Chapter 2 — worked answers and marking guidance
All chapter review and practice examination items
Chapter review solutions
Calculate 453,806 + 144,541.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Calculate 526,628 − 148,186.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 378,442 + 148,186 = 526,628.
Use compensation to add 273,644 and 76,432. What is the exact sum?
Increase 273,644 by 356 to a friendly multiple of 1,000, then subtract 356 from the sum to preserve the total.
Calculate 555,323 − 147,283 by adding 100 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (555,423) − (147,383) = 408,040.
A library had 150,403 books, received 161,677 more, then lent 80,838. How many books are there now?
First add 312,080. Then subtract 80,838 to obtain 231,242.
Calculate 645,936 + 111,442.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Calculate 622,758 − 146,027.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 476,731 + 146,027 = 622,758.
Use compensation to add 248,577 and 79,607. What is the exact sum?
Increase 248,577 by 23 to a friendly multiple of 100, then subtract 23 from the sum to preserve the total.
Calculate 206,180 − 89,136 by adding 100 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (206,280) − (89,236) = 117,044.
Alex and Bea have 651,246 points together. Bea has 1,988 more than Alex. How many does Alex have?
Remove the extra 1,988 from the whole and split the rest into two equal parts: (651,246 − 1,988) ÷ 2 = 324,629.
Calculate 203,785 + 145,068.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Calculate 294,944 − 56,837.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 238,107 + 56,837 = 294,944.
Use compensation to add 610,167 and 149,382. What is the exact sum?
Increase 610,167 by 833 to a friendly multiple of 1,000, then subtract 833 from the sum to preserve the total.
Calculate 802,395 − 168,272 by adding 10 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (802,405) − (168,282) = 634,123.
A library had 351,596 books, received 47,598 more, then lent 23,799. How many books are there now?
First add 399,194. Then subtract 23,799 to obtain 375,395.
Calculate 458,941 + 49,367.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Calculate 457,642 − 55,466.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 402,176 + 55,466 = 457,642.
Use compensation to add 13,672 and 134,166. What is the exact sum?
Increase 13,672 by 8 to a friendly multiple of 10, then subtract 8 from the sum to preserve the total.
Calculate 261,087 − 128,944 by adding 100 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (261,187) − (129,044) = 132,143.
A library had 675,818 books, received 3,625 more, then lent 1,812. How many books are there now?
First add 679,443. Then subtract 1,812 to obtain 677,631.
Calculate 468,484 + 97,268.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Calculate 653,244 − 5,595.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 647,649 + 5,595 = 653,244.
Use compensation to add 171,243 and 95,374. What is the exact sum?
Increase 171,243 by 7 to a friendly multiple of 10, then subtract 7 from the sum to preserve the total.
Calculate 420,512 − 1,772 by adding 50 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (420,562) − (1,822) = 418,740.
Alex and Bea have 583,143 points together. Bea has 5,823 more than Alex. How many does Alex have?
Remove the extra 5,823 from the whole and split the rest into two equal parts: (583,143 − 5,823) ÷ 2 = 288,660.
Calculate 607,738 + 160,526.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Calculate 793,605 − 132,238.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 661,367 + 132,238 = 793,605.
Find 45,900 + 820 without losing the hundreds and tens. Explain how place values align.
Add matching place-value units: 45,900+820=46,720.
Find 73,002 − 9 and explain how regrouping across zeros works.
Exchange from higher places so the ones column contains enough ones. The difference is 72,993.
A student says 299 + 70 = 300 + 70. Identify the compensation error and find the actual sum.
Increasing 299 to 300 adds one extra, which must be removed: 300 + 70 − 1 = 369.
Explain why 613 − 148 = 615 − 150. Find the common difference.
Adding two to both quantities preserves their difference: 465.
Two classes collect 12175 cans. Class B collects 1829 more than Class A. A student divides the total by 2 without removing the difference. Explain the mistake and find Class A.
Remove the extra 1829, then split the remaining 10346 equally: (12175 − 1829) ÷ 2 = 5173.
Find 15,900 + 770 without losing the hundreds and tens. Explain how place values align.
Add matching place-value units: 15,900+770=16,670.
Practice examination solutions and scoring
Calculate 162,507 + 49,324.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Use compensation to add 677,167 and 20,980. What is the exact sum?
Increase 677,167 by 33 to a friendly multiple of 100, then subtract 33 from the sum to preserve the total.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Alex and Bea have 342,122 points together. Bea has 1,170 more than Alex. How many does Alex have?
Remove the extra 1,170 from the whole and split the rest into two equal parts: (342,122 − 1,170) ÷ 2 = 170,476.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Calculate 696,950 − 85,716.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 611,234 + 85,716 = 696,950.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 642,793 − 180,114 by adding 50 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (642,843) − (180,164) = 462,679.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 748,366 − 209,823.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 538,543 + 209,823 = 748,366.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 237,011 − 78,935 by adding 50 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (237,061) − (78,985) = 158,076.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 559,691 + 137,222.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Use compensation to add 269,765 and 61,123. What is the exact sum?
Increase 269,765 by 5 to a friendly multiple of 10, then subtract 5 from the sum to preserve the total.
Scoring (2 points): Award one point for a sound method and one for the correct result.
A library had 268,681 books, received 203,486 more, then lent 101,743. How many books are there now?
First add 472,167. Then subtract 101,743 to obtain 370,424.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Use compensation to add 31,769 and 159,091. What is the exact sum?
Increase 31,769 by 31 to a friendly multiple of 100, then subtract 31 from the sum to preserve the total.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Alex and Bea have 1,235,597 points together. Bea has 1,793 more than Alex. How many does Alex have?
Remove the extra 1,793 from the whole and split the rest into two equal parts: (1,235,597 − 1,793) ÷ 2 = 616,902.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Calculate 492,429 − 96,432.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 395,997 + 96,432 = 492,429.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 244,012 − 14,336 by adding 100 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (244,112) − (14,436) = 229,676.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 193,914 + 214,331.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 251,947 − 44,959 by adding 50 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (251,997) − (45,009) = 206,988.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 455,874 + 52,640.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Use compensation to add 187,063 and 103,407. What is the exact sum?
Increase 187,063 by 37 to a friendly multiple of 100, then subtract 37 from the sum to preserve the total.
Scoring (2 points): Award one point for a sound method and one for the correct result.
A library had 326,030 books, received 174,022 more, then lent 87,011. How many books are there now?
First add 500,052. Then subtract 87,011 to obtain 413,041.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Calculate 803,861 − 139,232.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 664,629 + 139,232 = 803,861.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Alex and Bea have 454,917 points together. Bea has 5,429 more than Alex. How many does Alex have?
Remove the extra 5,429 from the whole and split the rest into two equal parts: (454,917 − 5,429) ÷ 2 = 224,744.
Scoring (3 points): Award one point for identifying the quantities or relationship, one for the correct method, and one for a correct justified result.
Calculate 394,493 − 14,107.
Subtract like place values, exchanging one higher unit for ten lower units when necessary. Check: 380,386 + 14,107 = 394,493.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 787,896 − 208,740 by adding 1000 to both quantities first. What answer do you get?
Equal adjustment preserves the difference: (788,896) − (209,740) = 579,156.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 404,021 + 216,541.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Use compensation to add 604,732 and 209,810. What is the exact sum?
Increase 604,732 by 8 to a friendly multiple of 10, then subtract 8 from the sum to preserve the total.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Calculate 150,574 + 168,177.
Align the place values; add ones, tens, hundreds and higher units, exchanging 10 of one unit for 1 of the next when needed.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Use compensation to add 143,953 and 45,359. What is the exact sum?
Increase 143,953 by 7 to a friendly multiple of 10, then subtract 7 from the sum to preserve the total.
Scoring (2 points): Award one point for a sound method and one for the correct result.
Find 41,800 + 270 without losing the hundreds and tens. Explain how place values align.
Add matching place-value units: 41,800+270=42,070.
Scoring (4 points): Award one point for a sound method and one for the correct result.
Find 67,002 − 9 and explain how regrouping across zeros works.
Exchange from higher places so the ones column contains enough ones. The difference is 66,993.
Scoring (4 points): Award one point for a sound method and one for the correct result.
A student says 299 + 36 = 300 + 36. Identify the compensation error and find the actual sum.
Increasing 299 to 300 adds one extra, which must be removed: 300 + 36 − 1 = 335.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
Explain why 565 − 193 = 567 − 195. Find the common difference.
Adding two to both quantities preserves their difference: 372.
Scoring (4 points): Award one point for a sound method and one for the correct result.
Two classes collect 12580 cans. Class B collects 2582 more than Class A. A student divides the total by 2 without removing the difference. Explain the mistake and find Class A.
Remove the extra 2582, then split the remaining 9998 equally: (12580 − 2582) ÷ 2 = 4999.
Scoring (4 points): Award one point for finding the mistaken step, one for a correct mathematical model, one for the corrected answer, and one for a clear explanation.
Find 24,800 + 260 without losing the hundreds and tens. Explain how place values align.
Add matching place-value units: 24,800+260=25,060.
Scoring (4 points): Award one point for a sound method and one for the correct result.