5{,}163 = 3{,}262 feet. Whole climb 8{,}425 - 2{,}947 = 5{,}478 feet, and 2{,}216 + 3{,}262 = 5{,}478.
4{,}966. The missing number is 9{,}352 - 4{,}386, by the fact family. Ones: tens 5 \to 4, 12 - 6 = 6. Tens: hundreds 3 \to 2, 14 - 8 = 6. Hundreds: thousands 9 \to 8, 12 - 3 = 9. Thousands 8 - 4 = 4. Check: 4{,}386 + 4{,}966 = 9{,}352.
A trade always lowers the place to the left. From the left: thousands 6 - 2 = 4 is written first. Then the hundreds, 4 - 7, run short and must take a thousand, so the 4 already written has to become 3. The same happens again at the tens and the ones. Starting at the right settles each trade before the place it changes is written, so nothing is rewritten.
After the trades the top holds 7 thousands, 14 hundreds, 11 tens and 13 ones, and 7{,}000 + 1{,}400 + 110 + 13 = 8{,}523. A trade takes one unit from a place and puts ten of the next smaller unit in its place, which is the same amount, so the total never changes.
A top digit is 0 or more, so after a trade it is 10 or more, while the digit below is at most 9; ten or more minus at most nine is never short. Adding back is fair because the difference is, by its meaning, the amount that joins the smaller number to make the larger, so the difference plus the smaller number must give the larger number; this is the fact family of the three numbers.