Add and subtract multi-digit numbers

Learning goals

Finds sums of multi-digit whole numbers.
Finds differences of multi-digit whole numbers.
Uses place value or number decompositions to add and subtract.
Checks results using inverse operations, estimation, or structure.

Difficulty: 1 – 5

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Add and subtract multi-digit numbers 1

Two large amusement parks compare attendance for the first six months of the year. Bluebird Adventure Park had $375{,}000$ visitors. Canyon Trail Park had $425{,}000$ visitors. How many visitors did the two parks have altogether?

Hints

- Which operation finds a total? - Think of the values as $375$ thousand and $425$ thousand. - Look for parts that combine to make a whole hundred thousand.

Solution

1. Add the two attendance numbers: $375{,}000 + 425{,}000$. 2. Think in thousands: $375 + 425 = 800$. 3. Therefore, the total attendance was $800{,}000$.

Answer

The two parks had $800{,}000$ visitors altogether.

Add and subtract multi-digit numbers 2

The vertical addition shows two addends, but the ones digit of the sum is hidden. Find the missing digit.

Hints

- Read the numbers from the vertical setup. - Start with the ones column.

Solution

1. Read the ones digits from the vertical setup: $4$ and $3$. 2. $4+3=7$, so the hidden ones digit is $7$.

Answer

$7$, because the ones column shows $4+3=7$.

Add and subtract multi-digit numbers 3

Find $2300+1400$ by adding thousands and hundreds separately. Explain your place-value steps.

Hints

- Break each addend into thousands and hundreds. - Add like place values. - Combine the place-value sums.

Solution

1. Add the thousands: $2000+1000=3000$. 2. Add the hundreds: $300+400=700$. 3. Combine them: $3000+700=3700$.

Answer

$3700$. The thousands make $3000$, and the hundreds make $700$.

Add and subtract multi-digit numbers 4

Find the sum and explain where the regrouping happens.

Hints

- Start in the ones column. - If the ones make $10$ or more, regroup one ten. - Then continue with the tens and hundreds.

Solution

1. Add the ones: $6+7=13$. Write $3$ ones and regroup $1$ ten. 2. Add the tens: $3+4+1=8$. 3. Add the hundreds: $1+2=3$. The sum is $383$.

Answer

$383$. The regrouping happens in the ones column because $6+7=13$.

Add and subtract multi-digit numbers 5

Add $1150$ to each number. Decide whether to work mentally or use a place-value strategy. a) $1250$ b) $3400$ c) $4750$ d) $6125$ e) $8875$

Hints

- Think about which place values change when you add. - It may help to add $1000$ first and then add the rest. - Watch for regrouping across a hundred or a thousand.

Solution

1. Add $1150$ to $1250$: $1250 + 1150 = 2400$. 2. Add $1150$ to $3400$: $3400 + 1150 = 4550$. 3. Add $1150$ to $4750$: $4750 + 1150 = 5900$. 4. Add $1150$ to $6125$: $6125 + 1150 = 7275$. 5. Add $1150$ to $8875$: $8875 + 1150 = 10{,}025$.

Answer

a) $2400$ b) $4550$ c) $5900$ d) $7275$ e) $10{,}025$

Add and subtract multi-digit numbers 6

Subtract $1250$ from each number. a) $2000$ b) $3250$ c) $5100$ d) $7450$ e) $9000$ For any two parts, also describe a mental strategy you could use. Different correct strategies are acceptable.

Hints

- Can you break $1250$ into parts that are easier to subtract? - Look closely at the place values. Some differences can be found mentally. - Use a written calculation when the subtraction is difficult to track mentally.

Solution

1. a) $2000-1250=750$. 2. b) $3250-1250=2000$. 3. c) $5100-1250=3850$. 4. d) $7450-1250=6200$. 5. e) $9000-1250=7750$. 6. One possible mental strategy is to subtract $1000$, then $250$. Another is to subtract $1200$, then $50$. Any correct described strategy for two parts is acceptable.

Answer

a) $750$ b) $2000$ c) $3850$ d) $6200$ e) $7750$ For two chosen parts, include a correct mental subtraction strategy; for example, subtract $1000$ and then $250$.

Add and subtract multi-digit numbers 7

Find each sum. a) $450{,}000+350{,}000$ b) $280{,}000+170{,}000$ c) $620{,}000+290{,}000$ d) $54{,}000+360{,}000$ For one part, explain a mental place-value strategy. For a different part, explain how you would align the digits in a written place-value calculation.

Hints

- Think of each number as a number of thousands. - How many thousands are there altogether? - In a written calculation, which digits must line up with each other?

Solution

1. a) $450{,}000+350{,}000=800{,}000$. 2. b) $280{,}000+170{,}000=450{,}000$. 3. c) $620{,}000+290{,}000=910{,}000$. 4. d) $54{,}000+360{,}000=414{,}000$. 5. For a), a mental strategy is to treat the numbers as thousands: $450+350=800$ thousand. For d), align digits with the same place value in the same column before adding and regrouping.

Answer

a) $800{,}000$. One mental strategy is $450+350=800$ thousand. b) $450{,}000$ c) $910{,}000$ d) $414{,}000$. In a written calculation, align digits with the same place value in the same column before adding.

Add and subtract multi-digit numbers 8

Find each missing number so that the equation is true. a) $640{,}000 + \square = 1{,}000{,}000$ b) $\square + 215{,}000 = 500{,}000$ c) $138{,}000 + 462{,}000 = \square$

Hints

- How much is needed to reach one million? - Subtract the known addend from the total to find a missing addend. - Think of the numbers as thousands.

Solution

1. For a), subtract the known addend from the total: $1{,}000{,}000 - 640{,}000 = 360{,}000$. 2. For b), $500{,}000 - 215{,}000 = 285{,}000$. 3. For c), $138 + 462 = 600$, so $138{,}000 + 462{,}000 = 600{,}000$.

Answer

a) $360{,}000$ b) $285{,}000$ c) $600{,}000$

Add and subtract multi-digit numbers 9

Find each difference mentally or with a place-value strategy. a) $100{,}000 - 1$ b) $100{,}000 - 10$ c) $100{,}000 - 100$ d) $100{,}000 - 1000$ e) $100{,}000 - 10{,}000$

Hints

- Notice which place value is being subtracted each time. - Track how regrouping changes the zeros. - Look for a pattern in the five differences.

Solution

1. Subtract $1$: $100{,}000 - 1 = 99{,}999$. 2. Subtract $10$: $100{,}000 - 10 = 99{,}990$. 3. Subtract $100$: $100{,}000 - 100 = 99{,}900$. 4. Subtract $1000$: $100{,}000 - 1000 = 99{,}000$. 5. Subtract $10{,}000$: $100{,}000 - 10{,}000 = 90{,}000$.

Answer

a) $99{,}999$ b) $99{,}990$ c) $99{,}900$ d) $99{,}000$ e) $90{,}000$

Add and subtract multi-digit numbers 10

Find each difference using an efficient mental or written place-value strategy. a) $640{,}000 - 270{,}000$ b) $910{,}000 - 85{,}000$

Hints

- Think of each number as a number of thousands. - Can you subtract to a nearby hundred thousand first? - Break the number being subtracted into convenient parts when helpful.

Solution

1. For a), subtract the thousands: $640 - 270 = 370$, so the difference is $370{,}000$. 2. For b), $910 - 85 = 825$, so the difference is $825{,}000$.

Answer

a) $370{,}000$ b) $825{,}000$

Add and subtract multi-digit numbers 11

The first sum is $123{,}456+876{,}543=999{,}999$. Without calculating the other three sums from scratch, determine each one and explain why the sums stay unchanged. a) $223{,}456+776{,}543$ b) $323{,}456+676{,}543$ c) $423{,}456+576{,}543$

Hints

- Compare how the two addends change from one line to the next. - Ask what happens to a sum when the same amount is added to one addend and subtracted from the other. - Use the given first sum rather than recomputing every column.

Solution

1. From the first sum to a), the first addend increases by $100{,}000$ while the second decreases by $100{,}000$. Equal opposite changes leave the total unchanged, so a) is $999{,}999$. 2. The same compensation occurs again in b) and c). 3. Therefore all three sums are $999{,}999$.

Answer

a) $999{,}999$ b) $999{,}999$ c) $999{,}999$ Each step adds $100{,}000$ to one addend and subtracts $100{,}000$ from the other, so the sum stays constant.

Add and subtract multi-digit numbers 12

Calculate each value. a) $15{,}673+8429+402$ b) $50{,}000-23{,}456$

Hints

- Line up digits with the same place value. - Record regrouping carefully in both operations. - In part b), regroup across several zeros before subtracting.

Solution

1. Line up the addends by place value and add, regrouping as needed: $15{,}673 + 8429 + 402 = 24{,}504$. 2. Line up the numbers by place value and subtract, regrouping across the zeros: $50{,}000 - 23{,}456 = 26{,}544$.

Answer

a) $24{,}504$ b) $26{,}544$

Add and subtract multi-digit numbers 13

Calculate mentally. Use an efficient strategy. a) $498 + 325$ b) $1230 - 990$ c) $75 + 180 + 25$ d) $1005 - 17$

Hints

- Can you adjust a number to make a multiple of $100$ or $1000$? - Do any addends combine to make a friendly number? - Can you break a difficult subtraction into smaller steps? - When you change one addend, how can you change the other addend to keep the sum the same?

Solution

1. For $498 + 325$, add $2$ to $498$ and subtract $2$ from $325$: $500 + 323 = 823$. 2. For $1230 - 990$, subtract $1000$, then add back $10$: $1230 - 1000 + 10 = 240$. 3. For $75 + 180 + 25$, combine $75$ and $25$: $100 + 180 = 280$. 4. For $1005 - 17$, subtract in parts: $1005 - 5 - 12 = 1000 - 12 = 988$.

Answer

a) $823$ b) $240$ c) $280$ d) $988$

Add and subtract multi-digit numbers 14

Find each sum or difference. a) $486 + 275$ b) $1205 - 318$ c) $8412 + 1588$ d) $10{,}000 - 4721$

Hints

- Line up digits with the same place value. - Regroup when a place does not have enough units to subtract. - Estimate first so you can check whether each answer is reasonable.

Solution

1. Add the ones, tens, and hundreds by place value: $486 + 275 = 761$. 2. Subtract by place value, regrouping as needed: $1205 - 318 = 887$. 3. Add the numbers: $8412 + 1588 = 10{,}000$. 4. Subtract the numbers: $10{,}000 - 4721 = 5279$.

Answer

a) $761$ b) $887$ c) $10{,}000$ d) $5279$

Add and subtract multi-digit numbers 15

Calculate each value. a) $56{,}782+9304+1557$ b) $400{,}000-123{,}456$

Hints

- Line up digits with the same place value. - Record each regrouping in the addition. - In the subtraction, regroup carefully across consecutive zeros.

Solution

1. Line up the addends by place value and add, regrouping as needed: $56{,}782 + 9304 + 1557 = 67{,}643$. 2. Subtract by place value, regrouping across the zeros: $400{,}000 - 123{,}456 = 276{,}544$.

Answer

a) $67{,}643$ b) $276{,}544$

Add and subtract multi-digit numbers 16

Write and calculate each expression. a) Add $27{,}640$ and $12{,}360$. b) Subtract $6789$ from $45{,}012$.

Hints

- Translate each sentence into an arithmetic expression. - Line up digits by place value before calculating. - Check the subtraction by adding the difference and $6789$.

Solution

1. $27{,}640 + 12{,}360 = 40{,}000$. 2. $45{,}012 - 6789 = 38{,}223$.

Answer

a) $27{,}640+12{,}360=40{,}000$ b) $45{,}012-6789=38{,}223$

Add and subtract multi-digit numbers 17

Write and calculate each expression. a) Subtract $1234$ from $10{,}000$. b) Add $11{,}112$ and $8888$.

Hints

- Pay attention to the order of the numbers in the subtraction words. - Line up equal place values before calculating. - The order of addends does not change a sum.

Solution

1. $10{,}000 - 1234 = 8766$. 2. $11{,}112 + 8888 = 20{,}000$.

Answer

a) $10{,}000-1234=8766$ b) $11{,}112+8888=20{,}000$

Add and subtract multi-digit numbers 18

Calculate mentally or by breaking the numbers into parts. Write only the result. a) $327 + 451$ b) $864 - 239$ c) $1540 + 2320$ d) $12{,}450 - 600$

Hints

- Break numbers apart by place value. - For subtraction, subtract the hundreds, tens, and ones in manageable steps. - Use friendly numbers to check whether your result is reasonable.

Solution

1. Break apart the addends: $300 + 400 = 700$, $20 + 50 = 70$, and $7 + 1 = 8$. Then $700 + 70 + 8 = 778$. 2. Subtract in parts: $864 - 200 = 664$, $664 - 30 = 634$, and $634 - 9 = 625$. 3. Break apart the addends: $1000 + 2000 = 3000$ and $540 + 320 = 860$. Then $3000 + 860 = 3860$. 4. Subtract $600$: $12{,}450 - 600 = 11{,}850$.

Answer

a) $778$ b) $625$ c) $3860$ d) $11{,}850$

Add and subtract multi-digit numbers 19

Find $120{,}050 - 98{,}765$.

Hints

- Line up digits by place value. - Regroup carefully across zeros. - Check by adding $98{,}765$ to your difference.

Solution

1. Line up equal place values. 2. Subtract with regrouping as needed: $120{,}050 - 98{,}765 = 21{,}285$.

Answer

$21{,}285$

Add and subtract multi-digit numbers 20

Find the sum of the three addends $45{,}231$, $7890$, and $123$.

Hints

- Addends are the numbers being added. - Line up digits with the same place value. - Record each regrouping before moving to the next place.

Solution

1. Write the addition expression: $45{,}231 + 7890 + 123$. 2. Add by place value: $45{,}231 + 7890 + 123 = 53{,}244$.

Answer

The sum is $53{,}244$.

Add and subtract multi-digit numbers 21

Calculate mentally by using a nearby multiple of $100$ or $1000$, then correcting the result. a) $456 + 199$ b) $1234 - 98$ c) $3780 + 995$ d) $5600 - 497$

Hints

- Replace a number close to $100$, $500$, or $1000$ with that friendly number. - If you subtract too much, add the extra amount back. - If you add too much, subtract the extra amount afterward.

Solution

1. Add $200$, then subtract $1$: $456 + 200 - 1 = 655$. 2. Subtract $100$, then add back $2$: $1234 - 100 + 2 = 1136$. 3. Add $1000$, then subtract $5$: $3780 + 1000 - 5 = 4775$. 4. Subtract $500$, then add back $3$: $5600 - 500 + 3 = 5103$.

Answer

a) $655$ b) $1136$ c) $4775$ d) $5103$

Add and subtract multi-digit numbers 22

Find the sum of $45{,}678$, $123{,}094$, and $9876$.

Hints

- Align the ones, tens, hundreds, and other places. - Include each regrouped amount in the next column. - Estimate the sum to check the final size.

Solution

1. Line up the three addends by place value. 2. Add from right to left, regrouping as needed. 3. The sum is $45{,}678 + 123{,}094 + 9876 = 178{,}648$.

Answer

$178{,}648$

Add and subtract multi-digit numbers 23

Calculate each sum mentally. State which two addends you combine first to make a multiple of $100$ or $1000$. a) $367 + 149 + 633$ b) $2450 + 871 + 550$ c) $125 + 789 + 875$ d) $444 + 99 + 556$

Hints

- Look at the ones and tens digits for pairs that make $100$ or $1000$. - You may change the order of addends. - Combine the most convenient pair before adding the remaining number.

Solution

1. For a), first add $367 + 633 = 1000$. Then $1000 + 149 = 1149$. 2. For b), first add $2450 + 550 = 3000$. Then $3000 + 871 = 3871$. 3. For c), first add $125 + 875 = 1000$. Then $1000 + 789 = 1789$. 4. For d), first add $444 + 556 = 1000$. Then $1000 + 99 = 1099$.

Answer

a) Combine $367$ and $633$ first; the sum is $1149$. b) Combine $2450$ and $550$ first; the sum is $3871$. c) Combine $125$ and $875$ first; the sum is $1789$. d) Combine $444$ and $556$ first; the sum is $1099$.

Add and subtract multi-digit numbers 24

Use a place-value strategy or the standard algorithm to find each sum or difference. a) $567 + 284$ b) $4321 - 567$ c) $2908 + 5192$ d) $7003 - 2456$

Hints

- Align digits by place value when using a written algorithm. - Regroup carefully in the addition and subtraction problems. - When subtracting from a number containing zeros, track each regrouping step. - Estimate first to check whether each result is reasonable.

Solution

1. For a), add the ones: $7 + 4 = 11$, so write $1$ and regroup $1$ ten. Add the tens: $6 + 8 + 1 = 15$, so write $5$ and regroup $1$ hundred. Add the hundreds: $5 + 2 + 1 = 8$. Therefore, $567 + 284 = 851$. 2. For b), regroup $1$ ten so $11 - 7 = 4$. Regroup $1$ hundred so $11 - 6 = 5$. Regroup $1$ thousand so $12 - 5 = 7$. Then $3 - 0 = 3$. Therefore, $4321 - 567 = 3754$. 3. For c), add the ones: $8 + 2 = 10$, so write $0$ and regroup $1$ ten. Add the tens: $0 + 9 + 1 = 10$, so write $0$ and regroup $1$ hundred. Add the hundreds: $9 + 1 + 1 = 11$, so write $1$ and regroup $1$ thousand. Add the thousands: $2 + 5 + 1 = 8$. Therefore, $2908 + 5192 = 8100$. 4. For d), regroup across the zeros so $7003$ becomes $6$ thousands, $9$ hundreds, $9$ tens, and $13$ ones. Then $13 - 6 = 7$, $9 - 5 = 4$, $9 - 4 = 5$, and $6 - 2 = 4$. Therefore, $7003 - 2456 = 4547$.

Answer

a) $851$ b) $3754$ c) $8100$ d) $4547$

Add and subtract multi-digit numbers 25

Find each sum or difference. a) $1{,}000{,}000 - 10$ b) $999{,}901 + 99$ c) $1{,}000{,}000 - 500$

Hints

- Think about counting backward from a full million. - For the addition problem, find the amount needed to reach the next thousand. - Use place-value steps to track changes across zeros.

Solution

1. For a), subtracting $10$ from one million gives $999{,}990$. 2. For b), $901 + 99 = 1000$, so $999{,}901 + 99 = 1{,}000{,}000$. 3. For c), subtract $500$ from the final thousand: $1{,}000{,}000 - 500 = 999{,}500$.

Answer

a) $999{,}990$ b) $1{,}000{,}000$ c) $999{,}500$

Add and subtract multi-digit numbers 26

Write $<$, $>$, or $=$ to make each statement true. Calculate mentally. a) $134 + 66 \quad \square \quad 250 - 45$ b) $800 - 125 \quad \square \quad 500 + 175$ c) $1250 + 750 \quad \square \quad 3000 - 900$ d) $1000 - 333 \quad \square \quad 444 + 222$

Hints

- Find the value of each expression before comparing. - Use place-value or adjustment strategies for mental calculation. - Check whether close results differ by only $1$.

Solution

1. $134 + 66 = 200$ and $250 - 45 = 205$, so $200 < 205$. 2. $800 - 125 = 675$ and $500 + 175 = 675$, so the values are equal. 3. $1250 + 750 = 2000$ and $3000 - 900 = 2100$, so $2000 < 2100$. 4. $1000 - 333 = 667$ and $444 + 222 = 666$, so $667 > 666$.

Answer

a) $<$ b) $=$ c) $<$ d) $>$

Add and subtract multi-digit numbers 27

Evaluate each expression mentally. a) $1240 + 560 - 300$ b) $5000 - (1200 + 800)$ c) $750 + 250 + 1300$ d) $2400 - 600 - 400$

Hints

- Look for numbers that combine to make a friendly value such as $100$ or $1000$. - Evaluate the quantity inside parentheses first. - Break each calculation into manageable steps.

Solution

1. a) $1240 + 560 = 1800$, and $1800 - 300 = 1500$. 2. b) Evaluate the parentheses first: $1200 + 800 = 2000$. Then $5000 - 2000 = 3000$. 3. c) Combine numbers that make a thousand: $750 + 250 = 1000$. Then $1000 + 1300 = 2300$. 4. d) Subtract from left to right: $2400 - 600 = 1800$, and $1800 - 400 = 1400$.

Answer

a) $1500$ b) $3000$ c) $2300$ d) $1400$

Add and subtract multi-digit numbers 28

Calculate each value. a) $67{,}892+154{,}309$ b) $302{,}100-45{,}678$ c) $12{,}450+8999+103{,}561$

Hints

- Align digits with the same place value. - Record regrouping in addition and subtraction. - For part c), all three addends can be aligned and added together.

Solution

1. Add by place value: $67{,}892 + 154{,}309 = 222{,}201$. 2. Subtract by place value, regrouping as needed: $302{,}100 - 45{,}678 = 256{,}422$. 3. Add all three numbers: $12{,}450 + 8999 + 103{,}561 = 125{,}010$.

Answer

a) $222{,}201$ b) $256{,}422$ c) $125{,}010$

Add and subtract multi-digit numbers 29

Calculate mentally using an efficient adjustment to a nearby multiple of $100$ or $1000$. a) $299 + 456$ b) $1002 - 75$ c) $4500 + 1999$ d) $10{,}000 - 5$

Hints

- Replace a number with a nearby friendly number. - Correct the result for the amount you added or subtracted. - For $1002 - 75$, think of subtracting $100$ and adding back the difference.

Solution

1. Replace $299$ with $300$, then subtract $1$: $300 + 456 - 1 = 755$. 2. Subtract $100$, then add back $25$: $1002 - 100 + 25 = 927$. 3. Add $2000$, then subtract $1$: $4500 + 2000 - 1 = 6499$. 4. Subtract directly across the place-value boundary: $10{,}000 - 5 = 9995$.

Answer

a) $755$ b) $927$ c) $6499$ d) $9995$

Add and subtract multi-digit numbers 30

Calculate mentally by breaking apart numbers or making a friendly number. a) $640 + 170$ b) $1250 - 350$ c) $880 + 121$ d) $500 - 248$

Hints

- Break numbers apart by hundreds, tens, and ones. - Look for parts that combine to make $100$ or $1000$. - For subtraction, use two manageable steps.

Solution

1. Add in parts: $640 + 100 + 70 = 810$. 2. Subtract in parts: $1250 - 250 - 100 = 900$. 3. Make $1000$: $880 + 120 + 1 = 1001$. 4. Subtract in parts: $500 - 200 - 48 = 252$.

Answer

a) $810$ b) $900$ c) $1001$ d) $252$

Add and subtract multi-digit numbers 31

Calculate each difference mentally. For each one, show how you split the number being subtracted so the first subtraction lands on a multiple of $100$, as in the example. $624-127=624-124-3=500-3=497$ a) $453-257$ b) $812-319$ c) $576-182$

Hints

- Find how much must be subtracted first to reach a multiple of $100$. - Split the number being subtracted into that amount and what remains. - Subtract the remaining part in the second step.

Solution

1. Break $257$ into $253 + 4$: $453 - 253 - 4 = 200 - 4 = 196$. 2. Break $319$ into $312 + 7$: $812 - 312 - 7 = 500 - 7 = 493$. 3. Break $182$ into $176 + 6$: $576 - 176 - 6 = 400 - 6 = 394$.

Answer

a) $453-253-4=200-4=196$ b) $812-312-7=500-7=493$ c) $576-176-6=400-6=394$

Add and subtract multi-digit numbers 32

The vertical addition has one hidden digit in the sum. Find the missing hundreds digit and explain how regrouping from the tens column affects the hundreds column.

Hints

- Start with the ones column and keep track of each regrouped amount. - The empty box is in the hundreds place. - Include the amount regrouped from the tens column when you determine that digit.

Solution

1. In the ones column, $6+9=15$, so write $5$ and regroup $1$ ten. 2. In the tens column, $8+4+1=13$, so write $3$ and regroup $1$ hundred. 3. In the hundreds column, $5+7+1=13$, so the missing hundreds digit is $3$ and $1$ thousand is regrouped. 4. The complete sum is $56{,}335$.

Answer

The missing digit is $3$. The tens column totals $8+4+1=13$, so it regroups $1$ hundred. Then the hundreds column is $5+7+1=13$, giving the hidden hundreds digit $3$.

Add and subtract multi-digit numbers 33

Find the difference. Then check it with addition.

Hints

- Start in the ones place and regroup carefully across the zeros. - Keep equal place values aligned. - Use addition to check the result.

Solution

1. Subtract by place value, regrouping across the zeros as needed: $5002-1786=3216$. 2. Check with addition: $3216+1786=5002$.

Answer

$3216$. Check: $3216+1786=5002$.

Add and subtract multi-digit numbers 34

Choose an efficient method for each calculation—mental math, a place-value strategy, or the standard algorithm. Find each result and briefly name the method you used. a) $6300 + 2700$ b) $9254 - 3876$ c) $4560 + 1999$ d) $7000 - 1250$

Hints

- Do any place values combine to make a new thousand? - Is one addend close to a multiple of $1000$? - When would lining up the place values be the most reliable method? - Can you break a subtraction into several easier steps?

Solution

1. For a), use mental math by combining thousands and hundreds: $6000 + 2000 = 8000$ and $300 + 700 = 1000$, so $6300 + 2700 = 9000$. 2. For b), use the standard subtraction algorithm with regrouping: $9254 - 3876 = 5378$. 3. For c), use compensation: $4560 + 2000 - 1 = 6559$. 4. For d), subtract in parts: $7000 - 1000 - 200 - 50 = 5750$.

Answer

a) $9000$ (mental math) b) $5378$ (standard algorithm) c) $6559$ (mental math with compensation) d) $5750$ (subtract in parts)

Add and subtract multi-digit numbers 35

Use the relationship between the two calculations in each pair to find both results efficiently. a) $463 - 298$ and $465 - 300$ b) $237 + 199$ and $236 + 200$ c) $751 - 349$ and $749 - 351$

Hints

- What happens to a difference when the same amount is added to both numbers? - What happens to a sum when an amount is moved from one addend to the other? - For part c), track how each number changes and combine both effects on the difference.

Solution

1. For a), adding $2$ to both numbers keeps the difference unchanged. Therefore, $463 - 298 = 465 - 300 = 165$. 2. For b), moving $1$ from one addend to the other keeps the sum unchanged. Therefore, $237 + 199 = 236 + 200 = 436$. 3. For c), $751 - 349 = 402$. In the second subtraction, the first number is $2$ smaller and the number being subtracted is $2$ larger, so the difference is $4$ smaller: $749 - 351 = 398$.

Answer

a) $165$ and $165$ b) $436$ and $436$ c) $402$ and $398$; the second difference is $4$ smaller.

Add and subtract multi-digit numbers 36

For each subtraction problem, choose mental math, a place-value strategy, or the standard algorithm as a reasonable method. Briefly explain your choice, then find the difference. a) $20{,}000 - 5$ b) $20{,}000 - 150$ c) $20{,}000 - 7432$

Hints

- Which problem can you evaluate almost immediately? - Which problem is convenient to break into hundreds and tens? - For the largest subtraction, consider whether a written layout would help you track regrouping.

Solution

1. For a), mental math is a reasonable choice because only $5$ is being subtracted: $20{,}000 - 5 = 19{,}995$. 2. For b), a place-value strategy is reasonable: $20{,}000 - 100 = 19{,}900$, and $19{,}900 - 50 = 19{,}850$. 3. For c), the standard algorithm is a reasonable choice. Regroup across the zeros and subtract by place value to get $20{,}000 - 7432 = 12{,}568$.

Answer

a) $19{,}995$ b) $19{,}850$ c) $12{,}568$ Method choices may vary if they are explained and used correctly.

Add and subtract multi-digit numbers 37

What number was subtracted from $1{,}000{,}000$ in each equation? a) $1{,}000{,}000 - \square = 999{,}999$ b) $1{,}000{,}000 - \square = 999{,}900$ c) $1{,}000{,}000 - \square = 990{,}000$ d) $1{,}000{,}000 - \square = 1$

Hints

- Use the inverse relationship: what must be added to the result to reach one million? - Compare the place values in the result with those in one million. - Check each missing number by adding it to the stated result.

Solution

1. Find each missing number by subtracting the stated result from $1{,}000{,}000$. 2. a) $1{,}000{,}000 - 999{,}999 = 1$. 3. b) $1{,}000{,}000 - 999{,}900 = 100$. 4. c) $1{,}000{,}000 - 990{,}000 = 10{,}000$. 5. d) $1{,}000{,}000 - 1 = 999{,}999$.

Answer

a) $1$ b) $100$ c) $10{,}000$ d) $999{,}999$

Add and subtract multi-digit numbers 38

Find each difference. For each one, choose either mental math or a written calculation and briefly explain why your choice is reasonable. a) $1{,}000{,}000 - 545{,}000$ b) $503{,}000 - 204{,}000$

Hints

- For a), consider counting up from the smaller number to one million. - For b), consider breaking the subtrahend into convenient place-value parts. - Check each exact difference by adding it back to the subtrahend.

Solution

1. For a), one reasonable mental strategy is to count up in thousands from $545$ to $1000$. The total increase is $455$ thousand, so the difference is $455{,}000$. 2. For b), one reasonable written place-value strategy is $503{,}000 - 200{,}000 = 303{,}000$, then $303{,}000 - 4000 = 299{,}000$.

Answer

a) $455{,}000$ b) $299{,}000$ Method choices may vary if they are explained and used correctly.

Add and subtract multi-digit numbers 39

Find the missing number in each subtraction equation. a) $820{,}000 - \square = 350{,}000$ b) $705{,}000 - \square = 698{,}000$

Hints

- Find the distance between the starting number and the result. - Rewrite the subtraction as an addition equation if that helps. - Check by subtracting your missing number from the first number.

Solution

1. For a), subtract the result from the starting number: $820{,}000 - 350{,}000 = 470{,}000$. 2. For b), $705{,}000 - 698{,}000 = 7000$.

Answer

a) $470{,}000$ b) $7000$

Add and subtract multi-digit numbers 40

Compare each pair without using the standard subtraction algorithm. Write $<$, $>$, or $=$. a) $10{,}000 - 10 \;\square\; 10{,}000 - 100$ b) $100{,}000 - 1000 \;\square\; 1{,}000{,}000 - 1000$ c) $100{,}000 - 100 \;\square\; 10{,}000 - 10$ d) $1000 - 1 \;\square\; 10{,}000 - 10$

Hints

- When the starting number is the same, compare how much is subtracted. - When the same amount is subtracted, compare the starting numbers. - Estimate the number of digits in each result.

Solution

1. For a), both expressions begin with $10{,}000$. Subtracting less gives the greater result, so $9990 > 9900$. 2. For b), the same amount is subtracted from two starting numbers, and $1{,}000{,}000$ is greater. Therefore, $99{,}000 < 999{,}000$. 3. For c), $100{,}000 - 100 = 99{,}900$, while $10{,}000 - 10 = 9990$. Therefore, the first value is greater. 4. For d), $1000 - 1 = 999$, while $10{,}000 - 10 = 9990$. Therefore, the first value is less.

Answer

a) $>$ b) $<$ c) $>$ d) $<$

Add and subtract multi-digit numbers 41

Complete the table by subtracting the number in each column heading from the number at the left. <table><tr><th>Number</th><th>$-1$</th><th>$-100$</th><th>$-10{,}000$</th></tr><tr><td>$325{,}400$</td><td>$\square$</td><td>$\square$</td><td>$\square$</td></tr><tr><td>$800{,}000$</td><td>$\square$</td><td>$\square$</td><td>$\square$</td></tr></table>

Hints

- Work one column at a time and identify the place value being subtracted. - Does every subtraction from $325{,}400$ require regrouping? - Why do several digits change when subtracting from $800{,}000$?

Solution

1. For $325{,}400$: $325{,}400 - 1 = 325{,}399$, $325{,}400 - 100 = 325{,}300$, and $325{,}400 - 10{,}000 = 315{,}400$. 2. For $800{,}000$: $800{,}000 - 1 = 799{,}999$, $800{,}000 - 100 = 799{,}900$, and $800{,}000 - 10{,}000 = 790{,}000$.

Answer

<table><tr><th>Number</th><th>$-1$</th><th>$-100$</th><th>$-10{,}000$</th></tr><tr><td>$325{,}400$</td><td>$325{,}399$</td><td>$325{,}300$</td><td>$315{,}400$</td></tr><tr><td>$800{,}000$</td><td>$799{,}999$</td><td>$799{,}900$</td><td>$790{,}000$</td></tr></table>

Add and subtract multi-digit numbers 42

For each starting number, add $8372$, then add $1628$. Record the intermediate result after the first addition and the final result. a) Starting number: $15{,}450$ b) Starting number: $27{,}891$ c) Starting number: $54{,}005$

Hints

- Perform the two additions in the stated order. - Check whether the two amounts being added combine to make a friendly number. - Use their combined value to check each final result.

Solution

1. Notice that $8372 + 1628 = 10{,}000$. 2. For a), $15{,}450 + 8372 = 23{,}822$, then $23{,}822 + 1628 = 25{,}450$. 3. For b), $27{,}891 + 8372 = 36{,}263$, then $36{,}263 + 1628 = 37{,}891$. 4. For c), $54{,}005 + 8372 = 62{,}377$, then $62{,}377 + 1628 = 64{,}005$.

Answer

a) Intermediate result: $23{,}822$; final result: $25{,}450$ b) Intermediate result: $36{,}263$; final result: $37{,}891$ c) Intermediate result: $62{,}377$; final result: $64{,}005$

Add and subtract multi-digit numbers 43

For each sum, choose mental math or the standard addition algorithm as a reasonable method. Briefly explain your choice, then find the sum. a) $12{,}500 + 7500$ b) $6105 + 1995$ c) $45{,}287 + 38{,}916$ d) $23{,}456 + 65{,}432$

Hints

- Look for addends that combine to make friendly thousands. - Is one addend close to a round number? - When several place values regroup, a written algorithm may help you organize the work.

Solution

1. For a), mental math is a reasonable choice: $12{,}500 + 7500 = 20{,}000$. 2. For b), mental compensation is convenient: $6105 + 2000 - 5 = 8100$. 3. For c), the standard algorithm is a reasonable choice because several place values regroup. The sum is $84{,}203$. 4. For d), adding by place value is convenient because no regrouping is needed: $23{,}456 + 65{,}432 = 88{,}888$.

Answer

a) $20{,}000$ b) $8100$ c) $84{,}203$ d) $88{,}888$ Method choices may vary if they are explained and used correctly.

Add and subtract multi-digit numbers 44

Choose an efficient strategy for each subtraction problem, and then find the difference. a) $5001 - 4998$ b) $10{,}000 - 2500$ c) $63{,}214 - 45{,}837$ d) $87{,}654 - 23{,}412$

Hints

- When two numbers are close, count up to find their difference. - Use large place-value steps with round numbers. - A written algorithm is useful when regrouping is needed in several places.

Solution

1. For a), use mental math and count up because the numbers are close: the difference is $3$. 2. For b), use a place-value strategy: $10{,}000 - 2500 = 7500$. 3. For c), use the standard subtraction algorithm. Regroup $1$ ten so $14 - 7 = 7$. Regroup $1$ hundred so $10 - 3 = 7$. Regroup $1$ thousand so $11 - 8 = 3$. Regroup $1$ ten-thousand so $12 - 5 = 7$. Finally, $5 - 4 = 1$. Therefore, $63{,}214 - 45{,}837 = 17{,}377$. 4. For d), use a place-value strategy without regrouping: $87{,}654 - 23{,}412 = 64{,}242$.

Answer

a) $3$ (mental math) b) $7500$ (place-value strategy) c) $17{,}377$ (standard algorithm) d) $64{,}242$ (place-value strategy)

Add and subtract multi-digit numbers 45

The two differences below involve numbers that are close together. a) Explain why counting up can be a useful strategy for $34{,}005 - 29{,}998$. b) Use counting up or another valid strategy to find $34{,}005 - 29{,}998$ and $82{,}003 - 79{,}995$.

Hints

- Count from the smaller number to the next convenient multiple of $1000$. - Continue from that multiple to the larger number. - Add the two increases to get the difference.

Solution

1. For a), counting up is useful because the numbers are close and the standard algorithm would require regrouping across several zeros. 2. From $29{,}998$ to $30{,}000$ is $2$, and from $30{,}000$ to $34{,}005$ is $4005$. The difference is $4007$. 3. From $79{,}995$ to $80{,}000$ is $5$, and from $80{,}000$ to $82{,}003$ is $2003$. The difference is $2008$.

Answer

a) Counting up is useful because the numbers are close and avoids lengthy regrouping. b) $34{,}005 - 29{,}998 = 4007$ $82{,}003 - 79{,}995 = 2008$

Add and subtract multi-digit numbers 46

In the vertical addition, one digit of the sum is hidden by an empty box. Find the missing digit. Then explain what happens in the hundreds column, including the regrouping into the thousands column.

Hints

- Work from the ones column toward the left. - Keep track of each regrouped amount before moving to the next column. - The empty box is in the hundreds place, so explain the column total that determines it.

Solution

1. Add the ones: $4+5+7=16$. Write $6$ and regroup $1$ ten. 2. Add the tens: $6+3+9+1=19$. Write $9$ and regroup $1$ hundred. 3. Add the hundreds: $7+8+4+1=20$. Therefore the hidden hundreds digit is $0$, and $2$ thousands are regrouped into the next column. 4. Continuing gives the complete sum $45{,}096$.

Answer

The missing digit is $0$. In the hundreds column the total is $20$ hundreds, so $0$ is written in the hundreds place and $2$ thousands are regrouped.

Add and subtract multi-digit numbers 47

Compare each pair of numerical expressions without using the standard algorithm. Write $<$, $>$, or $=$. a) $380 + 420 \quad \square \quad 250 + 550$ b) $1200 - 450 \quad \square \quad 1300 - 550$ c) $99 + 99 + 99 \quad \square \quad 300 - 3$ d) $1020 - 45 \quad \square \quad 950 + 30$

Hints

- Look for ways to rewrite one expression so it resembles the other. - In subtraction, changing both numbers by the same amount keeps the difference unchanged. - Think of $99$ as $100 - 1$.

Solution

1. Both sums equal $800$, so $380 + 420 = 250 + 550$. 2. Adding $100$ to both numbers in a subtraction keeps the difference unchanged, so $1200 - 450 = 1300 - 550$. 3. The left side is $3 \times 100 - 3 = 297$, and the right side is $300 - 3 = 297$, so the expressions are equal. 4. $1020 - 45 = 975$, while $950 + 30 = 980$. Therefore, $1020 - 45 < 950 + 30$.

Answer

a) $=$ b) $=$ c) $=$ d) $<$

Add and subtract multi-digit numbers 48

Start with $520$ and complete each step in order. 1. Add $180$. 2. Subtract $345$. 3. Add $1025$. 4. Subtract $99$. What is the final result?

Hints

- Complete the chain in order and record each intermediate result. - For the last step, subtract $100$ and then add back $1$. - Check each step because an early error affects every later result.

Solution

1. First step: $520 + 180 = 700$. 2. Second step: $700 - 345 = 355$. 3. Third step: $355 + 1025 = 1380$. 4. Fourth step: $1380 - 99 = 1281$.

Answer

The final result is $1281$.

Add and subtract multi-digit numbers 49

Calculate both results and determine which is greater. Calculation A: $135{,}246+84{,}754$ Calculation B: $310{,}000-89{,}989$

Hints

- Calculate both results separately. - Compare the results from the greatest place value to the least.

Solution

1. Calculation A is $135{,}246 + 84{,}754 = 220{,}000$. 2. Calculation B is $310{,}000 - 89{,}989 = 220{,}011$. 3. Since $220{,}011 > 220{,}000$, Calculation B has the greater result.

Answer

Calculation B has the greater result: $220{,}011$.

Add and subtract multi-digit numbers 50

Two calculations are described below. Calculation A: Add $13{,}400$ and $6600$. Calculation B: Subtract $5050$ from $25{,}000$. Which calculation has the greater result? How much greater is it?

Hints

- Translate each description into an addition or subtraction expression. - Calculate both results before comparing them. - Subtract the smaller result from the larger result to find how much greater it is.

Solution

1. Calculation A is $13{,}400 + 6600 = 20{,}000$. 2. Calculation B is $25{,}000 - 5050 = 19{,}950$. 3. Calculation A has the greater result. 4. $20{,}000 - 19{,}950 = 50$, so it is $50$ greater.

Answer

Calculation A has the greater result. It is $50$ greater than Calculation B.

Add and subtract multi-digit numbers 51

Calculate mentally by adjusting to a nearby friendly number or by counting up. a) $754 + 199$ b) $1320 - 298$ c) $456 + 457$ d) $2005 - 1998$

Hints

- Is one number close to a multiple of $100$ or $1000$? - Can you add or subtract a friendly number and then correct the result? - For two numbers that differ by $1$, try using a near-double. - For a small difference, count up from the smaller number.

Solution

1. Add $200$, then subtract $1$: $754 + 200 - 1 = 953$. 2. Subtract $300$, then add back $2$: $1320 - 300 + 2 = 1022$. 3. Use a near-double: $456 + 457 = 456 + 456 + 1 = 913$. 4. Count up from $1998$ to $2005$: $2$ to reach $2000$, then $5$ more. The difference is $2 + 5 = 7$.

Answer

a) $953$ b) $1022$ c) $913$ d) $7$

Add and subtract multi-digit numbers 52

Fill in the missing numbers to show each mental-math adjustment. a) $875 + 298 = 875 + 300 - \square = \square$ b) $1432 - 199 = 1432 - 200 + \square = \square$ c) $9997 + 543 = 10{,}000 + 543 - \square = \square$

Hints

- Find the difference between the original number and the friendly number used in the rewritten expression. - If you subtract too much, add the extra amount back. - If you replace an addend with a larger number, subtract the amount of the increase.

Solution

1. Since $298$ is $2$ less than $300$, subtract $2$: $875 + 300 - 2 = 1173$. 2. Subtracting $200$ removes $1$ too much, so add back $1$: $1432 - 200 + 1 = 1233$. 3. Since $9997 = 10{,}000 - 3$, subtract $3$: $10{,}000 + 543 - 3 = 10{,}540$.

Answer

a) $2$; $1173$ b) $1$; $1233$ c) $3$; $10{,}540$

Add and subtract multi-digit numbers 53

Kira wants to calculate $15{,}400 - 2995$ mentally. She says, “I will subtract $3000$ first because that is easier.” 1. Explain what Kira must do next to get the correct result. 2. Use Kira’s method to calculate the final result.

Hints

- Compare $2995$ with $3000$. - Did Kira subtract more or less than the original amount? - Decide how to correct the intermediate result.

Solution

1. Kira subtracts $3000$ instead of $2995$, so she subtracts $5$ too much. She must add $5$ back. 2. Calculate $15{,}400 - 3000 = 12{,}400$, then $12{,}400 + 5 = 12{,}405$.

Answer

1. Kira must add $5$ back because she subtracted $5$ too much. 2. $12{,}405$

Add and subtract multi-digit numbers 54

A fifth-grade class has a field-trip budget of $\$2500$. The class must pay these costs: - Bus: $\$845$ - Admission: $\$1120$ - Food: $\$378$ How much money remains after all costs are paid?

Hints

- Find the total cost first. - Subtract the total cost from the starting budget. - Line up digits by place value in both calculations.

Solution

1. Add the expenses: $\$845 + \$1120 + \$378 = \$2343$. 2. Subtract the total expenses from the budget: $\$2500 - \$2343 = \$157$.

Answer

$\$157$ remains.

Add and subtract multi-digit numbers 55

Use an efficient mental strategy for each calculation. Briefly describe your strategy. a) $5621 - 998$ b) $14{,}350 - 4050$ c) $2845 + 199$ d) $7231 - 102$

Hints

- Replace a number with a nearby friendly number and then adjust. - Break a number into parts that are easier to add or subtract mentally.

Solution

1. Subtract $1000$, then add back $2$: $5621 - 1000 + 2 = 4623$. 2. Subtract in parts: $14{,}350 - 4000 - 50 = 10{,}300$. 3. Add $200$, then subtract $1$: $2845 + 200 - 1 = 3044$. 4. Subtract $100$, then subtract $2$: $7231 - 100 - 2 = 7129$.

Answer

a) $4623$ b) $10{,}300$ c) $3044$ d) $7129$

Add and subtract multi-digit numbers 56

For each problem, choose a reasonable method: “mental math” or “standard algorithm.” Then calculate every result. a) $12{,}450 + 7550$ b) $8342 - 2765$ c) $9999 + 1$ d) $5678 + 321$ e) $10{,}005 - 10$

Hints

- Use mental math when the numbers form friendly combinations or require only a small adjustment. - Use the standard algorithm when several places require regrouping. - More than one method choice can be reasonable if you can justify it.

Solution

1. One reasonable choice is mental math for a), c), d), and e), and the standard algorithm for b). 2. For a), combine to make $20{,}000$: $12{,}450 + 7550 = 20{,}000$. 3. For b), subtract using the standard algorithm: $8342 - 2765 = 5577$. 4. For c), add $1$: $9999 + 1 = 10{,}000$. 5. For d), add in parts: $5678 + 300 + 21 = 5999$. 6. For e), subtract $10$: $10{,}005 - 10 = 9995$.

Answer

One reasonable method choice is: a), c), d), and e) mental math; b) standard algorithm. a) $20{,}000$ b) $5577$ c) $10{,}000$ d) $5999$ e) $9995$

Add and subtract multi-digit numbers 57

The first addend is $8705$. It is $1234$ greater than the second addend. Find the sum of the two numbers.

Hints

- Identify which addend is greater. - Find the second addend before finding the sum. - Check that the two addends differ by $1234$.

Solution

1. Find the second addend: $8705 - 1234 = 7471$. 2. Add the two numbers: $8705 + 7471 = 16{,}176$.

Answer

The sum is $16{,}176$.

Add and subtract multi-digit numbers 58

A library counted $38{,}764$ visits in spring and $27{,}895$ visits in summer. A later audit found $18{,}974$ duplicate scanner entries in the combined count. Use the two vertical layouts. a) Find the combined count before the correction. b) Subtract the duplicate entries to find the corrected count. c) In part b), explain the regrouping that determines the hidden hundreds digit.

Hints

- Complete the addition before using the combined count in the subtraction. - In part b, track what the tens-column regrouping does to the hundreds column. - The empty box is in the hundreds place, so explain the regrouping immediately before that subtraction.

Solution

1. Add the spring and summer counts: $38{,}764+27{,}895=66{,}659$. 2. Subtract the duplicate entries: $66{,}659-18{,}974=47{,}685$. 3. In the tens column, $5$ tens cannot subtract $7$ tens, so regroup $1$ hundred. That leaves $5$ hundreds and gives $15$ tens. 4. In the hundreds column, $5$ hundreds cannot subtract $9$ hundreds, so regroup $1$ thousand. Then $15-9=6$, so the hidden hundreds digit is $6$.

Answer

a) $66{,}659$ b) $47{,}685$ c) After the tens-column regrouping leaves $5$ hundreds, another regrouping makes $15$ hundreds; $15-9=6$, so the hidden hundreds digit is $6$.

Add and subtract multi-digit numbers 59

A bookstore begins the year with $15{,}480$ books. During the first six months, it receives $3765$ new books and removes $2894$ older books for donation. Find the new inventory.

Hints

- Decide which action increases the inventory and which decreases it. - Complete the calculations in chronological order. - Line up digits by place value.

Solution

1. Add the new books: $15{,}480 + 3765 = 19{,}245$. 2. Subtract the donated books: $19{,}245 - 2894 = 16{,}351$.

Answer

The bookstore now has $16{,}351$ books.

Add and subtract multi-digit numbers 60

Fill in the missing digits in each standard algorithm. a) Explain how one column tells you each missing digit. b) Explain the regrouping you need before finding the missing digits.

Hints

- Begin in the ones place and work left. - Include regrouped values in the next column. - For subtraction, track each regrouping across the zeros.

Solution

1. In a), the tens column gives $8+\square+1=14$, so the missing tens digit is $5$. The hundreds column then gives $\square+5+1=13$, so the missing hundreds digit is $7$. Thus $4782+1559=6341$. 2. In b), regroup across the zero in $8043$ before subtracting. The result is $5275$, so the missing thousands and ones digits are both $5$.

Answer

a) Missing digits: $7$ and $5$. The tens and hundreds columns determine them after including the carried $1$. b) Missing digits: $5$ and $5$. Regroup across the zero before subtracting.

Add and subtract multi-digit numbers 61

Evaluate both expressions. Find the difference between the results and identify the greater result. Expression A: $34{,}509+7812+655$ Expression B: $50{,}000-7025$

Hints

- Calculate each result separately. - Regroup carefully when subtracting from $50{,}000$. - Subtract the smaller result from the larger result.

Solution

1. Expression A is $34{,}509 + 7812 + 655 = 42{,}976$. 2. Expression B is $50{,}000 - 7025 = 42{,}975$. 3. The difference is $42{,}976 - 42{,}975 = 1$, so Expression A is greater by $1$.

Answer

Expression A equals $42{,}976$, and Expression B equals $42{,}975$. Expression A is greater by $1$.

Add and subtract multi-digit numbers 62

Compare the values of the expressions. Write $<$, $>$, or $=$ in each box. a) $1450 + 550 \quad \square \quad 3000 - 1050$ b) $6700 - 2300 \quad \square \quad 3100 + 1300$ c) $8240 - 1360 \quad \square \quad 5420 + 1470$

Hints

- Evaluate the expression on each side separately. - Compare the two resulting numbers. - Check regrouping carefully in the subtraction expressions.

Solution

1. For a), $1450 + 550 = 2000$ and $3000 - 1050 = 1950$. Therefore, $2000 > 1950$. 2. For b), $6700 - 2300 = 4400$ and $3100 + 1300 = 4400$. Therefore, the values are equal. 3. For c), $8240 - 1360 = 6880$ and $5420 + 1470 = 6890$. Therefore, $6880 < 6890$.

Answer

a) $>$ b) $=$ c) $<$

Add and subtract multi-digit numbers 63

First find the sum of $456{,}700$ and $123{,}456$. Then subtract that sum from $1{,}000{,}000$.

Hints

- Complete the operations in the stated order. - Regroup carefully across the zeros in $1{,}000{,}000$. - Estimate to check that the final result is close to $420{,}000$.

Solution

1. Add: $456{,}700 + 123{,}456 = 580{,}156$. 2. Subtract the sum from $1{,}000{,}000$: $1{,}000{,}000 - 580{,}156 = 419{,}844$.

Answer

$419{,}844$

Add and subtract multi-digit numbers 64

Compare the results of Calculations A and B. Write $<$, $>$, or $=$. A: $345{,}678 + 128{,}933$ B: $600{,}000 - 125{,}389$

Hints

- Calculate both expressions before comparing. - Regroup carefully in the subtraction. - Compare the exact results digit by digit.

Solution

1. Calculation A is $345{,}678 + 128{,}933 = 474{,}611$. 2. Calculation B is $600{,}000 - 125{,}389 = 474{,}611$. 3. The results are equal, so $A = B$.

Answer

$A = B$. Both results are $474{,}611$.

Add and subtract multi-digit numbers 65

Calculate mentally. Pay attention to place value and crossing a place-value boundary. a) $123{,}000 + 77{,}000$ b) $45{,}000 - 105$ c) $99{,}998 + 7$ d) $10{,}002 - 999$

Hints

- Think in groups of thousands for part a). - Break small subtrahends into manageable parts. - Notice how close $99{,}998$ is to $100{,}000$. - Replace $999$ with $1000 - 1$.

Solution

1. Add the thousands: $123{,}000 + 77{,}000 = 200{,}000$. 2. Subtract in parts: $45{,}000 - 100 - 5 = 44{,}895$. 3. Add $2$ to reach $100{,}000$, then add the remaining $5$: $99{,}998 + 7 = 100{,}005$. 4. Subtract $1000$, then add back $1$: $10{,}002 - 1000 + 1 = 9003$.

Answer

a) $200{,}000$ b) $44{,}895$ c) $100{,}005$ d) $9003$

Add and subtract multi-digit numbers 66

Round each number to the nearest hundred to give an estimate. Then find the exact value and compare it with the estimate. $4285-(1134+867)$

Hints

- Round each number to the nearest hundred for the estimate. - Evaluate the grouping symbols before subtracting. - Compare the exact value with the estimate.

Solution

1. Estimate: $4300 - (1100 + 900) = 4300 - 2000 = 2300$. 2. Find the exact sum in the grouping symbols: $1134 + 867 = 2001$. 3. Subtract: $4285 - 2001 = 2284$. 4. The exact value $2284$ is close to the estimate $2300$.

Answer

Estimate: $2300$ Exact value: $2284$

Add and subtract multi-digit numbers 67

Round each number to the nearest hundred to give an estimate. Then find the exact value and compare it with the estimate. $(6721-3456)+(1289-544)$

Hints

- Round each number to the nearest hundred before estimating. - Evaluate the two grouped differences separately. - Compare the exact value with the estimate.

Solution

1. Estimate: $(6700 - 3500) + (1300 - 500) = 3200 + 800 = 4000$. 2. Evaluate the first grouped difference: $6721 - 3456 = 3265$. 3. Evaluate the second grouped difference: $1289 - 544 = 745$. 4. Add: $3265 + 745 = 4010$. 5. The exact value $4010$ is close to the estimate $4000$.

Answer

Estimate: $4000$ Exact value: $4010$

Add and subtract multi-digit numbers 68

Lukas and Mia calculate $745 - 298$ mentally using different strategies. Lukas: $745 - 245 - 53$ Mia: $745 - 300 + 2$ 1. Calculate the result using each strategy. Do both strategies give the same result? 2. Which strategy is easier for this problem? Briefly explain. 3. Use Mia’s strategy to calculate $563 - 197$.

Hints

- Complete each strategy one step at a time. - Compare the number and difficulty of the intermediate calculations. - When you subtract a number that is too large, add the extra amount back.

Solution

1. Lukas’s strategy gives $745 - 245 = 500$, then $500 - 53 = 447$. Mia’s strategy gives $745 - 300 = 445$, then $445 + 2 = 447$. Both strategies give the same result. 2. One reasonable response is that Mia’s strategy is easier because $298$ is close to $300$, and correcting by $2$ is simple. 3. Subtract $200$, then add back $3$: $563 - 200 + 3 = 366$.

Answer

1. Yes. Both strategies give $447$. 2. Answers will vary. A valid explanation must compare the mental steps. 3. $366$

Add and subtract multi-digit numbers 69

Find the missing hundreds digit and explain how regrouping across the zeros in $60{,}002$ leads to it.

Hints

- Work from the ones place toward the left. - Because several middle digits of the top number are zero, track the regrouping chain carefully. - The empty box is in the hundreds place of the result.

Solution

1. Regroup across the zeros so the ones and tens columns can be subtracted. 2. After regrouping, the hundreds column has $9$ hundreds in the top number. 3. Since $9-7=2$, the missing hundreds digit is $2$, and the complete difference is $51{,}237$.

Answer

The missing digit is $2$. Regrouping across the zeros leaves $9$ hundreds in the top number, and $9-7=2$.

Add and subtract multi-digit numbers 70

One digit is missing in the vertical addition. Find the missing digit and explain how you know.

Hints

- You can reason column by column or use the inverse operation. - If you use subtraction, subtract the complete known addend from the sum. - Check that the completed addition gives the displayed total.

Solution

1. The addition is $5\square32 + 2487 = 8019$. 2. Subtract the known addend from the sum: $8019 - 2487 = 5532$. 3. Therefore, the missing hundreds digit is $5$.

Answer

The missing digit is $5$. The tens column gives $3+8=11$, so $1$ hundred is regrouped. The hundreds column must then satisfy $5+4+1=10$, producing the shown $0$ and regrouping $1$ thousand.

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Grade 5 math worksheets

Add and subtract multi-digit numbers

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