Free Math Worksheets

- One kilogram equals \(1000\) grams. - Find the number of complete thousands in \(4055\). - The remaining grams stay in the smaller unit.

1. One kilogram equals \(1000\) grams. 2. Split \(4055\,\text{g}\) into \(4000\,\text{g} + 55\,\text{g}\). 3. Since \(4000\,\text{g} = 4\,\text{kg}\), the mass is \(4\,\text{kg}\ 55\,\text{g}\).

The package weighs \(4\,\text{kg}\ 55\,\text{g}\).

- How many days are in one week? - Recall the multiples of \(7\). - Multiply the number of weeks by the number of days in each week.

1. One week has \(7\) days. 2. Multiply: \(9 \times 7 = 63\).

The project lasts \(63\) days.

- How many centimeters are in \(1\) meter? - Think about the place values when multiplying by \(100\). - A hundredth of a meter is \(1\) centimeter.

1. Since \(1\,\text{m} = 100\,\text{cm}\), multiply each number of meters by \(100\). 2. a) \(1.25 \times 100 = 125\), so the length is \(125\,\text{cm}\). 3. b) \(0.80 \times 100 = 80\), so the length is \(80\,\text{cm}\). 4. c) \(2.06 \times 100 = 206\), so the length is \(206\,\text{cm}\). 5. d) \(0.09 \times 100 = 9\), so the length is \(9\,\text{cm}\).

a) \(125\,\text{cm}\) b) \(80\,\text{cm}\) c) \(206\,\text{cm}\) d) \(9\,\text{cm}\)

- Convert all cards to meters. - One kilometer equals \(1000\) meters. - Pay close attention to place value and zeros.

1. Convert each mixed measurement to meters. 2. \(3\,\text{km}\ 200\,\text{m} = 3000\,\text{m} + 200\,\text{m} = 3200\,\text{m}\). 3. \(3\,\text{km}\ 20\,\text{m} = 3000\,\text{m} + 20\,\text{m} = 3020\,\text{m}\). 4. \(30\,\text{km}\ 200\,\text{m} = 30{,}000\,\text{m} + 200\,\text{m} = 30{,}200\,\text{m}\).

\(3200\,\text{m} = 3\,\text{km}\ 200\,\text{m}\) \(3020\,\text{m} = 3\,\text{km}\ 20\,\text{m}\) \(30{,}200\,\text{m} = 30\,\text{km}\ 200\,\text{m}\)

- Convert \(1\,\text{km}\) to meters first. - Subtract the given length from the total. - Think about completing each number to \(1000\).

1. Convert the target length: \(1\,\text{km} = 1000\,\text{m}\). 2. Subtract each given length from \(1000\,\text{m}\). 3. The differences are \(1000 - 600 = 400\), \(1000 - 150 = 850\), \(1000 - 880 = 120\), and \(1000 - 405 = 595\).

a) \(400\,\text{m}\) b) \(850\,\text{m}\) c) \(120\,\text{m}\) d) \(595\,\text{m}\)

- Convert \(1\,\text{km}\) to meters. - Subtract the known part from the total to find the missing part. - Check that the two distances in each part add to \(1000\,\text{m}\).

1. Convert the total: \(1\,\text{km} = 1000\,\text{m}\). 2. Subtract each given distance from \(1000\,\text{m}\): \(1000 - 310 = 690\), \(1000 - 685 = 315\), \(1000 - 505 = 495\), and \(1000 - 99 = 901\).

a) \(690\,\text{m}\) b) \(315\,\text{m}\) c) \(495\,\text{m}\) d) \(901\,\text{m}\)

- Convert both measurements to the same unit. - Change kilometers to meters before comparing. - One kilometer equals \(1000\) meters.

1. Convert the kilometer measurements to meters. 2. Compare \(1000\,\text{m}\) and \(900\,\text{m}\): \(1000 > 900\). 3. Compare \(1500\,\text{m}\) and \(2000\,\text{m}\): \(1500 < 2000\). 4. Compare \(1000\,\text{m}\) and \(1000\,\text{m}\): they are equal. 5. Compare \(750\,\text{m}\) and \(1000\,\text{m}\): \(750 < 1000\).

a) \(>\) b) \(<\) c) \(=\) d) \(<\)

- The tenths place represents tenths of a meter, and the hundredths place represents centimeters. - Use a zero as a placeholder when a place value has no units. - For a length less than \(10\,\text{cm}\), think carefully about which decimal places must contain zeros.

1. Since \(100\,\text{cm} = 1\,\text{m}\), divide each number of centimeters by \(100\). 2. a) \(408\,\text{cm} = 4.08\,\text{m}\). 3. b) \(560\,\text{cm} = 5.60\,\text{m}\). 4. c) \(90\,\text{cm} = 0.90\,\text{m}\). 5. d) \(7\,\text{cm} = 0.07\,\text{m}\).

a) \(4.08\,\text{m}\) b) \(5.60\,\text{m}\) c) \(0.90\,\text{m}\) d) \(0.07\,\text{m}\)

- How many centimeters are in \(1\) meter? - Convert each measurement to the requested unit before multiplying. - Keep the unit with each product.

1. Convert the meter measurements to centimeters: \(1.20\,\text{m} = 120\,\text{cm}\), \(0.75\,\text{m} = 75\,\text{cm}\), and \(2.15\,\text{m} = 215\,\text{cm}\). 2. a) \(4 \times 120\,\text{cm} = 480\,\text{cm}\). 3. b) \(5 \times 75\,\text{cm} = 375\,\text{cm}\). 4. c) \(3 \times 215\,\text{cm} = 645\,\text{cm}\).

a) \(480\,\text{cm}\) b) \(375\,\text{cm}\) c) \(645\,\text{cm}\)

- How many meters are in \(1\) kilometer? - Convert the kilometers first. - Then add the extra \(450\) meters.

1. Convert the kilometers to meters: \(128 \times 1000 = 128{,}000\,\text{m}\). 2. Add the remaining distance. Adding \(450\,\text{m}\) gives \(128{,}450\,\text{m}\).

The road is \(128{,}450\,\text{m}\) long.

- Convert both quantities to the same unit before comparing them. - How many pounds are in \(1\,\text{ton}\)? - How many ounces are in \(1\,\text{lb}\)? - What is half of the number of pounds in \(1\,\text{ton}\)?

1. a) Since \(1\,\text{ton}=2000\,\text{lb}\), \(3 \times 2000=6000\,\text{lb}\). Therefore, \(3\,\text{tons}=6000\,\text{lb}\). 2. b) Since \(1\,\text{lb}=16\,\text{oz}\), \(300 \times 16=4800\,\text{oz}\). Because \(4500<4800\), \(4500\,\text{oz}<300\,\text{lb}\). 3. c) Half of \(2000\,\text{lb}\) is \(1000\,\text{lb}\). Because \(1000>800\), \(\frac{1}{2}\,\text{ton}>800\,\text{lb}\). 4. d) Since \(1\,\text{lb}=16\,\text{oz}\) and \(16<1000\), \(1\,\text{lb}<1000\,\text{oz}\).

a) \(=\) b) \(<\) c) \(>\) d) \(<\)

- One hour equals \(60\) minutes. - To convert minutes to hours and minutes, find how many groups of \(60\) fit. - The remainder is the number of extra minutes.

1. For a), \(60 + 25 = 85\) minutes. 2. For b), \(3 \times 60 + 5 = 185\) minutes. 3. For c), \(85 = 60 + 25\), so the time is \(1\) hour \(25\) minutes. 4. For d), \(190 = 3 \times 60 + 10\), so the time is \(3\) hours \(10\) minutes.

a) \(85\,\text{min}\) b) \(185\,\text{min}\) c) \(1\,\text{hr}\ 25\,\text{min}\) d) \(3\,\text{hr}\ 10\,\text{min}\)

- Recall the number of minutes in one-fourth hour and one-half hour. - Convert whole hours and fractional hours separately. - For larger numbers of minutes or seconds, find how many groups of \(60\) there are.

1. Three-fourths of an hour is \(3 \times 15 = 45\) minutes. 2. Two hours is \(120\) minutes, and one-half hour is \(30\) minutes. The total is \(150\) minutes. 3. \(420 \div 60 = 7\) hours. 4. \(600 \div 60 = 10\) minutes.

a) \(45\,\text{min}\) b) \(150\,\text{min}\) c) \(7\,\text{hr}\) d) \(10\,\text{min}\)

- Convert both sides to the same unit. - One hour is \(60\) minutes. - One minute is \(60\) seconds. - One day is \(24\) hours.

1. \(1\) hour is \(60\) minutes, and \(60 < 75\). 2. \(2\) minutes is \(120\) seconds, so the measurements are equal. 3. \(1\) day is \(24\) hours, and \(24 > 20\). 4. \(1\) hour \(30\) minutes is \(90\) minutes, and \(90 < 95\). 5. Three-fourths of an hour is \(45\) minutes, so the measurements are equal.

a) \(<\) b) \(=\) c) \(>\) d) \(<\) e) \(=\)

- One hour equals \(60\) minutes. - For more than \(60\) minutes, separate out a full hour and find the remaining minutes. - Ask how many groups of \(60\) fit in the number of minutes.

1. \(1\) hour \(15\) minutes is \(60 + 15 = 75\) minutes. 2. \(90\) minutes is \(1\) hour \(30\) minutes. 3. \(1\) hour \(40\) minutes is \(60 + 40 = 100\) minutes. 4. \(110\) minutes is \(1\) hour \(50\) minutes. 5. \(2\) hours is \(2 \times 60 = 120\) minutes.

a) \(75\,\text{min}\) b) \(1\,\text{hr}\ 30\,\text{min}\) c) \(100\,\text{min}\) d) \(1\,\text{hr}\ 50\,\text{min}\) e) \(120\,\text{min}\)

- Recall the lengths of one-fourth of an hour, one-half of an hour, and three-fourths of an hour. - One day has \(24\) hours. - One minute has \(60\) seconds.

1. One-fourth of an hour is \(15\) minutes, and one-half of an hour is \(30\) minutes. Together, they are \(15 + 30 = 45\) minutes. 2. Three-fourths of an hour is \(45\) minutes, so \(60 - 45 = 15\) minutes are needed. 3. Two days have \(2 \times 24 = 48\) hours. 4. Since \(60\) seconds is \(1\) minute, \(80\) seconds is \(1\) minute \(20\) seconds.

a) \(45\) minutes b) \(15\) minutes c) \(48\) hours d) \(1\) minute \(20\) seconds

- Convert both amounts to the same unit before comparing them. - How many cups are in \(1\,\text{quart}\)? - How many cups are in \(1\,\text{gallon}\)?

1. Use \(1\,\text{quart}=4\,\text{cups}\) and \(1\,\text{gallon}=16\,\text{cups}\). 2. a) Since \(3<4\), \(3\,\text{cups}<1\,\text{quart}\). 3. b) \(2\,\text{quarts}=8\,\text{cups}\). Since \(10>8\), \(10\,\text{cups}>2\,\text{quarts}\). 4. c) Half of \(16\,\text{cups}\) is \(8\,\text{cups}\), so \(\frac{1}{2}\,\text{gallon}=8\,\text{cups}\). 5. d) \(3\,\text{quarts}=12\,\text{cups}\). Since \(12<14\), \(3\,\text{quarts}<14\,\text{cups}\).

a) \(<\) b) \(>\) c) \(=\) d) \(<\)

- How many fluid ounces are in \(1\,\text{cup}\)? - Multiply the number of cups by \(8\). - Think of each fraction of a cup as the same fraction of \(8\,\text{fl oz}\).

1. Use \(1\,\text{cup}=8\,\text{fl oz}\). 2. Multiply each number of cups by \(8\): \(3 \times 8=24\) \(\frac{1}{2} \times 8=4\) \(\frac{3}{4} \times 8=6\) \(1\frac{1}{2} \times 8=12\) \(\frac{1}{4} \times 8=2\)

\(24\,\text{fl oz}\); \(4\,\text{fl oz}\); \(6\,\text{fl oz}\); \(12\,\text{fl oz}\); \(2\,\text{fl oz}\)

- Count the days when the museum is open. - How many hours are in one day? - Multiply the hours in one day by the number of days in a week.

1. The museum is open on Thursday, Friday, Saturday, and Sunday, which is \(4\) days. 2. Find the weekly open hours: \(4 \times 10\,\text{hours} = 40\,\text{hours}\). 3. One day has \(24\) hours, so a full week has \(7 \times 24\,\text{hours} = 168\,\text{hours}\).

a) The museum is open \(40\) hours each week. b) A full week has \(168\) hours.

- How many hours are in \(1\) day? - Once you know the hours in one day, how can you find the hours in many days? - Which operation combines equal groups?

1. One day has \(24\) hours. 2. Multiply to find the total number of hours: \(42 \times 24=1008\).

The expedition lasts \(1008\,\text{hours}\).

- One dollar equals \(100\) cents. - One meter equals \(100\) centimeters. - In each decimal, use the digits before and after the decimal point to identify the two units.

1. In dollar notation, the whole-number part gives dollars and the two decimal places give cents. Therefore, \(\$6.02\) is \(6\) dollars \(2\) cents, and \(\$35.70\) is \(35\) dollars \(70\) cents. 2. Since \(100\) cents equals \(1\) dollar, \(915\) cents is \(9\) dollars \(15\) cents. 3. Since \(100\,\text{cm} = 1\,\text{m}\), the two decimal places in a meter measurement represent centimeters. Therefore, \(4.08\,\text{m} = 4\,\text{m}\ 8\,\text{cm}\) and \(12.55\,\text{m} = 12\,\text{m}\ 55\,\text{cm}\).

a) \(6\,\text{dollars}\ 2\,\text{cents}\) b) \(35\,\text{dollars}\ 70\,\text{cents}\) c) \(9\,\text{dollars}\ 15\,\text{cents}\) d) \(4\,\text{m}\ 8\,\text{cm}\) e) \(12\,\text{m}\ 55\,\text{cm}\)

- How many minutes are in \(1\) full hour? - Multiply the number of hours by \(60\), then add the remaining minutes.

1. Soccer: \(1 \times 60\,\text{min}+30\,\text{min}=90\,\text{min}\). 2. Swimming: \(1 \times 60\,\text{min}+15\,\text{min}=75\,\text{min}\). 3. Gymnastics: \(2 \times 60\,\text{min}+5\,\text{min}=125\,\text{min}\).

Soccer: \(90\,\text{min}\); swimming: \(75\,\text{min}\); gymnastics: \(125\,\text{min}\).

- One hour equals \(60\) minutes. - One minute equals \(60\) seconds. - To find a unit fraction of a duration, divide the whole duration by the denominator.

1. \(\frac{1}{2}\) hour is \(60\,\text{min} \div 2 = 30\,\text{min}\). 2. \(\frac{3}{4}\) hour is \((60\,\text{min} \div 4) \times 3 = 45\,\text{min}\). 3. One minute equals \(60\) seconds. 4. \(\frac{1}{4}\) minute is \(60\,\text{s} \div 4 = 15\,\text{s}\). 5. \(\frac{1}{6}\) minute is \(60\,\text{s} \div 6 = 10\,\text{s}\).

\(\frac{1}{2}\,\text{hr} = 30\,\text{min}\) \(\frac{3}{4}\,\text{hr} = 45\,\text{min}\) \(1\,\text{min} = 60\,\text{s}\) \(\frac{1}{4}\,\text{min} = 15\,\text{s}\) \(\frac{1}{6}\,\text{min} = 10\,\text{s}\)

- One pound equals \(16\) ounces. - One ton equals \(2000\) pounds. - Divide the whole-unit amount by the denominator of each unit fraction.

1. Since \(1\,\text{lb} = 16\,\text{oz}\), \(\frac{1}{2}\,\text{lb} = 16\,\text{oz} \div 2 = 8\,\text{oz}\). 2. \(\frac{1}{4}\,\text{lb} = 16\,\text{oz} \div 4 = 4\,\text{oz}\). 3. \(\frac{1}{8}\,\text{lb} = 16\,\text{oz} \div 8 = 2\,\text{oz}\). 4. Since \(1\,\text{ton} = 2000\,\text{lb}\), \(\frac{1}{10}\,\text{ton} = 2000\,\text{lb} \div 10 = 200\,\text{lb}\).

\(\frac{1}{2}\,\text{lb} = 8\,\text{oz}\) \(\frac{1}{4}\,\text{lb} = 4\,\text{oz}\) \(\frac{1}{8}\,\text{lb} = 2\,\text{oz}\) \(\frac{1}{10}\,\text{ton} = 200\,\text{lb}\)

- Identify how many smaller units make one larger unit. - Round to the place value represented by that conversion factor. - Then write the rounded value in the requested unit.

1. a) Since \(100\,\text{cm}=1\,\text{m}\), round \(932\,\text{cm}\) to the nearest \(100\,\text{cm}\): \(900\,\text{cm}=9\,\text{m}\). 2. b) Since \(1000\,\text{g}=1\,\text{kg}\), round \(15{,}480\,\text{g}\) to the nearest \(1000\,\text{g}\): \(15{,}000\,\text{g}=15\,\text{kg}\). 3. c) Since \(10\,\text{mm}=1\,\text{cm}\), round \(57\,\text{mm}\) to the nearest \(10\,\text{mm}\): \(60\,\text{mm}=6\,\text{cm}\). 4. d) Since \(1000\,\text{m}=1\,\text{km}\), round \(12{,}500\,\text{m}\) to the nearest \(1000\,\text{m}\): \(13{,}000\,\text{m}=13\,\text{km}\).

a) \(9\,\text{m}\) b) \(15\,\text{kg}\) c) \(6\,\text{cm}\) d) \(13\,\text{km}\)

- Decide whether the seconds are closer to \(0\) seconds or \(60\) seconds. - Half of one minute is \(30\) seconds. - At \(30\) seconds or more, round to the next minute.

1. A time with \(0\) through \(29\) seconds rounds down; a time with \(30\) through \(59\) seconds rounds up. 2. a) \(12\,\text{min}\ 15\,\text{s}\) rounds to \(12\,\text{min}\). 3. b) \(7\,\text{min}\ 54\,\text{s}\) rounds to \(8\,\text{min}\). 4. c) \(38\,\text{s}\) rounds to \(1\,\text{min}\). 5. d) \(25\,\text{min}\ 30\,\text{s}\) rounds to \(26\,\text{min}\). 6. e) \(4\,\text{min}\ 4\,\text{s}\) rounds to \(4\,\text{min}\).

a) \(12\,\text{min}\) b) \(8\,\text{min}\) c) \(1\,\text{min}\) d) \(26\,\text{min}\) e) \(4\,\text{min}\)

- Recall that \(1\,\text{kg}=1000\,\text{g}\). - Find half of a kilogram in grams. - Compare the number of grams with the halfway value.

1. Half of \(1\,\text{kg}\) is \(500\,\text{g}\). 2. a) Since \(610\,\text{g}\geq500\,\text{g}\), round \(8\,\text{kg}\ 610\,\text{g}\) up to \(9\,\text{kg}\). 3. b) Since \(490\,\text{g}<500\,\text{g}\), round \(12\,\text{kg}\ 490\,\text{g}\) down to \(12\,\text{kg}\). 4. c) Since \(500\,\text{g}\) is exactly half a kilogram, round up to \(4\,\text{kg}\). 5. d) Since \(550\,\text{g}\geq500\,\text{g}\), round up to \(1\,\text{kg}\).

a) \(9\,\text{kg}\) b) \(12\,\text{kg}\) c) \(4\,\text{kg}\) d) \(1\,\text{kg}\)

- Recall how many meters are in a kilometer and how many centimeters are in a meter. - Find half of each larger unit. - Compare the smaller-unit amount with the halfway value.

1. a) Half of \(1\,\text{km}\) is \(500\,\text{m}\). Since \(800\,\text{m}\geq500\,\text{m}\), round up to \(13\,\text{km}\). 2. b) Half of \(1\,\text{m}\) is \(50\,\text{cm}\). Since \(45\,\text{cm}<50\,\text{cm}\), round down to \(4\,\text{m}\). 3. c) Since \(500\,\text{m}\) is exactly half a kilometer, round up to \(10\,\text{km}\). 4. d) Since \(90\,\text{cm}\geq50\,\text{cm}\), round up to \(2\,\text{m}\).

a) \(13\,\text{km}\) b) \(4\,\text{m}\) c) \(10\,\text{km}\) d) \(2\,\text{m}\)

- How many hours are in a full day? - Separate the time into one day and the additional hours. - Add the hours in the day to the remaining hours.

1. Convert the day to hours: \(1\,\text{day} = 24\,\text{hr}\). 2. Add the remaining hours: \(24\,\text{hr} + 9\,\text{hr} = 33\,\text{hr}\).

The trip lasts \(33\) hours altogether.

- Convert both measurements to the same unit. - One meter equals \(100\) centimeters. - Compare only after both sides are in centimeters.

1. Use \(1\,\text{m} = 100\,\text{cm}\). 2. \(3\,\text{m} = 300\,\text{cm}\), so the measurements are equal. 3. \(5\,\text{m} = 500\,\text{cm}\), and \(500 < 520\). 4. \(7\,\text{m} = 700\,\text{cm}\), and \(800 > 700\).

a) \(=\) b) \(<\) c) \(>\)

- How many centimeters are in one meter? - Converting meters to centimeters makes the number larger. - Converting centimeters to meters makes the number smaller. - For a mixed measurement, convert the meters first and then add the centimeters.

1. Use \(1\,\text{m} = 100\,\text{cm}\). 2. For a), \(6 \times 100 = 600\,\text{cm}\). 3. For b), \(900 \div 100 = 9\,\text{m}\). 4. For c), \(4 \times 100 + 20 = 420\,\text{cm}\). 5. For d), \(10 \times 100 = 1000\,\text{cm}\).

a) \(600\,\text{cm}\) b) \(9\,\text{m}\) c) \(420\,\text{cm}\) d) \(1000\,\text{cm}\)

- Convert both sides to the same unit. - Change meters to centimeters before comparing. - Pay close attention to the tens digit in part c).

1. For a), \(5\,\text{m} = 500\,\text{cm}\), and \(500 > 50\). 2. For b), \(3\,\text{m} = 300\,\text{cm}\), so the measurements are equal. 3. For c), \(2\,\text{m}\ 5\,\text{cm} = 205\,\text{cm}\), and \(205 < 250\). 4. For d), \(8\,\text{m} = 800\,\text{cm}\), and \(800 < 801\).

a) \(>\) b) \(=\) c) \(<\) d) \(<\)

- One ton equals \(2000\) pounds. - When converting to a smaller unit, the numerical value becomes larger. - Split \(5000\) pounds into groups of \(2000\) and a remainder.

1. a) Since \(2000\,\text{lb} = 1\,\text{ton}\), \(8000 \div 2000 = 4\), so \(8000\,\text{lb} = 4\,\text{tons}\). 2. b) \(9 \times 2000\,\text{lb} = 18{,}000\,\text{lb}\). 3. c) \(5000\,\text{lb}\) contains \(4000\,\text{lb} = 2\,\text{tons}\), with \(1000\,\text{lb}\) remaining. 4. d) \(6 \times 2000\,\text{lb} + 80\,\text{lb} = 12{,}080\,\text{lb}\).

a) \(4\,\text{tons}\) b) \(18{,}000\,\text{lb}\) c) \(2\,\text{tons}\ 1000\,\text{lb}\) d) \(12{,}080\,\text{lb}\)

- Convert both sides to the same unit before comparing. - One ton equals \(2000\) pounds. - One pound equals \(16\) ounces.

1. a) \(3\,\text{tons} = 6000\,\text{lb}\). Since \(6000 > 300\), \(3\,\text{tons} > 300\,\text{lb}\). 2. b) \(2\,\text{tons}\ 1200\,\text{lb} = 4000\,\text{lb} + 1200\,\text{lb} = 5200\,\text{lb}\), so the weights are equal. 3. c) \(1\,\text{ton}\ 5\,\text{lb} = 2005\,\text{lb}\). Since \(2005 < 2050\), the first weight is less. 4. d) Since \(1\,\text{lb} = 16\,\text{oz}\), \(500\,\text{lb} = 500 \times 16\,\text{oz} = 8000\,\text{oz}\), so the weights are equal.

a) \(>\) b) \(=\) c) \(<\) d) \(=\)

- Can you compare the measurements directly, or must you first use the same unit? - It is often helpful to convert the larger unit to the smaller unit. - Recall how many centimeters are in a meter and how many millimeters are in a centimeter.

1. Convert the larger unit in each pair to the smaller unit. 2. a) \(5\,\text{km}=5000\,\text{m}\), and \(5000>500\), so \(5\,\text{km}>500\,\text{m}\). 3. b) \(2\,\text{km}=2000\,\text{m}\), so \(2000\,\text{m}=2\,\text{km}\). 4. c) \(60\,\text{m}=6000\,\text{cm}\), and \(6000>600\), so \(60\,\text{m}>600\,\text{cm}\). 5. d) \(10\,\text{cm}=100\,\text{mm}\), and \(100<1000\), so \(10\,\text{cm}<1000\,\text{mm}\). 6. e) \(4\,\text{km}=4000\,\text{m}\), and \(4000<4400\), so \(4\,\text{km}<4400\,\text{m}\).

a) \(>\) b) \(=\) c) \(>\) d) \(<\) e) \(<\)

- How many meters are in \(1\,\text{km}\)? - Does the numerical value become larger or smaller when you convert kilometers to meters? - Use place value to multiply or divide by \(1000\).

1. Convert kilometers to meters by multiplying by \(1000\): a) \(7 \times 1000=7000\) b) \(26 \times 1000=26{,}000\) c) \(400 \times 1000=400{,}000\) 2. Convert meters to kilometers by dividing by \(1000\): d) \(3000 \div 1000=3\) e) \(80{,}000 \div 1000=80\)

a) \(7000\,\text{m}\) b) \(26{,}000\,\text{m}\) c) \(400{,}000\,\text{m}\) d) \(3\,\text{km}\) e) \(80\,\text{km}\)

- Use \(100\,\text{cm} = 1\,\text{m}\). - Convert each part of a mixed measurement separately. - Check whether the requested unit is larger or smaller than the given unit.

1. a) \(407\,\text{cm}\) contains \(4\) groups of \(100\,\text{cm}\) with \(7\,\text{cm}\) remaining, so \(407\,\text{cm} = 4\,\text{m}\ 7\,\text{cm}\). 2. b) \(2\,\text{m} = 200\,\text{cm}\). Adding \(50\,\text{cm}\) gives \(250\,\text{cm}\). 3. c) \(1\,\text{m} = 100\,\text{cm}\). Adding \(34\,\text{cm}\) gives \(134\,\text{cm}\). 4. d) Since \(100\,\text{cm} = 1\,\text{m}\), \(800\,\text{cm} = 8\,\text{m}\).

a) \(4\,\text{m}\ 7\,\text{cm}\) b) \(250\,\text{cm}\) c) \(134\,\text{cm}\) d) \(8\,\text{m}\)

- Use \(100\,\text{cm} = 1\,\text{m}\) and \(1000\,\text{m} = 1\,\text{km}\). - Divide by the conversion factor to find the number of larger units. - The remainder stays in the smaller unit.

1. a) Since \(100\,\text{cm} = 1\,\text{m}\), \(504\,\text{cm} = 5\,\text{m}\ 4\,\text{cm}\). 2. b) Since \(100\,\text{cm} = 1\,\text{m}\), \(820\,\text{cm} = 8\,\text{m}\ 20\,\text{cm}\). 3. c) Since \(1000\,\text{m} = 1\,\text{km}\), \(1530\,\text{m} = 1\,\text{km}\ 530\,\text{m}\). 4. d) Since \(1000\,\text{m} = 1\,\text{km}\), \(2009\,\text{m} = 2\,\text{km}\ 9\,\text{m}\).

a) \(5\,\text{m}\ 4\,\text{cm}\) b) \(8\,\text{m}\ 20\,\text{cm}\) c) \(1\,\text{km}\ 530\,\text{m}\) d) \(2\,\text{km}\ 9\,\text{m}\)

- One liter equals \(1000\) milliliters. - Divide the number of milliliters into groups of \(1000\). - The remainder stays in milliliters.

1. Use \(1000\,\text{mL} = 1\,\text{L}\). 2. a) \(3600\,\text{mL}\) has \(3\) groups of \(1000\,\text{mL}\) and \(600\,\text{mL}\) remaining, so it is \(3\,\text{L}\ 600\,\text{mL}\). 3. b) \(5900\,\text{mL} = 5\,\text{L}\ 900\,\text{mL}\). 4. c) \(1240\,\text{mL} = 1\,\text{L}\ 240\,\text{mL}\). 5. d) \(700\,\text{mL} = 0\,\text{L}\ 700\,\text{mL}\).

a) \(3\,\text{L}\ 600\,\text{mL}\) b) \(5\,\text{L}\ 900\,\text{mL}\) c) \(1\,\text{L}\ 240\,\text{mL}\) d) \(0\,\text{L}\ 700\,\text{mL}\)

- One meter equals \(100\) centimeters. - Count the complete groups of \(100\) centimeters. - The remainder is the number of centimeters.

1. Use \(100\,\text{cm} = 1\,\text{m}\). 2. a) \(678\,\text{cm} = 6\,\text{m}\ 78\,\text{cm}\). 3. b) \(409\,\text{cm} = 4\,\text{m}\ 9\,\text{cm}\). 4. c) \(1050\,\text{cm} = 10\,\text{m}\ 50\,\text{cm}\). 5. d) Because \(82\,\text{cm}\) is less than \(1\) meter, it is \(0\,\text{m}\ 82\,\text{cm}\).

a) \(6\,\text{m}\ 78\,\text{cm}\) b) \(4\,\text{m}\ 9\,\text{cm}\) c) \(10\,\text{m}\ 50\,\text{cm}\) d) \(0\,\text{m}\ 82\,\text{cm}\)

- One kilometer equals \(1000\) meters. - Complete groups of \(1000\) meters become kilometers. - Keep zeros in their correct place values.

1. Use \(1\,\text{km} = 1000\,\text{m}\). 2. \(6842\,\text{m}\) contains \(6\) kilometers with \(842\) meters remaining, so it is \(6\,\text{km}\ 842\,\text{m}\). 3. \(4030\,\text{m} = 4\,\text{km}\ 30\,\text{m}\). 4. \(9009\,\text{m} = 9\,\text{km}\ 9\,\text{m}\). 5. \(15\,\text{km}\ 75\,\text{m} = 15 \times 1000\,\text{m} + 75\,\text{m} = 15{,}075\,\text{m}\).

a) \(6\,\text{km}\ 842\,\text{m}\) b) \(4\,\text{km}\ 30\,\text{m}\) c) \(9\,\text{km}\ 9\,\text{m}\) d) \(15{,}075\,\text{m}\)

- Identify whether each length already uses one unit or combines two units. - For a mixed-unit length, convert the larger unit and add the remaining smaller units.

1. \(7\,\text{km}\) already uses one unit, so it remains \(7\,\text{km}\). 2. \(3\,\text{m}\ 50\,\text{cm}=300\,\text{cm}+50\,\text{cm}=350\,\text{cm}\). 3. \(12\,\text{cm}\ 4\,\text{mm}=120\,\text{mm}+4\,\text{mm}=124\,\text{mm}\). 4. \(450\,\text{m}\) already uses one unit, so it remains \(450\,\text{m}\). 5. \(1\,\text{km}\ 5\,\text{m}=1000\,\text{m}+5\,\text{m}=1005\,\text{m}\).

\(7\,\text{km}\); \(350\,\text{cm}\); \(124\,\text{mm}\); \(450\,\text{m}\); \(1005\,\text{m}\)

- Convert both sides of each comparison to the smaller unit. - Recall the conversion factors for pounds and ounces, hours and minutes, feet and inches, and tons and pounds. - Pay close attention to place value when the smaller-unit amount has only one or two digits.

1. a) \(3\,\text{lb}\ 4\,\text{oz}=48\,\text{oz}+4\,\text{oz}=52\,\text{oz}\). Since \(52<56\), \(3\,\text{lb}\ 4\,\text{oz}<56\,\text{oz}\). 2. b) \(2\,\text{hr}\ 5\,\text{min}=120\,\text{min}+5\,\text{min}=125\,\text{min}\), so the measurements are equal. 3. c) \(5\,\text{ft}\ 7\,\text{in.}=60\,\text{in.}+7\,\text{in.}=67\,\text{in.}\), so the measurements are equal. 4. d) \(1\,\text{ton}\ 500\,\text{lb}=2000\,\text{lb}+500\,\text{lb}=2500\,\text{lb}\). Since \(2500>2050\), \(1\,\text{ton}\ 500\,\text{lb}>2050\,\text{lb}\).

a) \(3\,\text{lb}\ 4\,\text{oz}<56\,\text{oz}\) b) \(2\,\text{hr}\ 5\,\text{min}=125\,\text{min}\) c) \(5\,\text{ft}\ 7\,\text{in.}=67\,\text{in.}\) d) \(1\,\text{ton}\ 500\,\text{lb}>2050\,\text{lb}\)

- How many grams are in \(1\,\text{kg}\)? - For a mixed-unit mass, convert the kilograms first and then add the remaining grams. - What does one-fourth of a kilogram mean?

1. Use \(1\,\text{kg}=1000\,\text{g}\). 2. a) \(7 \times 1000\,\text{g}=7000\,\text{g}\). 3. b) \(15 \times 1000\,\text{g}=15{,}000\,\text{g}\). 4. c) \(3 \times 1000\,\text{g}+250\,\text{g}=3250\,\text{g}\). 5. d) \(20 \times 1000\,\text{g}+5\,\text{g}=20{,}005\,\text{g}\). 6. e) Since \(1000\,\text{g} \div 4=250\,\text{g}\), \(\frac{1}{4}\,\text{kg}=250\,\text{g}\).

a) \(7000\,\text{g}\) b) \(15{,}000\,\text{g}\) c) \(3250\,\text{g}\) d) \(20{,}005\,\text{g}\) e) \(250\,\text{g}\)

- How many pounds are in \(1\,\text{ton}\)? - How many ounces are in \(1\,\text{lb}\)? - Does converting to a smaller unit make the numerical value larger or smaller? - For a mixed-unit weight, convert the larger unit first and then add.

1. Convert tons to pounds by multiplying by \(2000\): a) \(9 \times 2000=18{,}000\) b) \(14 \times 2000=28{,}000\) 2. Convert pounds to tons by dividing by \(2000\): c) \(12{,}000 \div 2000=6\) d) \(64{,}000 \div 2000=32\) 3. Convert the mixed-unit weight to ounces: \(5\,\text{lb}=80\,\text{oz}\), and \(80+2=82\).

a) \(18{,}000\,\text{lb}\) b) \(28{,}000\,\text{lb}\) c) \(6\,\text{tons}\) d) \(32\,\text{tons}\) e) \(82\,\text{oz}\)

- How many pounds are in \(1\,\text{ton}\)? - Use zeros as place-value placeholders when you write each product. - Convert the tons first, then add the remaining pounds.

1. Use \(1\,\text{ton}=2000\,\text{lb}\). 2. Multiply the number of tons by \(2000\), then add the remaining pounds. 3. a) \(4 \times 2000\,\text{lb}+250\,\text{lb}=8250\,\text{lb}\). 4. b) \(6 \times 2000\,\text{lb}+80\,\text{lb}=12{,}080\,\text{lb}\). 5. c) \(2 \times 2000\,\text{lb}+5\,\text{lb}=4005\,\text{lb}\). 6. d) \(15 \times 2000\,\text{lb}+15\,\text{lb}=30{,}015\,\text{lb}\).

a) \(8250\,\text{lb}\) b) \(12{,}080\,\text{lb}\) c) \(4005\,\text{lb}\) d) \(30{,}015\,\text{lb}\)

- Each conversion uses a factor of \(1000\). - Find the number of complete groups of \(1000\). - The remainder stays in the smaller unit.

1. Use \(1000\,\text{m} = 1\,\text{km}\), \(1000\,\text{g} = 1\,\text{kg}\), and \(1000\,\text{mL} = 1\,\text{L}\). 2. a) \(3750\,\text{m} = 3\,\text{km}\ 750\,\text{m}\). 3. b) \(8005\,\text{g} = 8\,\text{kg}\ 5\,\text{g}\). 4. c) \(12{,}040\,\text{mL} = 12\,\text{L}\ 40\,\text{mL}\). 5. d) \(50{,}002\,\text{m} = 50\,\text{km}\ 2\,\text{m}\).

a) \(3\,\text{km}\ 750\,\text{m}\) b) \(8\,\text{kg}\ 5\,\text{g}\) c) \(12\,\text{L}\ 40\,\text{mL}\) d) \(50\,\text{km}\ 2\,\text{m}\)

- One ton equals \(2000\) pounds. - Convert tons to pounds before adding the remaining pounds. - For the last equation, think about how many pounds must be added to reach one ton.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\). 2. \(3 \times 2000\,\text{lb} + 5\,\text{lb} = 6005\,\text{lb}\). 3. \(8070\,\text{lb}\) contains \(8000\,\text{lb} = 4\) tons with \(70\,\text{lb}\) remaining. 4. \(10 \times 2000\,\text{lb} + 10\,\text{lb} = 20{,}010\,\text{lb}\). 5. \(2000\,\text{lb} - 1200\,\text{lb} = 800\,\text{lb}\).

1. \(6005\,\text{lb}\) 2. \(4\,\text{tons}\ 70\,\text{lb}\) 3. \(20{,}010\,\text{lb}\) 4. \(800\,\text{lb}\)

- How many ounces are in \(1\,\text{lb}\)? - How many pounds are in \(1\,\text{ton}\)? - When you convert to a larger unit, the numerical value becomes smaller. Which divisor should you use?

1. Convert ounces to pounds by dividing by \(16\): \(144 \div 16=9\), so \(144\,\text{oz}=9\,\text{lb}\). 2. \(384 \div 16=24\), so \(384\,\text{oz}=24\,\text{lb}\). 3. Convert pounds to tons by dividing by \(2000\): \(10{,}000 \div 2000=5\), so \(10{,}000\,\text{lb}=5\,\text{tons}\). 4. \(220{,}000 \div 2000=110\), so \(220{,}000\,\text{lb}=110\,\text{tons}\).

a) \(9\,\text{lb}\) b) \(24\,\text{lb}\) c) \(5\,\text{tons}\) d) \(110\,\text{tons}\)

- One kilogram equals \(1000\) grams. - Separate each number into thousands and the remainder. - Pay close attention to zeros in the tens and hundreds places.

1. Use \(1000\,\text{g} = 1\,\text{kg}\). Complete groups of \(1000\) become kilograms, and the remainder stays in grams. 2. a) \(1405\,\text{g} = 1\,\text{kg}\ 405\,\text{g}\). 3. b) \(3060\,\text{g} = 3\,\text{kg}\ 60\,\text{g}\). 4. c) \(7008\,\text{g} = 7\,\text{kg}\ 8\,\text{g}\). 5. d) \(12{,}500\,\text{g} = 12\,\text{kg}\ 500\,\text{g}\). 6. e) \(20{,}030\,\text{g} = 20\,\text{kg}\ 30\,\text{g}\).

a) \(1\,\text{kg}\ 405\,\text{g}\) b) \(3\,\text{kg}\ 60\,\text{g}\) c) \(7\,\text{kg}\ 8\,\text{g}\) d) \(12\,\text{kg}\ 500\,\text{g}\) e) \(20\,\text{kg}\ 30\,\text{g}\)

- One meter equals \(100\) centimeters. - One kilometer equals \(1000\) meters. - Keep zeros in their correct place values.

1. a) \(408\,\text{cm}\) contains \(4\) complete meters and \(8\) centimeters, so \(408\,\text{cm} = 4\,\text{m}\ 8\,\text{cm}\). 2. b) \(1050\,\text{m}\) contains \(1\) kilometer and \(50\) meters, so \(1050\,\text{m} = 1\,\text{km}\ 50\,\text{m}\). 3. c) \(7\,\text{m}\ 3\,\text{cm} = 7 \times 100\,\text{cm} + 3\,\text{cm} = 703\,\text{cm}\). 4. d) \(5\,\text{km}\ 20\,\text{m} = 5 \times 1000\,\text{m} + 20\,\text{m} = 5020\,\text{m}\).

a) \(4\,\text{m}\ 8\,\text{cm}\) b) \(1\,\text{km}\ 50\,\text{m}\) c) \(703\,\text{cm}\) d) \(5020\,\text{m}\)

- One liter equals \(1000\) milliliters. - Separate the number into thousands and the remainder. - The complete thousands are liters.

1. Use \(1000\,\text{mL} = 1\,\text{L}\). 2. The amount \(15{,}700\,\text{mL}\) contains \(15\) complete liters with \(700\,\text{mL}\) remaining. 3. Therefore, \(15{,}700\,\text{mL} = 15\,\text{L}\ 700\,\text{mL}\).

The container holds \(15\,\text{L}\ 700\,\text{mL}\).

- One ton equals \(2000\) pounds. - Find the greatest multiple of \(2000\) that does not exceed \(24{,}040\). - The difference is the remaining number of pounds.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\). 2. Twelve tons equals \(12 \times 2000\,\text{lb} = 24{,}000\,\text{lb}\). 3. The remaining weight is \(24{,}040\,\text{lb} - 24{,}000\,\text{lb} = 40\,\text{lb}\). 4. Therefore, the load weighs \(12\,\text{tons}\ 40\,\text{lb}\).

The load weighs \(12\,\text{tons}\ 40\,\text{lb}\).

- How many months are in one year? - Convert the full years to months first. - Add the extra months.

1. Anna: \(2 \times 12 + 3 = 24 + 3 = 27\) months. 2. Ben: \(4 \times 12 + 8 = 48 + 8 = 56\) months. 3. Clara: \(6 \times 12 + 1 = 72 + 1 = 73\) months.

Anna: \(27\) months Ben: \(56\) months Clara: \(73\) months

- How many years are in \(1\) century? - What number should you multiply by to convert centuries to years? - Compare your result with the student's number.

1. One century is \(100\) years, so multiply: \(12 \times 100=1200\). 2. Therefore, \(12\) centuries is \(1200\) years. 3. The student likely multiplied by \(10\) instead of \(100\), or omitted one zero.

\(12\) centuries is \(1200\,\text{years}\). The student likely multiplied by \(10\) instead of \(100\).

- How many days are in one week? - Find the number of days in the full weeks. - Add the remaining days.

1. Convert the weeks to days: \(7 \times 7 = 49\) days. 2. Add the extra days: \(49 + 4 = 53\) days.

The circus stays for \(53\) days.

- How many seconds are in \(1\) minute? - First find the number of seconds in the full minutes. - Add the remaining seconds at the end. - You can break apart a product such as \(12 \times 60\) into easier partial products.

1. Use \(1\,\text{min}=60\,\text{s}\). 2. Convert the full minutes to seconds, then add the remaining seconds. 3. a) \(5 \times 60\,\text{s}+20\,\text{s}=300\,\text{s}+20\,\text{s}=320\,\text{s}\). 4. b) \(12 \times 60\,\text{s}+45\,\text{s}=720\,\text{s}+45\,\text{s}=765\,\text{s}\). 5. c) \(20 \times 60\,\text{s}+8\,\text{s}=1200\,\text{s}+8\,\text{s}=1208\,\text{s}\).

a) \(320\,\text{s}\) b) \(765\,\text{s}\) c) \(1208\,\text{s}\)

- One hour equals \(60\) minutes. - Find the greatest multiple of \(60\) that does not exceed each number. - The difference is the remaining number of minutes.

1. Divide the number of minutes by \(60\). The quotient is the number of complete hours, and the remainder is the number of minutes. 2. a) \(145 = 2 \times 60 + 25\), so \(145\,\text{min} = 2\,\text{hr}\ 25\,\text{min}\). 3. b) \(310 = 5 \times 60 + 10\), so \(310\,\text{min} = 5\,\text{hr}\ 10\,\text{min}\). 4. c) \(605 = 10 \times 60 + 5\), so \(605\,\text{min} = 10\,\text{hr}\ 5\,\text{min}\).

a) \(2\,\text{hr}\ 25\,\text{min}\) b) \(5\,\text{hr}\ 10\,\text{min}\) c) \(10\,\text{hr}\ 5\,\text{min}\)

- Recall how many smaller time units are in each larger unit. - Write the conversion factor for hours, minutes, seconds, months, or years. - Use multiplication or division to check each statement.

1. a) Since \(180 \div 60=3\), the conversion is true. 2. b) Since \(2 \times 12=24\), \(2\) years is \(24\) months. The conversion is false. 3. c) Since \(5 \times 60=300\), \(5\) minutes is \(300\) seconds. The conversion is false. 4. d) Since \(72 \div 24=3\), the conversion is true.

a) True b) False c) False d) True

- What is the next larger unit after seconds? After minutes? - How many groups of \(60\) are in the given seconds or minutes? - How many hours are in \(1\) day? - How many months are in \(1\) year?

1. Convert seconds to minutes: \(420 \div 60=7\), so \(420\,\text{s}=7\,\text{min}\). 2. Convert minutes to hours: \(600 \div 60=10\), so \(600\,\text{min}=10\,\text{hr}\). 3. Convert hours to days: \(96 \div 24=4\), so \(96\,\text{hr}=4\,\text{days}\). 4. Convert months to years: \(120 \div 12=10\), so \(120\,\text{months}=10\,\text{years}\).

a) \(7\,\text{min}\) b) \(10\,\text{hr}\) c) \(4\,\text{days}\) d) \(10\,\text{years}\)

- One year has \(12\) months. - Find the number of complete groups of \(12\). - The remainder is the number of additional months.

1. Divide each number of months by \(12\). The quotient gives complete years, and the remainder gives additional months. 2. a) \(32 = 2 \times 12 + 8\), so \(32\) months is \(2\) years \(8\) months. 3. b) \(50 = 4 \times 12 + 2\), so \(50\) months is \(4\) years \(2\) months. 4. c) \(75 = 6 \times 12 + 3\), so \(75\) months is \(6\) years \(3\) months. 5. d) \(100 = 8 \times 12 + 4\), so \(100\) months is \(8\) years \(4\) months.

a) \(2\) years \(8\) months b) \(4\) years \(2\) months c) \(6\) years \(3\) months d) \(8\) years \(4\) months

- One meter equals \(100\) centimeters. - Convert the brother's claim back to centimeters. - Pay attention to the zero in \(405\).

1. Emma's brother's measurement equals \(4\,\text{m}\ 50\,\text{cm} = 400\,\text{cm} + 50\,\text{cm} = 450\,\text{cm}\). 2. Since \(450\,\text{cm} \ne 405\,\text{cm}\), he is not correct. 3. The measurement \(405\,\text{cm}\) contains \(4\) complete meters with \(5\) centimeters remaining, so \(405\,\text{cm} = 4\,\text{m}\ 5\,\text{cm}\).

He is not correct. \(405\,\text{cm} = 4\,\text{m}\ 5\,\text{cm}\), not \(4\,\text{m}\ 50\,\text{cm}\).

- First identify how many smaller units make one whole unit. - Divide the whole-unit amount by the fraction's denominator. - One dollar equals \(100\) cents.

1. Since \(1\,\text{m} = 100\,\text{cm}\), \(\frac{1}{2}\,\text{m} = 100\,\text{cm} \div 2 = 50\,\text{cm}\). 2. Since \(1\,\text{kg} = 1000\,\text{g}\), \(\frac{1}{4}\,\text{kg} = 1000\,\text{g} \div 4 = 250\,\text{g}\). 3. Since one dollar equals \(100\) cents, \(\frac{1}{10}\) of a dollar is \(100\,\text{cents} \div 10 = 10\,\text{cents}\).

a) \(50\,\text{cm}\) b) \(250\,\text{g}\) c) \(10\,\text{cents}\)

- Choose the unit that makes each measurement reasonable. - Then use the next smaller unit from the given list. - Millimeters are already the smallest unit in the list.

1. An elephant’s shoulder height is reasonably measured in meters: \(4\,\text{m} = 400\,\text{cm}\). 2. A wood ant’s length is reasonably measured in millimeters: \(6\,\text{mm}\). Millimeters are already the smallest unit in the list. 3. A blue whale’s length is reasonably measured in meters: \(33\,\text{m} = 3300\,\text{cm}\). 4. A long migration distance is reasonably measured in kilometers: more than \(10{,}000\,\text{km}\), which is more than \(10{,}000{,}000\,\text{m}\).

a) \(4\,\text{m} = 400\,\text{cm}\) b) \(6\,\text{mm}\) c) \(33\,\text{m} = 3300\,\text{cm}\) d) More than \(10{,}000\,\text{km}\), which is more than \(10{,}000{,}000\,\text{m}\)

- One liter equals \(1000\) milliliters. - One hour equals \(60\) minutes. - Divide by the denominator, then multiply by the numerator when needed.

1. Since \(1\,\text{L} = 1000\,\text{mL}\), \(\frac{1}{2}\,\text{L} = 500\,\text{mL}\), \(\frac{1}{4}\,\text{L} = 250\,\text{mL}\), and \(\frac{1}{10}\,\text{L} = 100\,\text{mL}\). 2. Since \(1\,\text{hr} = 60\,\text{min}\), \(\frac{1}{2}\,\text{hr} = 30\,\text{min}\), \(\frac{1}{4}\,\text{hr} = 15\,\text{min}\), and \(\frac{3}{4}\,\text{hr} = 3 \times 15\,\text{min} = 45\,\text{min}\).

a) \(500\,\text{mL}\); \(250\,\text{mL}\); \(100\,\text{mL}\) b) \(30\,\text{min}\); \(15\,\text{min}\); \(45\,\text{min}\)

- How many centimeters are in \(1\,\text{m}\)? - Convert the meters first, then add the remaining centimeters. - Pay close attention to place value when the centimeter amount is less than \(10\).

1. Use \(1\,\text{m}=100\,\text{cm}\). 2. a) \(3 \times 100\,\text{cm}+15\,\text{cm}=315\,\text{cm}\). 3. b) \(5 \times 100\,\text{cm}+4\,\text{cm}=504\,\text{cm}\). 4. c) \(10 \times 100\,\text{cm}+60\,\text{cm}=1060\,\text{cm}\). 5. d) \(2 \times 100\,\text{cm}+8\,\text{cm}=208\,\text{cm}\).

a) \(315\,\text{cm}\) b) \(504\,\text{cm}\) c) \(1060\,\text{cm}\) d) \(208\,\text{cm}\)

- Recall how many smaller units are in one larger unit. - Multiply the larger-unit amount by the conversion factor, then add the remaining smaller units. - Use place value carefully when adding a small remainder.

1. a) Since \(1\,\text{yd}=3\,\text{ft}\), \(12 \times 3\,\text{ft}+2\,\text{ft}=38\,\text{ft}\). 2. b) Since \(1\,\text{ft}=12\,\text{in.}\), \(40 \times 12\,\text{in.}+8\,\text{in.}=488\,\text{in.}\). 3. c) Since \(1\,\text{ton}=2000\,\text{lb}\), \(5 \times 2000\,\text{lb}+70\,\text{lb}=10{,}070\,\text{lb}\). 4. d) Since \(1\,\text{lb}=16\,\text{oz}\), \(10 \times 16\,\text{oz}+5\,\text{oz}=165\,\text{oz}\).

a) \(38\,\text{ft}\) b) \(488\,\text{in.}\) c) \(10{,}070\,\text{lb}\) d) \(165\,\text{oz}\)

- Recall how many meters are in \(1\) kilometer. - Separate the full thousands of meters from the remaining meters. - What do the first two digits represent when the distance is split into kilometers and meters?

1. Use \(1000\,\text{m} = 1\,\text{km}\). 2. Separate the distance into \(42{,}000\,\text{m}\) and the remaining \(195\,\text{m}\). 3. Since \(42{,}000\,\text{m} = 42\,\text{km}\), the full distance is \(42\,\text{km}\,195\,\text{m}\).

\(42\,\text{km}\,195\,\text{m}\)

- Find how many smaller units make one of the next larger units. - Divide to find complete larger units. - Write any remainder in the smaller unit.

1. a) Since \(100\,\text{cm} = 1\,\text{m}\), \(940\,\text{cm} = 9\,\text{m}\ 40\,\text{cm}\). 2. b) Since \(1000\,\text{m} = 1\,\text{km}\), \(6200\,\text{m} = 6\,\text{km}\ 200\,\text{m}\). 3. c) Since \(1000\,\text{g} = 1\,\text{kg}\), \(5080\,\text{g} = 5\,\text{kg}\ 80\,\text{g}\). 4. d) Since \(2000\,\text{lb} = 1\,\text{ton}\), \(24{,}400\,\text{lb} = 12\,\text{tons}\ 400\,\text{lb}\).

a) \(9\,\text{m}\ 40\,\text{cm}\) b) \(6\,\text{km}\ 200\,\text{m}\) c) \(5\,\text{kg}\ 80\,\text{g}\) d) \(12\,\text{tons}\ 400\,\text{lb}\)

- Recall how many kilograms equal one metric ton. - Separate each number into complete thousands and leftover kilograms. - Do not treat missing place values as extra kilograms.

1. Use \(1000\,\text{kg} = 1\) metric ton. The number of complete thousands gives the metric tons, and the remaining kilograms stay as kilograms. 2. \(12{,}450\,\text{kg} = 12\) metric tons \(450\,\text{kg}\). 3. \(9070\,\text{kg} = 9\) metric tons \(70\,\text{kg}\). 4. \(25{,}004\,\text{kg} = 25\) metric tons \(4\,\text{kg}\).

\(12\) metric tons \(450\,\text{kg}\); \(9\) metric tons \(70\,\text{kg}\); \(25\) metric tons \(4\,\text{kg}\)

- Recall how many meters are in \(1\) kilometer. - Separate each number into full groups of \(1000\) meters and the remaining meters. - Pay close attention to zeros inside the numbers.

1. Use \(1000\,\text{m} = 1\,\text{km}\). 2. \(14{,}320\,\text{m} = 14\,\text{km}\,320\,\text{m}\). 3. \(5080\,\text{m} = 5\,\text{km}\,80\,\text{m}\). 4. \(40{,}009\,\text{m} = 40\,\text{km}\,9\,\text{m}\).

\(14\,\text{km}\,320\,\text{m}\) \(5\,\text{km}\,80\,\text{m}\) \(40\,\text{km}\,9\,\text{m}\)

- Identify whether the conversion factor is \(100\) or \(1000\). - Complete groups become the larger unit. - The remainder stays in the smaller unit.

1. Since \(1000\,\text{m} = 1\,\text{km}\), \(12{,}005\,\text{m} = 12\,\text{km}\ 5\,\text{m}\). 2. Since \(1000\,\text{mL} = 1\,\text{L}\), \(2080\,\text{mL} = 2\,\text{L}\ 80\,\text{mL}\). 3. Since \(100\) cents equals one dollar, \(505\) cents is \(5\) dollars \(5\) cents. 4. Since \(1000\,\text{g} = 1\,\text{kg}\), \(8070\,\text{g} = 8\,\text{kg}\ 70\,\text{g}\).

\(12\,\text{km}\ 5\,\text{m}\); \(2\,\text{L}\ 80\,\text{mL}\); \(5\,\text{dollars}\ 5\,\text{cents}\); \(8\,\text{kg}\ 70\,\text{g}\)

- Convert every load to pounds. - Remember that a freight train has many cargo cars. - Order the four numerical values from least to greatest.

1. Convert the ton measurements to pounds: \(25\,\text{tons}=25\times 2000=50{,}000\,\text{lb}\), and \(2000\,\text{tons}=4{,}000{,}000\,\text{lb}\). 2. Compare the loads: \(40\,\text{lb}<3000\,\text{lb}<50{,}000\,\text{lb}<4{,}000{,}000\,\text{lb}\). 3. Therefore, the order is B, D, C, A.

B, D, C, A

- How many grams are in \(1\,\text{kg}\)? - Should you multiply or divide when converting from the larger unit to the smaller unit? - How does multiplying or dividing by \(1000\) change the place values?

1. Use \(1\,\text{kg}=1000\,\text{g}\). 2. \(8\,\text{kg}=8 \times 1000=8000\,\text{g}\). 3. \(15{,}000\,\text{g}=15{,}000 \div 1000=15\,\text{kg}\). 4. \(250\,\text{kg}=250 \times 1000=250{,}000\,\text{g}\). 5. \(900{,}000\,\text{g}=900{,}000 \div 1000=900\,\text{kg}\).

The missing values are \(8000\,\text{g}\), \(15\,\text{kg}\), \(250{,}000\,\text{g}\), and \(900\,\text{kg}\).

- One meter equals \(100\) centimeters. - Convert each part of a mixed measurement before adding. - When converting centimeters to meters, count complete groups of \(100\).

1. a) \(15 \times 100\,\text{cm} = 1500\,\text{cm}\). 2. b) \(3 \times 100\,\text{cm} + 7\,\text{cm} = 307\,\text{cm}\). 3. c) \(240\,\text{cm}\) contains \(2\) complete meters with \(40\) centimeters remaining, so it is \(2\,\text{m}\ 40\,\text{cm}\). 4. d) \(10 \times 100\,\text{cm} + 50\,\text{cm} = 1000\,\text{cm} + 50\,\text{cm} = 1050\,\text{cm}\).

a) \(1500\,\text{cm}\) b) \(307\,\text{cm}\) c) \(2\,\text{m}\ 40\,\text{cm}\) d) \(1050\,\text{cm}\)

- Convert each whole unit to the requested smaller unit. - Divide by the denominator. - Multiply by the numerator when it is greater than \(1\).

1. Use \(1\,\text{m} = 100\,\text{cm}\), \(1\,\text{km} = 1000\,\text{m}\), and \(1\,\text{L} = 1000\,\text{mL}\). 2. a) \(100\,\text{cm} \div 2 = 50\,\text{cm}\). 3. b) \(1000\,\text{m} \div 5 = 200\,\text{m}\). 4. c) \(1000\,\text{mL} \div 4 = 250\,\text{mL}\), and \(3 \times 250\,\text{mL} = 750\,\text{mL}\).

a) \(50\,\text{cm}\) b) \(200\,\text{m}\) c) \(750\,\text{mL}\)

- One dollar equals \(100\) cents. - The last two digits show the number of cents. - Use a zero in the tenths place when the remaining number of cents has only one digit.

1. One dollar equals \(100\) cents, so separate each amount into groups of \(100\) cents and the remaining cents. 2. a) \(804\) cents is \(8\) dollars and \(4\) cents, or \(\$8.04\). 3. b) \(2560\) cents is \(25\) dollars and \(60\) cents, or \(\$25.60\). 4. c) \(7003\) cents is \(70\) dollars and \(3\) cents, or \(\$70.03\). 5. d) \(15{,}020\) cents is \(150\) dollars and \(20\) cents, or \(\$150.20\).

a) \(\$8.04\) b) \(\$25.60\) c) \(\$70.03\) d) \(\$150.20\)

- Convert all measurements to the smallest unit shown. - One centimeter equals \(10\) millimeters. - One meter equals \(100\) centimeters. - Compare the numerical values after the units match.

1. For a), convert to millimeters: \(8\), \(80\), \(18\), \(8000\), and \(880\). Order these values from least to greatest. 2. For b), convert to millimeters: \(25\), \(250\), \(52\), \(2000\), and \(50\). Order these values from least to greatest.

a) \(8\,\text{mm} < 18\,\text{mm} < 8\,\text{cm} < 88\,\text{cm} < 8\,\text{m}\) b) \(2\,\text{cm}\ 5\,\text{mm} < 5\,\text{cm} < 52\,\text{mm} < 25\,\text{cm} < 2\,\text{m}\)

- Convert both sides of each comparison to the same unit. - Pay close attention to place value when converting meters. - Check for zeros or reversed digits.

1. For a), \(3\,\text{m}\ 5\,\text{cm} = 305\,\text{cm}\), and \(305 < 350\). 2. For b), \(6\,\text{cm} = 60\,\text{mm}\), so the measurements are equal. 3. For c), \(4\,\text{cm}\ 2\,\text{mm} = 42\,\text{mm}\), and \(420 > 42\). 4. For d), \(1\,\text{m}\ 10\,\text{cm} = 110\,\text{cm}\), and \(110 > 101\).

a) \(<\) b) \(=\) c) \(>\) d) \(>\)

- Make a list with all lengths in the same unit. - Use everyday experience to estimate the order before calculating. - Pay attention to the difference between centimeters and millimeters.

1. Convert all lengths to millimeters: paper clip \(32\,\text{mm}\), toothpick \(65\,\text{mm}\), paintbrush \(180\,\text{mm}\), shoelace \(800\,\text{mm}\), and belt \(1050\,\text{mm}\). 2. Compare the values: \(32 < 65 < 180 < 800 < 1050\).

Paper clip, toothpick, paintbrush, shoelace, belt

- Multiply the numbers while keeping the unit meters. - Every \(1000\) meters makes \(1\) kilometer. - Split each result into thousands and the remaining meters.

1. a) \(4 \times 800\,\text{m} = 3200\,\text{m}\). Since \(3200 = 3000 + 200\), this is \(3\,\text{km}\ 200\,\text{m}\). 2. b) \(3 \times 1200\,\text{m} = 3600\,\text{m}\). Since \(3600 = 3000 + 600\), this is \(3\,\text{km}\ 600\,\text{m}\). 3. c) \(5 \times 450\,\text{m} = 2250\,\text{m}\). Since \(2250 = 2000 + 250\), this is \(2\,\text{km}\ 250\,\text{m}\).

a) \(3200\,\text{m} = 3\,\text{km}\ 200\,\text{m}\) b) \(3600\,\text{m} = 3\,\text{km}\ 600\,\text{m}\) c) \(2250\,\text{m} = 2\,\text{km}\ 250\,\text{m}\)

- Convert \(1\,\text{km}\) to meters. - Ask how many equal shorter distances make \(1000\,\text{m}\). - You can use division or a related multiplication fact.

1. Convert the total distance: \(1\,\text{km} = 1000\,\text{m}\). 2. Divide \(1000\) by each shorter distance: \(1000 \div 250 = 4\), \(1000 \div 100 = 10\), \(1000 \div 500 = 2\), and \(1000 \div 200 = 5\).

a) \(4\) times b) \(10\) times c) \(2\) times d) \(5\) times

- Convert each kilometer measurement to meters first. - Complete each multiplication before finding the missing addend. - Check whether the equation requires subtraction or addition.

1. Use \(1\,\text{km} = 1000\,\text{m}\). 2. For a), \(1000 - 350 = 650\). 3. For b), \(2 \times 400 = 800\), then \(1000 - 800 = 200\). 4. For c), \(1000 - 720 = 280\). 5. For d), \(3 \times 200 = 600\), then \(1000 - 600 = 400\).

a) \(650\,\text{m}\) b) \(200\,\text{m}\) c) \(280\,\text{m}\) d) \(400\,\text{m}\)

- Convert both lengths to the same unit before comparing. - Write each meter measurement through the hundredths place. - Compare the ones, then tenths, then hundredths.

1. a) \(160\,\text{cm} = 1.60\,\text{m}\). Since \(1.60 > 1.06\), \(160\,\text{cm} > 1.06\,\text{m}\). 2. b) \(203\,\text{cm} = 2.03\,\text{m}\). Since \(2.03 < 2.30\), \(203\,\text{cm} < 2.30\,\text{m}\). 3. c) \(80\,\text{cm} = 0.80\,\text{m}\), so the lengths are equal. 4. d) \(405\,\text{cm} = 4.05\,\text{m}\). Since \(4.50 > 4.05\), \(4.50\,\text{m} > 405\,\text{cm}\).

a) \(>\) b) \(<\) c) \(=\) d) \(>\)

- Measurements must use the same unit before you add or subtract them. - Decide whether each expression requires addition or subtraction. - Use \(1\,\text{m} = 100\,\text{cm}\).

1. Convert the meter measurements: \(2.40\,\text{m} = 240\,\text{cm}\), \(3.05\,\text{m} = 305\,\text{cm}\), and \(1.12\,\text{m} = 112\,\text{cm}\). 2. a) \(135\,\text{cm} + 240\,\text{cm} = 375\,\text{cm}\). 3. b) \(305\,\text{cm} - 80\,\text{cm} = 225\,\text{cm}\). 4. c) \(112\,\text{cm} + 88\,\text{cm} = 200\,\text{cm}\).

a) \(375\,\text{cm}\) b) \(225\,\text{cm}\) c) \(200\,\text{cm}\)

- Multiply before adding or subtracting. - Express both terms in the same unit before combining them. - To convert centimeters to meters, use \(100\,\text{cm} = 1\,\text{m}\).

1. a) Multiply first: \(8 \times 25\,\text{cm} = 200\,\text{cm}\). Convert: \(200\,\text{cm} = 2\,\text{m}\). Then add: \(2\,\text{m} + 1.75\,\text{m} = 3.75\,\text{m}\). 2. b) Multiply first: \(5 \times 1.20\,\text{m} = 6.00\,\text{m}\). Convert: \(350\,\text{cm} = 3.50\,\text{m}\). Then subtract: \(6.00\,\text{m} - 3.50\,\text{m} = 2.50\,\text{m}\).

a) \(3.75\,\text{m}\) b) \(2.50\,\text{m}\)

- Remember that \(1\,\text{km} = 1000\,\text{m}\). - After converting, where do the \(8\) meters belong in the place-value positions? - A place-value chart can help you keep the needed zeros.

1. Convert the kilometers to meters: \(14 \times 1000 = 14{,}000\,\text{m}\). 2. Add the remaining distance. Adding \(8\,\text{m}\) gives \(14{,}008\,\text{m}\).

The bike path will be \(14{,}008\,\text{m}\) long.

- Multiply the mass of one crate by \(5\). - Recall that \(1000\,\text{kg}\) equals \(1\) metric ton. - Recall that \(1\,\text{kg} = 1000\,\text{g}\).

1. Find the total mass in kilograms: \(5 \times 200\,\text{kg} = 1000\,\text{kg}\). 2. Since \(1000\,\text{kg} = 1\) metric ton, the total is \(1\) metric ton. 3. Each kilogram is \(1000\,\text{g}\). Converting \(1000\,\text{kg}\) gives \(1{,}000{,}000\,\text{g}\).

a) \(1000\,\text{kg}\) b) \(1\) metric ton c) \(1{,}000{,}000\,\text{g}\)

- Convert both times in each pair to the same unit. - Use the smaller unit when it makes comparison easier. - One minute equals \(60\) seconds.

1. \(180\,\text{s} = 3\,\text{min}\), so the measurements are equal. 2. \(1\) hour \(15\) minutes is \(75\) minutes, and \(75 < 80\). 3. \(2\) minutes \(10\) seconds is \(130\) seconds, and \(130 > 120\). 4. \(3\) hours \(10\) minutes is \(190\) minutes, and \(200 > 190\).

a) \(=\) b) \(<\) c) \(>\) d) \(>\)

- How many seconds are in \(1\) minute? - Express both times in seconds before comparing. - Subtract to find the difference.

1. Convert Lucas's time to seconds: \(1\,\text{min} = 60\,\text{s}\), so \(60\,\text{s} + 15\,\text{s} = 75\,\text{s}\). 2. Compare the times: \(85\,\text{s} > 75\,\text{s}\), so Emma jumps longer. 3. Find the difference: \(85\,\text{s} - 75\,\text{s} = 10\,\text{s}\).

Emma jumps longer by \(10\,\text{s}\).

- How many seconds are in one minute? - Convert both lap times to the same unit before comparing them. - For a race time, what does “faster” mean about the number of seconds?

1. Convert the second-lap time to seconds: \(1\,\text{minute} = 60\,\text{seconds}\), so \(60\,\text{seconds} + 15\,\text{seconds} = 75\,\text{seconds}\). 2. Compare the times. Since \(75\,\text{seconds} < 85\,\text{seconds}\), the second lap is faster. 3. Find the difference: \(85\,\text{seconds} - 75\,\text{seconds} = 10\,\text{seconds}\).

a) The second lap takes \(75\) seconds. b) The second lap is faster by \(10\) seconds.

- How many groups of \(60\) minutes fit into \(230\) minutes? - To compare the two times, convert them to the same unit.

1. Divide \(230\) minutes by \(60\): \(230 \div 60 = 3\) remainder \(50\). Therefore, \(230\) minutes is \(3\) hours \(50\) minutes. 2. Convert the second train’s time to minutes: \(4 \times 60 + 10 = 250\) minutes. 3. Compare the times: \(230\,\text{minutes} < 250\,\text{minutes}\), so the express train is faster.

a) \(230\) minutes is \(3\) hours \(50\) minutes. b) The express train is faster.

- First calculate each total in the smaller unit. - Use \(1000\,\text{g} = 1\,\text{kg}\), \(1000\,\text{mL} = 1\,\text{L}\), and \(100\,\text{cm} = 1\,\text{m}\). - Separate each total into the greatest possible number of larger units and the remainder.

1. a) \(5 \times 300\,\text{g} = 1500\,\text{g} = 1\,\text{kg}\ 500\,\text{g}\). 2. b) \(8 \times 125\,\text{g} = 1000\,\text{g} = 1\,\text{kg}\ 0\,\text{g}\). 3. c) \(4 \times 450\,\text{mL} = 1800\,\text{mL} = 1\,\text{L}\ 800\,\text{mL}\). 4. d) \(6 \times 50\,\text{cm} = 300\,\text{cm} = 3\,\text{m}\ 0\,\text{cm}\).

a) \(1\,\text{kg}\ 500\,\text{g}\) b) \(1\,\text{kg}\ 0\,\text{g}\) c) \(1\,\text{L}\ 800\,\text{mL}\) d) \(3\,\text{m}\ 0\,\text{cm}\)

- Convert both time intervals to minutes before comparing. - One hour equals \(60\) minutes.

1. a) \(1\,\text{hr}\ 20\,\text{min} = 60\,\text{min} + 20\,\text{min} = 80\,\text{min}\), so the intervals are equal. 2. b) \(2\,\text{hr}\ 15\,\text{min} = 120\,\text{min} + 15\,\text{min} = 135\,\text{min}\). Since \(130 < 135\), the first interval is shorter. 3. c) \(1\,\text{hr}\ 55\,\text{min} = 60\,\text{min} + 55\,\text{min} = 115\,\text{min}\). Since \(115 > 110\), the first interval is longer.

a) \(80\,\text{min} = 1\,\text{hr}\ 20\,\text{min}\) b) \(130\,\text{min} < 2\,\text{hr}\ 15\,\text{min}\) c) \(1\,\text{hr}\ 55\,\text{min} > 110\,\text{min}\)

- Use \(1\,\text{m} = 100\,\text{cm}\). - Convert the fractional and decimal meter measurements to centimeters. - Put all measurements in one unit before sorting them.

1. Since \(1\,\text{m} = 100\,\text{cm}\), \(\frac{1}{2}\,\text{m} = 0.50\,\text{m} = 50\,\text{cm}\). 2. \(\frac{1}{10}\,\text{m} = 0.10\,\text{m} = 10\,\text{cm}\). 3. \(\frac{1}{5}\,\text{m} = 0.20\,\text{m} = 20\,\text{cm}\).

Group 1: \(\frac{1}{2}\,\text{m} = 0.50\,\text{m} = 50\,\text{cm}\) Group 2: \(\frac{1}{10}\,\text{m} = 0.10\,\text{m} = 10\,\text{cm}\) Group 3: \(\frac{1}{5}\,\text{m} = 0.20\,\text{m} = 20\,\text{cm}\)

- Think about the range of gram amounts that round to \(6000\,\text{g}\). - Use the hundreds digit to decide whether the thousands digit stays the same or increases.

1. Since \(1000\,\text{g}=1\,\text{kg}\), round each amount to the nearest \(1000\,\text{g}\). 2. \(5500\,\text{g}\) rounds to \(6000\,\text{g}=6\,\text{kg}\). 3. \(6490\,\text{g}\) rounds to \(6000\,\text{g}=6\,\text{kg}\). 4. \(5495\,\text{g}\) rounds to \(5000\,\text{g}=5\,\text{kg}\). 5. \(6505\,\text{g}\) rounds to \(7000\,\text{g}=7\,\text{kg}\).

\(5500\,\text{g}\) and \(6490\,\text{g}\)

- Express each quantity in the smaller unit first. - Identify the place value that corresponds to one whole larger unit. - For part c), decide whether \(250\,\text{m}\) is closer to \(0\,\text{km}\) or \(1\,\text{km}\).

1. a) \(\$18.60\) rounds to \(\$19\) because \(60\) cents is at least half a dollar. 2. b) \(4\,\text{kg}\ 501\,\text{g}=4501\,\text{g}\), which rounds to \(5000\,\text{g}=5\,\text{kg}\). 3. c) \(250\,\text{m}\) is less than halfway to \(1000\,\text{m}\), so it rounds to \(0\,\text{km}\). 4. d) \(12\,\text{cm}\ 4\,\text{mm}=124\,\text{mm}\), which rounds to \(120\,\text{mm}=12\,\text{cm}\).

a) \(\$19\) b) \(5\,\text{kg}\) c) \(0\,\text{km}\) d) \(12\,\text{cm}\)

- Test each option separately. - A time can round up to \(15\) minutes or round down to \(15\) minutes. - Use \(30\) seconds as the halfway point.

1. A rounds to \(14\,\text{min}\) because \(25<30\). 2. B rounds to \(15\,\text{min}\) because \(40\geq30\). 3. C rounds to \(15\,\text{min}\) because \(5<30\). 4. D rounds to \(16\,\text{min}\) because \(30\geq30\). 5. E rounds to \(15\,\text{min}\) because \(59\geq30\).

B, C, and E

- Find half of the unit to which you are rounding. - Round up when the smaller unit is at least halfway to the next larger unit. - For a time written with three units, first convert the larger units to the requested unit.

1. a) Half a minute is \(30\,\text{s}\). Since \(42\,\text{s}\geq30\,\text{s}\), round up to \(15\,\text{min}\). 2. b) Half an hour is \(30\,\text{min}\). Since \(15\,\text{min}<30\,\text{min}\), round down to \(7\,\text{h}\). 3. c) First convert the hour: \(1\,\text{h}=60\,\text{min}\), so the time is \(85\,\text{min}\ 50\,\text{s}\). Since \(50\,\text{s}\geq30\,\text{s}\), round up to \(86\,\text{min}\). 4. d) Since \(28\,\text{s}<30\,\text{s}\), round down to \(5\,\text{min}\).

a) \(15\,\text{min}\) b) \(7\,\text{h}\) c) \(86\,\text{min}\) d) \(5\,\text{min}\)

- Try starting with the largest available weight. - For \(300\,\text{g}\), look for one combination with two weights and another with more weights. - Remember how many grams are in \(1\) kilogram. - In part b, use each weight no more times than it appears in the list.

1. Two combinations for \(300\,\text{g}\) are \(200\,\text{g} + 100\,\text{g} = 300\,\text{g}\) and \(200\,\text{g} + 50\,\text{g} + 50\,\text{g} = 300\,\text{g}\). 2. One combination for \(850\,\text{g}\) is \(500\,\text{g} + 200\,\text{g} + 100\,\text{g} + 50\,\text{g} = 850\,\text{g}\). 3. The total of all six weights is \(500\,\text{g} + 200\,\text{g} + 100\,\text{g} + 100\,\text{g} + 50\,\text{g} + 50\,\text{g} = 1000\,\text{g}\). Since \(1000\,\text{g} = 1\,\text{kg}\), they weigh \(1\,\text{kg}\) altogether.

a) For example, \(200\,\text{g} + 100\,\text{g}\) and \(200\,\text{g} + 50\,\text{g} + 50\,\text{g}\) b) \(500\,\text{g}\), \(200\,\text{g}\), \(100\,\text{g}\), and \(50\,\text{g}\) c) \(1\,\text{kg}\)

- Break \(387\) into hundreds, tens, and ones to find suitable weights. - Look closely at the ones digit of \(387\). - For part b, add all the weights carefully. - How many grams would the total need to equal \(1\,\text{kg}\)?

1. Decompose \(387\,\text{g}\) as \(300\,\text{g} + 80\,\text{g} + 7\,\text{g}\). Use \(200\,\text{g} + 100\,\text{g}\), \(50\,\text{g} + 20\,\text{g} + 10\,\text{g}\), and \(5\,\text{g} + 2\,\text{g}\). 2. Add all ten weights: \(1 + 2 + 2 + 5 + 10 + 20 + 50 + 100 + 200 + 500 = 890\), so the total is \(890\,\text{g}\). 3. Since \(1\,\text{kg} = 1000\,\text{g}\) and \(890\,\text{g} < 1000\,\text{g}\), Lucas is not correct. The set is \(110\,\text{g}\) short of \(1\,\text{kg}\).

a) Use \(200\,\text{g}\), \(100\,\text{g}\), \(50\,\text{g}\), \(20\,\text{g}\), \(10\,\text{g}\), \(5\,\text{g}\), and one \(2\,\text{g}\) weight. b) No. All ten weights total \(890\,\text{g}\), which is \(110\,\text{g}\) less than \(1\,\text{kg}\).

- Convert every weight to pounds. - One ton equals \(2000\) pounds. - Pay attention to the difference between \(1\) ton \(100\) pounds and \(3000\) pounds.

1. Convert every weight to pounds using \(1\,\text{ton} = 2000\,\text{lb}\). 2. \(2\,\text{tons} = 4000\,\text{lb}\). 3. \(1\,\text{ton}\ 100\,\text{lb} = 2100\,\text{lb}\). 4. \(2\,\text{tons}\ 5\,\text{lb} = 4005\,\text{lb}\). 5. The pound values are \(1800 < 2100 < 3000 < 4000 < 4005\), so place the original measurements in that order.

\(1800\,\text{lb} < 1\,\text{ton}\ 100\,\text{lb} < 3000\,\text{lb} < 2\,\text{tons} < 2\,\text{tons}\ 5\,\text{lb}\)

- One kilogram equals \(1000\) grams. - Check whether the kilograms match the thousands in the gram measurement. - Convert each mixed-unit result back to grams as a check.

1. Conversion 2 is incorrect. Since \(1000\,\text{g} = 1\,\text{kg}\), \(2500\,\text{g} = 2\,\text{kg}\ 500\,\text{g}\), not \(25\,\text{kg}\). 2. Conversion 4 is incorrect. Since \(406\,\text{g}\) is less than \(1000\,\text{g}\), it is \(0\,\text{kg}\ 406\,\text{g}\), not \(4\,\text{kg}\ 6\,\text{g}\). 3. Conversions 1 and 3 are correct.

The incorrect conversions are 2 and 4. 2. \(2500\,\text{g} = 2\,\text{kg}\ 500\,\text{g}\) 4. \(406\,\text{g} = 0\,\text{kg}\ 406\,\text{g}\)

- Convert both lengths to centimeters before comparing. - Use \(1\,\text{m} = 100\,\text{cm}\). - Convert every part of a mixed measurement.

1. a) \(4\,\text{m}\ 7\,\text{cm} = 400\,\text{cm} + 7\,\text{cm} = 407\,\text{cm}\). Since \(407 > 405\), the correct symbol is \(>\). 2. b) \(2\,\text{m}\ 30\,\text{cm} = 200\,\text{cm} + 30\,\text{cm} = 230\,\text{cm}\), so the lengths are equal. 3. c) \(15\,\text{m}\ 2\,\text{cm} = 1500\,\text{cm} + 2\,\text{cm} = 1502\,\text{cm}\). Since \(1502 > 1500\), the correct symbol is \(>\). 4. d) \(6\,\text{m} = 600\,\text{cm}\), so the lengths are equal.

a) \(>\) b) \(=\) c) \(>\) d) \(=\)

- Convert all lengths to meters before ordering them. - One kilometer equals \(1000\) meters. - Pay special attention to the zeros in \(4\) kilometers \(5\) meters.

1. Convert every length to meters. 2. \(4\,\text{km}\ 5\,\text{m} = 4005\,\text{m}\). 3. \(4\,\text{km}\ 500\,\text{m} = 4500\,\text{m}\). 4. The meter values are \(450 < 4005 < 4050 < 4500\). 5. Write the original measurements in that order.

\(450\,\text{m} < 4\,\text{km}\ 5\,\text{m} < 4050\,\text{m} < 4\,\text{km}\ 500\,\text{m}\)

- How many ounces are in \(1\,\text{lb}\)? - How many pounds are in \(1\,\text{ton}\)? - Convert all measurements to the same unit. - Then compare the numerical values.

1. Convert the weights to pounds: \(16\,\text{oz}=1\,\text{lb}\) \(160\,\text{oz}=10\,\text{lb}\) \(30\,\text{lb}=30\,\text{lb}\) \(300\,\text{lb}=300\,\text{lb}\) \(3\,\text{tons}=6000\,\text{lb}\) 2. Compare the values: \(1<10<30<300<6000\). 3. Write the original measurements in order: \(16\,\text{oz}<160\,\text{oz}<30\,\text{lb}<300\,\text{lb}<3\,\text{tons}\).

\(16\,\text{oz}\), \(160\,\text{oz}\), \(30\,\text{lb}\), \(300\,\text{lb}\), \(3\,\text{tons}\)

- Convert mixed units to grams before calculating. - One kilogram equals \(1000\) grams. - Convert the final gram total back to kilograms and grams.

1. a) \(4650\,\text{g} + 2350\,\text{g} = 7000\,\text{g} = 7\,\text{kg}\ 0\,\text{g}\). 2. b) Convert \(10\,\text{kg}\) to \(10{,}000\,\text{g}\). Then \(10{,}000\,\text{g} - 4200\,\text{g} = 5800\,\text{g} = 5\,\text{kg}\ 800\,\text{g}\). 3. c) \(6 \times 500\,\text{g} = 3000\,\text{g} = 3\,\text{kg}\ 0\,\text{g}\). 4. d) Convert \(1\,\text{kg}\) to \(1000\,\text{g}\). Then \(1000\,\text{g} \div 8 = 125\,\text{g} = 0\,\text{kg}\ 125\,\text{g}\).

a) \(7\,\text{kg}\ 0\,\text{g}\) b) \(5\,\text{kg}\ 800\,\text{g}\) c) \(3\,\text{kg}\ 0\,\text{g}\) d) \(0\,\text{kg}\ 125\,\text{g}\)

- One ton equals \(2000\) pounds. - Convert the whole tons to pounds first. - Subtract the pounds in the whole tons from the total.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\). 2. a) Four tons is \(8000\,\text{lb}\), leaving \(8025 - 8000 = 25\,\text{lb}\). 3. b) \(14{,}105\,\text{lb}\) contains \(14{,}000\,\text{lb} = 7\) tons and \(105\,\text{lb}\). 4. c) Fifteen tons is \(30{,}000\,\text{lb}\), leaving \(8\,\text{lb}\).

a) \(25\) b) \(7\) c) \(8\)

- Convert both weights in each statement to the same unit. - How many pounds are in half a ton? - Pay close attention to comparison words such as “heavier,” “lighter,” and “the same weight.”

1. a) Since \(48\,\text{oz}=3\,\text{lb}\), the statement is false. The weights are equal. 2. b) Since \(1\,\text{ton}=2000\,\text{lb}\), \(5 \times 2000=10{,}000\). The statement is true. 3. c) \(192\,\text{oz}=12\,\text{lb}\). Since \(12<100\), the statement is true. 4. d) Half a ton is \(1000\,\text{lb}\). Since \(400<1000\), the statement is false. \(400\,\text{lb}\) is lighter than half a ton.

a) False; \(48\,\text{oz}=3\,\text{lb}\). b) True c) True d) False; \(400\,\text{lb}\) is lighter than half a ton.

- Convert both sides to the same unit. - One ton equals \(2000\) pounds, and one pound equals \(16\) ounces. - Use subtraction to find a missing addend or subtrahend.

1. Use \(1\,\text{ton} = 2000\,\text{lb}\) and \(1\,\text{lb} = 16\,\text{oz}\). 2. a) \(2000\,\text{lb} - 900\,\text{lb} = 1100\,\text{lb}\). 3. b) \(12\,\text{lb} = 192\,\text{oz}\), so \(192\,\text{oz} - 176\,\text{oz} = 16\,\text{oz}\). 4. c) \(117\,\text{oz}\) contains \(112\,\text{oz} = 7\,\text{lb}\), with \(5\,\text{oz}\) remaining. 5. d) \(2\,\text{tons} = 4000\,\text{lb}\), and \(4000\,\text{lb} - 3000\,\text{lb} = 1000\,\text{lb}\). 6. e) \(640\,\text{oz} \div 16 = 40\,\text{lb}\).

a) \(1100\,\text{lb}\) b) \(16\,\text{oz}\) c) \(7\,\text{lb}\ 5\,\text{oz}\) d) \(1000\,\text{lb}\) e) \(40\,\text{lb}\)

- Convert all measurements to grams. - Use place value carefully, especially when zeros appear. - Compare the resulting whole numbers.

1. Convert each mixed measurement to grams. 2. a) \(3\,\text{kg}\ 50\,\text{g} = 3050\,\text{g}\), and \(3050 < 3500\). 3. b) \(6\,\text{kg}\ 40\,\text{g} = 6040\,\text{g}\), and \(6004 < 6040\). 4. c) \(2\,\text{kg}\ 7\,\text{g} = 2007\,\text{g}\), so the masses are equal. 5. d) \(10\,\text{kg}\ 200\,\text{g} = 10{,}200\,\text{g}\), and \(10{,}200 > 1200\). 6. e) \(4\,\text{kg}\ 8\,\text{g} = 4008\,\text{g}\), and \(4080 > 4008\).

a) \(<\) b) \(<\) c) \(=\) d) \(>\) e) \(>\)

- Convert both sides to seconds. - One minute equals \(60\) seconds. - Multiply the minutes by \(60\), then add the remaining seconds.

1. Convert each mixed time interval to seconds using \(1\,\text{min} = 60\,\text{s}\). 2. a) \(4 \times 60\,\text{s} + 15\,\text{s} = 255\,\text{s}\). Since \(260 > 255\), the correct symbol is \(>\). 3. b) \(8 \times 60\,\text{s} + 20\,\text{s} = 500\,\text{s}\), so the intervals are equal. 4. c) \(10 \times 60\,\text{s} + 5\,\text{s} = 605\,\text{s}\). Since \(605 > 600\), the correct symbol is \(>\). 5. d) \(12 \times 60\,\text{s} + 40\,\text{s} = 760\,\text{s}\), so the intervals are equal.

a) \(>\) b) \(=\) c) \(>\) d) \(=\)

- Convert both quantities to grams. - Find each fraction of its whole amount. - Compare the two results.

1. Find half of \(600\,\text{g}\): \(600\,\text{g} \div 2 = 300\,\text{g}\). 2. Convert \(1\,\text{kg}\) to \(1000\,\text{g}\). 3. Find one-fourth of \(1000\,\text{g}\): \(1000\,\text{g} \div 4 = 250\,\text{g}\). 4. Since \(300\,\text{g} > 250\,\text{g}\), half of \(600\,\text{g}\) is heavier.

\(\frac{1}{2}\) of \(600\,\text{g}\) is heavier because \(300\,\text{g} > 250\,\text{g}\).

- Convert each expression to the unit on the right. - Use place value to separate complete larger units from the remainder. - Use \(1\,\text{ton} = 2000\,\text{lb}\) for part d).

1. a) Six meters equals \(600\,\text{cm}\), so the missing amount is \(609 - 600 = 9\,\text{cm}\). 2. b) \(5012\,\text{m}\) contains \(5000\,\text{m} = 5\,\text{km}\) and \(12\,\text{m}\), so the missing number is \(5\). 3. c) \(4\,\text{kg}\ 5\,\text{g} = 4000\,\text{g} + 5\,\text{g} = 4005\,\text{g}\). 4. d) \(3\,\text{tons} = 6000\,\text{lb}\). Adding \(20\,\text{lb}\) gives \(6020\,\text{lb}\).

a) \(9\) b) \(5\) c) \(4005\) d) \(6020\)

- How many cents equal one dollar? - First find how many cents are in \(\$15\). - Then add the \(6\) cents and compare your result with \(156\).

1. One dollar equals \(100\) cents. 2. Convert the whole dollars: \(15 \times 100 = 1500\) cents. 3. Add the remaining \(6\) cents: \(1500 + 6 = 1506\) cents. 4. Since \(1506 \ne 156\), Lucas is not correct. He likely treated \(\$15\) as \(150\) cents or left out a zero when writing the amount.

Lucas is not correct. \(\$15.06\) equals \(1506\) cents. He probably forgot that \(\$15\) alone equals \(1500\) cents.

- Convert each left-hand measurement to the unit on the right. - Use \(1\,\text{km} = 1000\,\text{m}\), \(1\,\text{m} = 100\,\text{cm}\), and \(1\,\text{ton} = 2000\,\text{lb}\). - Compare the resulting whole numbers.

1. a) \(20\,\text{km}\ 5\,\text{m} = 20{,}005\,\text{m}\). Since \(20{,}005 > 2500\), the correct symbol is \(>\). 2. b) \(8\,\text{m}\ 40\,\text{cm} = 840\,\text{cm}\). Since \(840 > 804\), the correct symbol is \(>\). 3. c) \(4\,\text{tons}\ 50\,\text{lb} = 8000\,\text{lb} + 50\,\text{lb} = 8050\,\text{lb}\). Since \(8050 < 8500\), the correct symbol is \(<\).

a) \(>\) b) \(>\) c) \(<\)

- Identify the conversion factor for each pair of units. - Keep zeros in their correct place values. - For part d), convert the mixed weight to pounds before comparing.

1. a) \(3050\,\text{mL}\) contains \(3\) liters with \(50\,\text{mL}\) remaining. 2. b) \(9\,\text{km}\ 4\,\text{m} = 9000\,\text{m} + 4\,\text{m} = 9004\,\text{m}\). 3. c) \(15{,}005\,\text{g}\) contains \(15\) kilograms with \(5\,\text{g}\) remaining. 4. d) \(4\,\text{tons}\ 20\,\text{lb} = 8020\,\text{lb}\). Since \(8020 < 8200\), the correct symbol is \(<\).

a) \(3\,\text{L}\ 50\,\text{mL}\) b) \(9004\,\text{m}\) c) \(15\,\text{kg}\ 5\,\text{g}\) d) \(<\)

- Convert each lettered measurement to the unit used by one of the numbered choices. - Keep zeros in their correct place values. - One ton equals \(2000\) pounds.

1. A: \(3\,\text{km}\ 5\,\text{m} = 3000\,\text{m} + 5\,\text{m} = 3005\,\text{m}\), so A matches 1. 2. B: \(350\,\text{cm} = 3\,\text{m}\ 50\,\text{cm}\), so B matches 3. 3. C: \(3\,\text{kg}\ 50\,\text{g} = 3050\,\text{g}\), so C matches 4. 4. D: \(35\,\text{tons} = 35 \times 2000\,\text{lb} = 70{,}000\,\text{lb}\), so D matches 2.

A-1, B-3, C-4, D-2

- Start with the object that one person can move. - Convert the pound values to tons when helpful. - Distinguish an object's mass from a vehicle's cargo capacity.

1. A packed suitcase can weigh about \(50\,\text{lb}\), so 1 matches c. 2. Since \(12{,}000\,\text{lb}\div 2000=6\,\text{tons}\), that is a reasonable mass for an adult male African elephant. Therefore, 2 matches d. 3. A large dump truck can carry about \(25\,\text{tons}\), so 3 matches a. 4. Since \(4{,}000{,}000\,\text{lb}\div 2000=2000\,\text{tons}\), that value fits a loaded river barge. Therefore, 4 matches b.

1) c 2) d 3) a 4) b

- Convert both sides of each comparison to the same unit. - Evaluate each addition expression before comparing. - How many grams are in half a kilogram?

1. Convert all masses to grams. 2. a) \(4\,\text{kg}=4000\,\text{g}\). Since \(4000>400\), use \(>\). 3. b) \(12\,\text{kg}=12{,}000\,\text{g}\), so use \(=\). 4. c) \(3\,\text{kg}+50\,\text{g}=3000\,\text{g}+50\,\text{g}=3050\,\text{g}\). Since \(3050<3500\), use \(<\). 5. d) \(750\,\text{g}+250\,\text{g}=1000\,\text{g}=1\,\text{kg}\), so use \(=\). 6. e) \(\frac{1}{2}\,\text{kg}=500\,\text{g}\). Since \(600>500\), use \(>\).

a) \(>\) b) \(=\) c) \(<\) d) \(=\) e) \(>\)

- Put all three times in the same form before comparing them. - How many minutes are in \(1\) hour? - Compare the hours first and then the minutes.

1. Convert Leo's time: \(215 \div 60 = 3\) remainder \(35\), so \(215\) minutes is \(3\) hours \(35\) minutes. 2. Convert Noah's time: \(220 \div 60 = 3\) remainder \(40\), so \(220\) minutes is \(3\) hours \(40\) minutes. 3. Compare the times: \(3\,\text{hours}\,35\,\text{minutes} < 3\,\text{hours}\,40\,\text{minutes} < 3\,\text{hours}\,45\,\text{minutes}\). 4. The order is Leo, Noah, Mia.

Leo: \(3\) hours \(35\) minutes Noah: \(3\) hours \(40\) minutes Mia: \(3\) hours \(45\) minutes Order: Leo, Noah, Mia.

- Identify how many smaller units make one whole unit. - Divide by the denominator. - Multiply the result by the numerator.

1. One hour equals \(60\) minutes, one day equals \(24\) hours, and one dollar equals \(100\) cents. 2. a) \(60\,\text{min} \div 3 = 20\,\text{min}\), and \(2 \times 20\,\text{min} = 40\,\text{min}\). 3. b) \(24\,\text{hr} \div 8 = 3\,\text{hr}\), and \(3 \times 3\,\text{hr} = 9\,\text{hr}\). 4. c) \(100\,\text{cents} \div 5 = 20\,\text{cents}\), and \(4 \times 20\,\text{cents} = 80\,\text{cents}\).

a) \(40\,\text{min}\) b) \(9\,\text{hr}\) c) \(80\,\text{cents}\)

- Convert unfamiliar measurements to units you can picture. - Check both the numerical value and its unit. - Compare each measurement with a familiar benchmark.

1. A height of \(6\,\text{ft}\) is reasonable for an adult. 2. A stick of butter weighs about \(4\,\text{oz}\), or \(0.25\,\text{lb}\), so \(2.5\,\text{lb}\) is not reasonable. 3. Convert seconds to minutes: \(2700\div 60=45\). A \(45\)-minute class period is reasonable. 4. Six hundred feet is only a little more than one-tenth of a mile. A long trip between cities could reasonably be about \(600\,\text{mi}\), not \(600\,\text{ft}\).

a) Reasonable. b) Not reasonable; about \(4\,\text{oz}\), or \(0.25\,\text{lb}\), is reasonable. c) Reasonable. d) Not reasonable; about \(600\,\text{mi}\) is reasonable for a long city-to-city trip.

- Convert both movie lengths to the same unit. - How many minutes are in \(1\) hour? - Subtract the shorter time from the longer time.

1. Convert Movie A's length to minutes: \(1\,\text{hour}\,35\,\text{minutes} = 60 + 35 = 95\) minutes. 2. Compare the lengths: \(110 > 95\), so Movie B is longer. 3. Find the difference: \(110 - 95 = 15\) minutes.

Movie B is longer by \(15\) minutes.