Place Value and Decimals: practice solutions, Grade 5 – download the PDF

1 Value of each digit ★★★

(a) The 8 is in the tens place: its value is \(80\).

(b) The 3 is in the tenths place: its value is \(0.3\).

(c) The 1 is in the hundredths place: its value is \(0.01\).

(d) The 7 is in the thousandths place: its value is \(0.007\).

2 Ten times and one tenth ★★★

(a) \(7 = 10 \times 0.7\). To go one place right, divide by 10.

(b) \(0.08 = 10 \times 0.008\).

(c) \(0.4\) is \(\dfrac{1}{10}\) of \(4\), because \(4 \div 10 = 0.4\).

(d) \(600\) is \(\dfrac{1}{10}\) of \(6{,}000\), because \(6{,}000 \div 10 = 600\).

3 Powers of ten ★★★

(a) \(100\) has 2 zeros, so \(100 = 10^2\). \(100{,}000\) has 5 zeros, so it is \(10^5\). And \(10 = 10^1\).

(b) \(10^4 = 10 \times 10 \times 10 \times 10 = 10{,}000\) and \(10^6 = 1{,}000{,}000\).

4 Words to numbers ★★★

(a) \(9.427\)

(b) \(0.63\)

(c) \(0.005\): two placeholder zeros hold the tenths and hundredths places.

(d) \(12.8\)

5 Numbers to words ★★★

(a) thirty and two hundred five thousandths.

(b) nine hundredths.

(c) seven and eighteen thousandths.

6 Greater, less or equal ★★★

(a) \(0.80 > 0.79\), since \(8 > 7\) in the hundredths place: \(>\).

(b) A zero at the end changes nothing: \(=\).

(c) \(0.099 < 0.100\): \(<\).

(d) The tenths give \(0 < 6\), so \(12.06 < 12.6\): \(<\).

7 Round to the nearest tenth ★★★

\(6.37\): the hundredths digit is 7, so round up: \(6.4\).

\(0.82\): the hundredths digit is 2, so keep: \(0.8\).

\(2.04\): the hundredths digit is 4, so keep: \(2.0\).

\(9.96\): the hundredths digit is 6, so round up and carry: \(10.0\).

8 Expanded form ★★★

(a) \(73.406 = 7 \times 10 + 3 \times 1 + 4 \times 0.1 + 0 \times 0.01 + 6 \times 0.001 = 70 + 3 + 0.4 + 0.006\).

(b) \(200 + 0.5 + 0.009 = 200.509\); the tens, ones and hundredths places hold zeros.

(c) \(6 + 0.8 + 0.03 = 6.83\).

9 Order from least to greatest ★★★

Write all with three decimal places: \(4.052\), \(4.250\), \(4.500\), \(4.205\), \(4.025\). Compare tenths first: \(4.052\) and \(4.025\) have tenths 0, then \(4.205\) has 2, \(4.250\) has 2, and \(4.500\) has 5.

Among the two with tenths 0: \(4.025 < 4.052\). Among the two with tenths 2: \(4.205 < 4.250\).

Answer: \(4.025 < 4.052 < 4.205 < 4.25 < 4.5\).

10 Reading a number line ★★★

(a) The ticks go \(4.20, 4.21, 4.22, \dots\). \(A = 4.22\), \(B = 4.26\) (one tick after the halfway mark \(4.25\)), \(C = 4.29\) (one tick before \(4.30\)).

(b) \(4.22 \approx 4.2\) (hundredths digit 2, keep); \(4.26 \approx 4.3\) (digit 6, round up); \(4.29 \approx 4.3\) (digit 9, round up).

11 Multiplying by powers of 10 ★★★

(a) One place right: \(34.2\).

(b) Two places right: \(6.8\).

(c) Three places right, with two placeholder zeros: \(5{,}700\).

(d) \(10^2 = 100\), so \(0.4 \times 100 = 40\).

(e) \(10^3 = 1{,}000\), so \(12.5 \times 1{,}000 = 12{,}500\).

12 Dividing by powers of 10 ★★★

(a) One place left: \(0.45\).

(b) The decimal point of 62 is after the 2; two places left gives \(0.62\).

(c) Three places left with placeholder zeros: \(0.0083\).

(d) \(730 \div 100 = 7.3\).

(e) \(0.9 \div 100 = 0.009\).

13 Sprint times ★★★

The smallest time wins. Write every time with three decimals: \(11.480\), \(11.400\), \(11.520\), \(11.399\). The ones and tens match, so compare tenths: three times have tenths 4 and one has 5 (Cleo, last).

Among \(11.480\), \(11.400\) and \(11.399\): hundredths of \(11.399\) are 9... but tenths of \(11.399\) are 3, which is less than 4, so it is the smallest. Then \(11.400 < 11.480\).

Order: Dev (\(11.399\)), Ben (\(11.4\)), Ava (\(11.48\)), Cleo (\(11.52\)). Dev won.

14 Rounding to different places ★★★

(a) The tenths digit is 4, so keep: \(7\).

(b) The hundredths digit is 9, so round up: \(7.5\).

(c) The thousandths digit is 6, so round up: \(7.49 + 0.01 = 7.50\).

(d) The ten-thousandths digit is 8, so round up: \(7.497\).

15 Metric conversions ★★★

(a) \(1\) m \(= 100\) cm, so \(3.5 \times 100 = 350\) cm.

(b) \(1\) L \(= 1{,}000\) mL, so \(2{,}450 \div 1{,}000 = 2.45\) L.

(c) \(7.2 \div 100 = 0.072\) kg. Since \(1\) kg \(= 1{,}000\) g, that is \(0.072 \times 1{,}000 = 72\) g. Each packet holds \(0.072\) kg, or \(72\) g.

16 True or false? ★★★

(a) False. \(0.30 = 0.3\) (equivalent decimals).

(b) False. \(5.6 \times 100 = 560\), not \(5{,}600\) (that would be \(5.6 \times 10^3\)).

(c) True. Each place to the left is worth 10 times as much: \(0.007 \times 10 = 0.07\).

(d) False. \(3.09 < 3.10\), because the tenths are \(0 < 1\).

17 Mystery number ★★★

(a) Tens 2, ones 7, tenths \(7 - 3 = 4\), hundredths 0, thousandths \(4 + 1 = 5\): the number is \(27.405\).

(b) Twenty-seven and four hundred five thousandths.

(c) The hundredths digit is 0, so keep the tenths digit: \(27.4\).

(d) \(27.405 \times 100 = 2{,}740.5\).

18 Find the mistake ★★★

First mistake: Leo thought that multiplying by \(10^3\) means adding three zeros after the number. Zeros at the end of a decimal change nothing, so \(2.5000 = 2.5\), which cannot be 1,000 times larger. The digits must move 3 places left: \(2.5 \times 1{,}000 = 2{,}500\).

Second mistake: dividing by 10 makes the number smaller, but \(4.5\) is larger than \(0.45\). Leo moved the point the wrong way. Correct: \(0.45 \div 10 = 0.045\).

19 Digits and numbers in between ★★★

(a) Greatest: put the largest digits in the highest places: \(7.530\). Least: \(0.357\) (the 0 goes in the ones place, then 3, 5, 7 in order).

(b) Write both as thousandths: \(0.470\) and \(0.480\). For example \(0.471\), \(0.475\), \(0.479\).

(c) They are \(0.471, 0.472, \dots, 0.479\): that makes \(9\) numbers.

20 Under the microscope ★★★

(a) \(0.0042 \times 10^3 = 0.0042 \times 1{,}000 = 4.2\) cm.

(b) \(0.0042 \times 100 = 0.42\) cm.

(c) \(4.2\) is \(10\) times \(0.42\) (one place to the left), since \(10^3 \div 10^2 = 10\).

21 Capsule factory ★★★

(a) \(4.6 \times 1{,}000 = 4{,}600\) g. Dividing by \(1{,}000\) to get kilograms: \(4{,}600 \div 1{,}000 = 4.6\) kg.

(b) \(4{,}600 \div 100 = 46\) g in each jar.

22 Comparing places ★★★

(a) \(500\) and \(50\): \(500 = 10 \times 50\), so 10 times.

(b) \(50 = \dfrac{1}{10} \times 500\), so \(\dfrac{1}{10}\).

(c) The hundredths 5 is worth \(0.05\). \(500 \div 0.05 = 10{,}000 = 10^4\): the hundreds 5 is worth \(10^4\) times as much (four places to the left).