Dividing Whole Numbers: practice solutions, Grade 5 – download the PDF

1 Mental division with zeros ★★★

1. \(36 \div 9 = 4\), so \(3{,}600 \div 9 = 400\).
2. \(54 \div 9 = 6\), so \(540 \div 90 = 6\).
3. Cross out one zero: \(280 \div 4 = 70\).
4. \(63 \div 3 = 21\), so \(630 \div 30 = 21\).
5. Cross out one zero: \(4{,}500 \div 5 = 900\).

Answers: 400, 6, 70, 21 and 900.

2 Estimate with compatible numbers ★★★

1. \(2{,}415\) is close to \(2{,}400\), and \(2{,}400 \div 8 = 300\). Estimate: about 300.
2. \(5{,}987 \approx 6{,}000\), and \(6{,}000 \div 3 = 2{,}000\). Estimate: about 2,000.
3. \(4{,}130 \approx 4{,}200\), and \(4{,}200 \div 7 = 600\). Estimate: about 600.
4. \(812 \approx 800\), and \(800 \div 4 = 200\). Estimate: about 200.

Other reasonable compatible numbers are accepted if they are easy to divide.

3 Partial quotients with 848 ★★★

\(4 \times 200 = 800\), and \(848 - 800 = 48\).
\(4 \times 12 = 48\), and \(48 - 48 = 0\).

Add the partial quotients: \(200 + 12 = 212\). Check: \(4 \times 212 = 848\). So \(848 \div 4 = 212\).

4 Complete the area model ★★★

\(3 \times 100 = 300\), and \(435 - 300 = 135\). Next, \(3 \times 40 = 120\), and \(135 - 120 = 15\). Finally \(3 \times 5 = 15\).

The quotient is \(100 + 40 + 5 = 145\).

5 Long division by 4 ★★★

\(12 \div 4 = 3\), and \(12 - 12 = 0\). Bring down 9: \(9 \div 4 = 2\), and \(4 \times 2 = 8\), so the remainder is 1. Bring down 6 to make 16: \(16 \div 4 = 4\), with nothing left.

The quotient is 324. Check: \(4 \times 324 = 1{,}296\).

6 Sticker sharing ★★★

We share equally, so we divide: \(252 \div 6\). \(6 \times 40 = 240\), and \(252 - 240 = 12\). \(6 \times 2 = 12\). So \(40 + 2 = 42\).

Each friend gets 42 stickers. Check: \(6 \times 42 = 252\).

7 True or false? ★★★

1. True. \(420 \div 6 = 70\), and \(60 \times 70 = 4{,}200\).
2. False. \(540 \div 9 = 60\), so \(5{,}400 \div 90 = 60\). Check: \(90 \times 600 = 54{,}000\), which is too big.
3. False. \(7 \times 105 = 735\), not 7,350. The correct quotient is \(1{,}050\), because \(7 \times 1{,}050 = 7{,}350\).

8 A division with a remainder ★★★

\(34 \div 6 = 5\), and \(6 \times 5 = 30\), so the remainder is 4. Bring down 5 to make 45: \(45 \div 6 = 7\), and \(6 \times 7 = 42\), remainder 3. Bring down 7 to make 37: \(37 \div 6 = 6\), and \(6 \times 6 = 36\), remainder 1.

\(3{,}457 \div 6 = 576\) R 1. Check: \(6 \times 576 + 1 = 3{,}456 + 1 = 3{,}457\).

9 A first 2-digit divisor ★★★

12 is less than 24, so start with 124: \(124 \div 24 = 5\), because \(24 \times 5 = 120\), and \(124 - 120 = 4\). Bring down 8 to make 48: \(48 \div 24 = 2\), with nothing left.

The quotient is 52. Check: \(24 \times 52 = 1{,}248\).

10 Area model with 22 ★★★

\(22 \times 40 = 880\), and \(902 - 880 = 22\). Then \(22 \times 1 = 22\). The two pieces of the missing side are 40 and 1.

The quotient is \(40 + 1 = 41\).

11 Trays of muffins ★★★

Divide: \(1{,}260 \div 15\). With partial quotients: \(15 \times 80 = 1{,}200\), and \(1{,}260 - 1{,}200 = 60\). \(15 \times 4 = 60\). So \(80 + 4 = 84\).

The cafe fills 84 trays. Check: \(15 \times 84 = 1{,}260\).

12 Find the error ★★★

1. Leo used the basic fact \(30 \div 5 = 6\) but forgot a zero. After crossing out one zero in each number, \(3{,}000 \div 50 = 300 \div 5 = 60\). Check: \(50 \times 60 = 3{,}000\).
2. Priya forgot the zero placeholders. \(9 \div 9 = 1\) thousand; the 0 in the hundreds and the 7 in the tens are too small to divide by 9, so we write 0 in both places; then \(72 \div 9 = 8\). The quotient is \(1{,}008\). Check: \(9 \times 1{,}008 = 9{,}072\).

13 Bags of rolls ★★★

\(275 \div 12\): \(12 \times 20 = 240\), and \(275 - 240 = 35\). \(12 \times 2 = 24\), and \(35 - 24 = 11\). So \(275 \div 12 = 22\) R 11.

1. The store can make 22 full bags.
2. 11 rolls are left over, because 11 is less than 12 and \(12 \times 22 + 11 = 275\).

14 Long division by 36 ★★★

Estimate: \(7{,}200 \div 36 = 200\), so the answer should be a little more than 200.

Divide: 78 is the first big enough group; \(78 \div 36 = 2\), and \(36 \times 2 = 72\), remainder 6. Bring down 4 to make 64: \(64 \div 36 = 1\), remainder 28. Bring down 8 to make 288: \(288 \div 36 = 8\), because \(36 \times 8 = 288\).

The quotient is 218, close to the estimate. Check: \(36 \times 218 = 7{,}848\).

15 Cutting a rope ★★★

Divide: \(2{,}450 \div 35\). \(35 \times 60 = 2{,}100\), and \(2{,}450 - 2{,}100 = 350\). \(35 \times 10 = 350\). So \(60 + 10 = 70\).

Each piece is 70 centimeters long. In inches: \(70 \div 2.54 \approx 27.6\), so each piece is about 28 inches.

16 Three questions, one remainder ★★★

\(83 \div 6 = 13\) R 5, because \(6 \times 13 = 78\) and \(83 - 78 = 5\).

1. Only full boxes count, so we ignore the remainder: 13 full boxes.
2. Every person needs a seat, so we round up: 14 vans (13 full vans and one van for the last 5 people).
3. Each cousin gets 13 dollars, and 5 dollars are left over.

17 Work backward from a quotient ★★★

Use \(\text{divisor} \times \text{quotient} + \text{remainder} = \text{dividend}\). First \(27 \times 45 = 1{,}215\). Then \(1{,}215 + 13 = 1{,}228\).

The number is 1,228. Check: \(1{,}228 \div 27\) gives 45 groups of 27 (that is 1,215) and \(1{,}228 - 1{,}215 = 13\), which is less than 27.

18 Estimate, then divide by 18 ★★★

1. \(6{,}372 \approx 6{,}300\), and \(18 \times 350 = 6{,}300\). The estimate is about 350.
2. 63 is the first group big enough: \(63 \div 18 = 3\), and \(18 \times 3 = 54\), remainder 9. Bring down 7 to make 97: \(97 \div 18 = 5\), and \(18 \times 5 = 90\), remainder 7. Bring down 2 to make 72: \(72 \div 18 = 4\). The quotient is 354.
3. 354 is very close to 350, so the answer is reasonable. Check: \(18 \times 354 = 6{,}372\).

19 Packs of pencils ★★★

Divide: \(4{,}860 \div 45\). 48 is the first group big enough: \(48 \div 45 = 1\), remainder 3. Bring down 6 to make 36. Since 36 is less than 45, write 0 in the quotient. Bring down 0 to make 360: \(360 \div 45 = 8\), because \(45 \times 8 = 360\).

The quotient is 108, so the school ordered 108 packs. In the tens place we write 0 because 36 cannot be divided by 45. Check: \(45 \times 108 = 4{,}860\).

20 Order without long division ★★★

The dividend is the same, so a bigger divisor gives a smaller quotient. Order the divisors: \(60 < 80 < 90\). So the quotients go the other way: \(7{,}200 \div 90 < 7{,}200 \div 80 < 7{,}200 \div 60\).

Check with basic facts: \(720 \div 9 = 80\), \(720 \div 8 = 90\), \(720 \div 6 = 120\). From least to greatest: 80, 90, 120.