Aimathic – Add and subtract multi-digit numbers

Add and subtract multi-digit numbers

Click problems to add them to your worksheet.

Difficulty: 1 – 5

5160375

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) $10{,}025$

5160385

Subtract $1250$ from each number. Choose the most efficient calculation method for each value. a) $2000$ b) $3250$ c) $5100$ d) $7450$ e) $9000$

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. Subtract $1250$ from $2000$: $2000 - 1250 = 750$. 2. Subtract $1250$ from $3250$: $3250 - 1250 = 2000$. 3. Subtract $1250$ from $5100$: $5100 - 1250 = 3850$. 4. Subtract $1250$ from $7450$: $7450 - 1250 = 6200$. 5. Subtract $1250$ from $9000$: $9000 - 1250 = 7750$.

Answer

a) $750$ b) $2000$ c) $3850$ d) $6200$ e) $7750$

5164845

Find each sum with numbers up to one million. Decide whether mental math or a written place-value strategy is more efficient. a) $450{,}000 + 350{,}000$ b) $280{,}000 + 170{,}000$ c) $620{,}000 + 290{,}000$ d) $54{,}000 + 360{,}000$

Hints

- Think of each number as a number of thousands. - How many thousands are there altogether? - Break the numbers into hundred-thousands and ten-thousands when helpful.

Solution

1. For a), use mental math. Add the thousands: $450 + 350 = 800$, so the sum is $800{,}000$. 2. For b), use mental math. Since $280 + 170 = 450$, the sum is $450{,}000$. 3. For c), use mental math. Since $620 + 290 = 910$, the sum is $910{,}000$. 4. For d), use mental math. Since $54 + 360 = 414$, the sum is $414{,}000$.

Answer

a) $800{,}000$ (mental math) b) $450{,}000$ (mental math) c) $910{,}000$ (mental math) d) $414{,}000$ (mental math)

5164855

Two large amusement parks compare attendance for the first six months of the year. Park A had $375{,}000$ visitors. Park B 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.

5164965

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) $90{,}000$

5164995

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) $825{,}000$

5170575

Use the standard addition algorithm for each sum. What pattern do you see in the results? 1. $111{,}111 + 222{,}222$ 2. $222{,}222 + 222{,}222$ 3. $333{,}333 + 222{,}222$ 4. $444{,}444 + 222{,}222$

Hints

- Compare corresponding digits in the addends. - Notice how the first addend changes while the second stays fixed. - Examine both the value and the digit pattern of each sum.

Solution

1. Align the addends by place value. In the first sum, every column gives $1 + 2 = 3$, so $111{,}111 + 222{,}222 = 333{,}333$. 2. In the second sum, every column gives $2 + 2 = 4$, so $222{,}222 + 222{,}222 = 444{,}444$. 3. In the third sum, every column gives $3 + 2 = 5$, so $333{,}333 + 222{,}222 = 555{,}555$. 4. In the fourth sum, every column gives $4 + 2 = 6$, so $444{,}444 + 222{,}222 = 666{,}666$. 5. Each result has six identical digits. From one problem to the next, the repeated digit increases by $1$ because the first addend increases by $111{,}111$ while the second addend stays fixed.

Answer

1. $333{,}333$ 2. $444{,}444$ 3. $555{,}555$ 4. $666{,}666$ Pattern: Each result has six equal digits, and that digit increases by $1$ each time.

5173545

Use the standard algorithm for each calculation. 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) $26{,}544$

5177375

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) $988$

5177405

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) $5279$

5177455

Use the standard algorithm for each calculation. 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) $276{,}544$

5177675

Use the given terms to write and calculate each expression. a) The first addend is $27{,}640$, and the second addend is $12{,}360$. b) The minuend is $45{,}012$, and the subtrahend is $6789$.

Hints

- Addends are the numbers being added. - In subtraction, the subtrahend is subtracted from the minuend. - Line up digits by place value before calculating.

Solution

1. Add the two addends: $27{,}640 + 12{,}360 = 40{,}000$. 2. Subtract the subtrahend from the minuend: $45{,}012 - 6789 = 38{,}223$.

Answer

a) $40{,}000$ b) $38{,}223$

5177685

Use the mathematical terms carefully to write and calculate each expression. a) The subtrahend is $1234$, and the minuend is $10{,}000$. b) The second addend is $8888$, and the first addend is $11{,}112$.

Hints

- The order in the sentence may differ from the order in the subtraction expression. - Use the structure: minuend minus subtrahend equals difference. - The order of addends does not change a sum.

Solution

1. Subtract the subtrahend from the minuend: $10{,}000 - 1234 = 8766$. 2. Add the two addends: $11{,}112 + 8888 = 20{,}000$.

Answer

a) $8766$ b) $20{,}000$

5177825

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) $11{,}850$

5177935

Find the difference when the minuend is $120{,}050$ and the subtrahend is $98{,}765$.

Hints

- Subtract the subtrahend from the minuend. - Line up digits by place value. - Regroup carefully across zeros.

Solution

1. Write the subtraction expression: $120{,}050 - 98{,}765$. 2. Subtract by place value, regrouping as needed: $120{,}050 - 98{,}765 = 21{,}285$.

Answer

The difference is $21{,}285$.

5177945

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 $53{,}244$.

5178055

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) $5103$

5178105

Use the standard addition algorithm to 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

$178{,}648$

5178135

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 $1099$.

5212245

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) $4547$

5213185

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) $999{,}500$

5217365

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) $>$

5217385

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) $1400$

5217485

Use standard algorithms. Align all digits by place 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) $125{,}010$

5217545

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) $9995$

5217555

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) $252$

5229575

Calculate each difference mentally. Break apart the subtrahend so the first subtraction lands on a multiple of $100$, as shown. $624 - 127 = 624 - 124 - 3 = 500 - 3 = 497$ a) $453 - 257$ b) $812 - 319$ c) $576 - 182$

Hints

- Find how much must be subtracted from the minuend to reach a multiple of $100$. - Split the subtrahend into that amount and a remainder. - Subtract the remainder 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) $196$ b) $493$ c) $394$

5160395

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)

5160785

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? - Compare how both numbers change from the first calculation to the second.

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), $751 - 349 = 751 - 350 + 1 = 402$. Also, $749 - 351 = 749 - 350 - 1 = 398$.

Answer

a) $165$ and $165$ b) $436$ and $436$ c) $402$ and $398$

5164865

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) $600{,}000$

5164975

For each subtraction problem, decide whether mental math, a place-value strategy, or the standard algorithm is most efficient. Then find the difference. a) $20{,}000 - 5$ b) $20{,}000 - 150$ c) $20{,}000 - 7432$

Hints

- Which problem requires regrouping across several place values? - Which difference can you see almost immediately? - Break the number being subtracted into convenient parts when helpful.

Solution

1. For a), mental math is efficient: $20{,}000 - 5 = 19{,}995$. 2. For b), use a place-value strategy: $20{,}000 - 100 = 19{,}900$, and $19{,}900 - 50 = 19{,}850$. 3. For c), use the standard algorithm. Regroup across the zeros so the place values become $1$ ten-thousand, $9$ thousands, $9$ hundreds, $9$ tens, and $10$ ones. Then $10 - 2 = 8$, $9 - 3 = 6$, $9 - 4 = 5$, $9 - 7 = 2$, and $1 - 0 = 1$. Therefore, $20{,}000 - 7432 = 12{,}568$.

Answer

a) $19{,}995$ (mental math) b) $19{,}850$ (place-value strategy) c) $12{,}568$ (standard algorithm)

5164985

What number was subtracted from $1{,}000{,}000$ in each equation? Find each missing subtrahend. 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 subtrahend by adding it to the stated result.

Solution

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

Answer

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

5165005

Find each difference. First decide whether mental math or a written calculation is more efficient. a) $1{,}000{,}000 - 545{,}000$ b) $503{,}000 - 204{,}000$

Hints

- For a), try counting up from the smaller number to one million. - Break the subtrahend into convenient place-value parts. - Compensation can help when one part is close to a friendly number.

Solution

1. For a), use mental math and count up in thousands from $545$ to $1000$: $545 + 5 = 550$, $550 + 50 = 600$, and $600 + 400 = 1000$. The total increase is $455$ thousand, so the difference is $455{,}000$. 2. For b), use a written place-value strategy: $503{,}000 - 200{,}000 = 303{,}000$, and $303{,}000 - 4000 = 299{,}000$.

Answer

a) $455{,}000$ (mental math) b) $299{,}000$ (written place-value strategy)

5165015

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

Hints

- Use the relationship among the minuend, subtrahend, and difference. - Find the distance between the first number and the result. - Rewrite the equation as an addition equation if that helps.

Solution

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

Answer

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

5165045

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) $<$

5165485

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></td> <td></td> <td></td> </tr> <tr> <td>$800{,}000$</td> <td></td> <td></td> <td></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>

5166685

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: $64{,}005$

5168655

For each sum, decide whether mental math or the standard addition algorithm is more efficient. Then find the sum. a) $12{,}500 + 7500$ b) $6105 + 1995$ c) $45{,}287 + 38{,}916$ d) $23{,}456 + 65{,}432$

Hints

- Look for place values that combine to make friendly thousands. - Is an addend close to a round number? - Use the standard algorithm when several place values require regrouping.

Solution

1. For a), use mental math and combine the thousands and hundreds: $12{,}500 + 7500 = 20{,}000$. 2. For b), use mental math with compensation: $6105 + 2000 - 5 = 8100$. 3. For c), use the standard addition algorithm. Add the ones: $7 + 6 = 13$, so write $3$ and regroup $1$ ten. Add the tens: $8 + 1 + 1 = 10$, so write $0$ and regroup $1$ hundred. Add the hundreds: $2 + 9 + 1 = 12$, so write $2$ and regroup $1$ thousand. Add the thousands: $5 + 8 + 1 = 14$, so write $4$ and regroup $1$ ten-thousand. Add the ten-thousands: $4 + 3 + 1 = 8$. Therefore, the sum is $84{,}203$. 4. For d), use mental math by place value because no regrouping is needed: $23{,}456 + 65{,}432 = 88{,}888$.

Answer

a) $20{,}000$ (mental math) b) $8100$ (mental math) c) $84{,}203$ (standard addition algorithm) d) $88{,}888$ (mental math)

5168665

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)

5168675

Some subtraction problems look difficult but can be solved quickly with an efficient mental strategy. a) Why is mental math more efficient than the standard algorithm for $34{,}005 - 29{,}998$? b) Use counting up or another efficient strategy to find $34{,}005 - 29{,}998$ and $82{,}003 - 79{,}995$.

Hints

- Count from the smaller number to the next multiple of $1000$. - Then count from that multiple of $1000$ to the larger number. - Think about how many regrouping steps the standard algorithm would require.

Solution

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

Answer

a) Counting up is faster because the numbers are close, while the standard algorithm would require regrouping across several zeros. b) $34{,}005 - 29{,}998 = 4007$ $82{,}003 - 79{,}995 = 2008$

5170585

Use the standard addition algorithm for each sum. Compare how the two addends change and explain what happens to the sums. a) $123{,}456 + 876{,}543$ b) $223{,}456 + 776{,}543$ c) $323{,}456 + 676{,}543$ d) $423{,}456 + 576{,}543$

Hints

- Calculate the sums before comparing them. - Find the change in the first addend from one line to the next. - Find the change in the second addend and compare its size and direction.

Solution

1. For a), align the digits. Each corresponding column sums to $9$, so $123{,}456 + 876{,}543 = 999{,}999$. 2. For b), each corresponding column again sums to $9$, so $223{,}456 + 776{,}543 = 999{,}999$. 3. For c), each corresponding column sums to $9$, so $323{,}456 + 676{,}543 = 999{,}999$. 4. For d), each corresponding column sums to $9$, so $423{,}456 + 576{,}543 = 999{,}999$. 5. From one problem to the next, the first addend increases by $100{,}000$, while the second addend decreases by $100{,}000$. These equal and opposite changes cancel, so the sum remains constant.

Answer

a) $999{,}999$ b) $999{,}999$ c) $999{,}999$ d) $999{,}999$ The first addend increases by $100{,}000$ while the second decreases by $100{,}000$, so every sum stays the same.

5177395

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 a subtraction expression, changing the minuend and subtrahend 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 the minuend and the subtrahend does not change the difference, 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. The left side is $1020 - 45 = 975$, and the right side is $950 + 30 = 980$. Therefore, $1020 - 45 < 950 + 30$.

Answer

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

5177425

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 $1281$.

5177465

Use the standard algorithm to determine which result 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: $220{,}011$.

5177695

Two calculations are described below. Calculation A: The addends are $13{,}400$ and $6600$. Calculation B: The minuend is $25{,}000$, and the subtrahend is $5050$. 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. - To find how much greater one result is, subtract the smaller result from the larger result.

Solution

1. Calculation A is $13{,}400 + 6600 = 20{,}000$. 2. Calculation B is $25{,}000 - 5050 = 19{,}950$. 3. Since $20{,}000 > 19{,}950$, Calculation A has the greater result. 4. The difference between the results is $20{,}000 - 19{,}950 = 50$.

Answer

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

5177845

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) $7$

5178065

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) $3$; $10{,}540$

5178075

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. $12{,}405$

5178125

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? Use standard addition and subtraction algorithms.

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.

5178145

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) $7129$

5178155

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) $9995$

5179085

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 $16{,}176$.

5185175

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. Use standard algorithms to 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.

5185195

Use standard algorithms to 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 $1$.

5212255

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

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) $<$

5217495

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

$419{,}844$

5217505

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 $474{,}611$.

5217565

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) $9003$

5217635

First estimate by rounding each number to the nearest hundred. Then find the exact value. $4285 - (1134 + 867)$

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: $2284$

5217645

First estimate by rounding each number to the nearest hundred. Then find the exact value. $(6721 - 3456) + (1289 - 544)$

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: $4010$

5217655

First estimate by rounding each number to the nearest hundred. Then find the exact value. $8123 - [4567 - (2134 - 899)]$

Hints

- Round every number to the nearest hundred for the estimate. - Work from the innermost grouping symbols outward. - Compare the exact result with the estimate.

Solution

1. Estimate: $8100 - [4600 - (2100 - 900)] = 8100 - (4600 - 1200) = 8100 - 3400 = 4700$. 2. Evaluate the innermost grouping symbols exactly: $2134 - 899 = 1235$. 3. Evaluate the brackets: $4567 - 1235 = 3332$. 4. Subtract: $8123 - 3332 = 4791$. 5. The exact value $4791$ is reasonably close to the estimate $4700$.

Answer

Estimate: $4700$ Exact value: $4791$

5229585

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. $366$

5179575

Find the greatest four-digit number with no repeated digits. From it, subtract the sum of the four least odd three-digit numbers.

Hints

- Use the greatest available digit in each place of the four-digit number. - Begin with the least three-digit number and list odd numbers in order. - Find the sum before performing the final subtraction.

Solution

1. Arrange the four greatest digits in decreasing order to get $9876$. 2. The four least odd three-digit numbers are $101$, $103$, $105$, and $107$. 3. Their sum is $101 + 103 + 105 + 107 = 416$. 4. Subtract: $9876 - 416 = 9460$.

Answer

$9460$

5185185

Fill in the missing digits in each standard algorithm. a) $\begin{array}{r} 4\square82 \\ + 15\square9 \\ \hline 6341 \end{array}$ b) $\begin{array}{r} 8043 \\ - 2768 \\ \hline \square27\square \end{array}$

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. For a), add from right to left. The tens column gives $8 + \square + 1 = 14$, so the missing tens digit is $5$. The hundreds column gives $\square + 5 + 1 = 13$, so the missing hundreds digit is $7$. The completed addition is $4782 + 1559 = 6341$. 2. For b), subtract with regrouping: $8043 - 2768 = 5275$. The missing thousands and ones digits are both $5$.

Answer

a) The missing digits are $7$ and $5$; $4782 + 1559 = 6341$. b) The missing digits are $5$ and $5$; $8043 - 2768 = 5275$.

All problems may be used, copied and printed free of charge for school and tutoring, including paid tutoring. Commercial adaptations as well as publication or redistribution on the internet are not permitted.

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