- Compare \(2500\,\text{mL}\) with \(1000\,\text{mL}\). - The answer should be a little more than \(2\,\text{L}\).

1. Since \(1000\,\text{mL}=1\,\text{L}\), divide \(2500\) by \(1000\). 2. \(2500\div1000=2.5\).

\(2.5\,\text{L}\)

- Use the conversion relationship given in the problem. - If \(1\,\text{dm}^3\) is \(1\,\text{L}\), think about what \(3\) equal cubic decimeters represent.

1. Each cubic decimeter is one liter. 2. Therefore, \(3\,\text{dm}^3=3\,\text{L}\).

\(3\,\text{L}\)

- Think about how many groups of \(1000\) are in each number. Each full group is \(1\,\text{L}\). - What happens to the remaining hundreds when you write the amount in liters? - Dividing by \(1000\) moves the decimal point three places to the left.

1. Use \(1000\,\text{mL}=1\,\text{L}\). 2. Divide each number of milliliters by \(1000\): \(5000 \div 1000=5.0\) \(800 \div 1000=0.8\) \(1500 \div 1000=1.5\) \(200 \div 1000=0.2\) \(600 \div 1000=0.6\)

\(5.0\,\text{L}\); \(0.8\,\text{L}\); \(1.5\,\text{L}\); \(0.2\,\text{L}\); \(0.6\,\text{L}\)

- Comparisons are easier when both quantities use the same unit. - Pay close attention to the milliliters in the mixed-unit amount. - Use familiar liter-to-milliliter benchmarks to check each comparison.

1. Convert both quantities in each comparison to the same unit. 2. For a), \(0.5\,\text{L}=500\,\text{mL}\), so the quantities are equal. 3. For b), \(1.5\,\text{L}=1500\,\text{mL}\). Since \(1200<1500\), the correct symbol is \(<\). 4. For c), \(1\,\text{L}\ 20\,\text{mL}=1020\,\text{mL}\), and \(1.2\,\text{L}=1200\,\text{mL}\). Since \(1020<1200\), the correct symbol is \(<\). 5. For d), \(0.3\,\text{L}=300\,\text{mL}\). Since \(300>30\), the correct symbol is \(>\). 6. For e), \(0.25\,\text{L}=250\,\text{mL}\), so the quantities are equal.

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

- Use \(1000\,\text{mL}=1\,\text{L}\). - Convert all measurements to milliliters. - Least to greatest means the values increase from left to right.

1. \(2\,\text{L}\ 500\,\text{mL}=2000\,\text{mL}+500\,\text{mL}=2500\,\text{mL}\). 2. \(2050\,\text{mL}\) remains \(2050\,\text{mL}\). 3. \(2.5\,\text{L}=2500\,\text{mL}\). 4. \(25\,\text{mL}\) remains \(25\,\text{mL}\). 5. Compare: \(25<2050<2500=2500\).

Conversions: \(2500\,\text{mL}\), \(2050\,\text{mL}\), \(2500\,\text{mL}\), \(25\,\text{mL}\) Order: \(25\,\text{mL}<2050\,\text{mL}<2\,\text{L}\ 500\,\text{mL}=2.5\,\text{L}\)

- Convert every liter measurement to milliliters. - Use \(1000\,\text{mL}=1\,\text{L}\). - Compare close values carefully by writing all milliliter amounts.

1. Convert to milliliters: \(1250\,\text{mL}\); \(1.2\,\text{L}=1200\,\text{mL}\); \(0.125\,\text{L}=125\,\text{mL}\); \(1025\,\text{mL}\); \(1.205\,\text{L}=1205\,\text{mL}\); and \(12\,\text{L}=12{,}000\,\text{mL}\). 2. Compare: \(125<1025<1200<1205<1250<12{,}000\).

\(0.125\,\text{L}<1025\,\text{mL}<1.2\,\text{L}<1.205\,\text{L}<1250\,\text{mL}<12\,\text{L}\)

- Use the edge length marked on the cube. - Volume uses three dimensions: length, width, and height. - Use the edge length three times in the volume calculation.

1. Each edge is \(10\,\text{cm}\). 2. The cube's volume is \(10\times10\times10=1000\,\text{cm}^3\). 3. Therefore, \(1\,\text{dm}^3=1000\,\text{cm}^3\).

\(1\,\text{dm}^3=1000\,\text{cm}^3\)

- Convert the cubic decimeters to cubic centimeters before comparing. - Be careful: you are converting cubic units, not just lengths. - Once the units match, compare and then subtract.

1. Since \(1\,\text{dm}^3=1000\,\text{cm}^3\), \(2\,\text{dm}^3=2000\,\text{cm}^3\). 2. Compare: \(2000>1800\), so \(2\,\text{dm}^3\) is greater. 3. Find the difference: \(2000-1800=200\).

\(2\,\text{dm}^3\) is greater by \(200\,\text{cm}^3\).

- Convert all three amounts to liters before rounding. - Use the conversion relationships given in the problem. - After converting, compare each decimal with the nearest whole liters.

1. Container A: \(2400\,\text{cm}^3=2.4\,\text{L}\), which rounds to \(2\,\text{L}\). 2. Container B: \(2350\,\text{mL}=2.35\,\text{L}\), which rounds to \(2\,\text{L}\). 3. Container C: \(2.6\,\text{dm}^3=2.6\,\text{L}\), which rounds to \(3\,\text{L}\). 4. Since \(3\,\text{L}>2\,\text{L}\), Container C has the greatest rounded volume.

Container C, with a rounded volume of \(3\,\text{L}\).

- Convert both amounts to fluid ounces. - Recall the number of fluid ounces in a quart and in a cup. - Subtract the amount used from the amount in the carton.

1. Convert both amounts to fluid ounces. One quart is \(32\) fluid ounces, so \(\frac{3}{8}\times32=12\) fluid ounces are needed. 2. Two cups equal \(16\) fluid ounces. 3. Subtract: \(16-12=4\). Therefore, \(4\) fluid ounces remain.

\(4\) fluid ounces remain.

- Add the amounts already in the dispenser. - Subtract that total from the dispenser's capacity. - How many cups are in one gallon? - Compare the available space with the amount to be added.

1. Add the amounts already in the dispenser: \(1 \frac{1}{2}+\frac{3}{4}+1 \frac{1}{2}+\frac{3}{4}=4 \frac{1}{2}\) gallons. 2. Find the remaining capacity: \(5 \frac{1}{4}-4 \frac{1}{2}=\frac{3}{4}\) gallon. 3. Convert the remaining capacity to cups. Since \(1\) gallon is \(16\) cups, \(\frac{3}{4}\) gallon is \(12\) cups. 4. Since \(13\) cups is more than \(12\) cups, the cherry juice would overflow the dispenser.

No. Only \(12\) cups of space remain, so adding \(13\) cups would overflow the dispenser.

- Convert both measurements to liters. - One cubic decimeter equals one liter. - How many cubic centimeters are in one cubic decimeter? - How many liters are in one cubic meter?

1. Convert Tank A to liters: \(0.06\,\text{m}^3=60\,\text{L}\). 2. Convert Tank B to liters: \(55{,}000\,\text{cm}^3=55\,\text{L}\). 3. Since \(60\,\text{L}>55\,\text{L}\), Tank A holds more. 4. The difference is \(60\,\text{L}-55\,\text{L}=5\,\text{L}\).

Tank A holds more water. The difference is \(5\,\text{L}\).

- Recall the relationship between capacity units and cubic volume units. - How many milliliters are in one liter? - What is the conversion factor from cubic decimeters to cubic centimeters?

1. Since \(1\,\text{dm}^3=1\,\text{L}\), \(2.5\,\text{dm}^3=2.5\,\text{L}\). 2. Since \(1\,\text{L}=1000\,\text{mL}\), \(2.5\,\text{L}=2500\,\text{mL}\). 3. Since \(1\,\text{dm}^3=1000\,\text{cm}^3\), \(2.5\,\text{dm}^3=2500\,\text{cm}^3\). 4. Liters and cubic decimeters have matching numerical values, and milliliters and cubic centimeters have matching numerical values.

a) \(2.5\,\text{L}\), \(2500\,\text{mL}\), and \(2500\,\text{cm}^3\) b) \(1\,\text{dm}^3=1\,\text{L}\) and \(1\,\text{cm}^3=1\,\text{mL}\), so each pair has the same numerical value.

- Think about how many smaller unit cubes fit inside one larger unit cube. - Adjacent cubic metric units differ by a factor of \(1000\). - When the unit becomes smaller, the numerical value becomes larger.

1. Convert cubic meters to cubic decimeters by multiplying by \(1000\): \(0.045\times1000=45\). Therefore, \(0.045\,\text{m}^3=45\,\text{dm}^3\). 2. Convert cubic decimeters to cubic centimeters by multiplying by \(1000\) again: \(45\times1000=45{,}000\). Therefore, \(45\,\text{dm}^3=45{,}000\,\text{cm}^3\).

\(45\,\text{dm}^3\) and \(45{,}000\,\text{cm}^3\)

- Convert both measurements to the same unit before comparing. - One liter equals one cubic decimeter. - How many liters are in one cubic meter?

1. Convert \(2500\,\text{mL}\) to liters: \(2500\,\text{mL}=2.5\,\text{L}\). 2. Convert \(0.03\,\text{m}^3\) to liters: \(0.03\,\text{m}^3=30\,\text{L}\). 3. Since \(30\,\text{L}>2.5\,\text{L}\), \(0.03\,\text{m}^3\) is greater. 4. The difference is \(30\,\text{L}-2.5\,\text{L}=27.5\,\text{L}\).

\(0.03\,\text{m}^3\) is greater. The difference is \(27.5\,\text{L}\).

- Convert both addends to the same unit first. - Which unit is more convenient for the addition? - Give the final result in both requested units.

1. Convert \(1.2\,\text{dm}^3\) to cubic centimeters: \(1.2\times1000=1200\,\text{cm}^3\). 2. Add: \(1200\,\text{cm}^3+800\,\text{cm}^3=2000\,\text{cm}^3\). 3. Since \(1000\,\text{cm}^3=1\,\text{L}\), \(2000\,\text{cm}^3=2\,\text{L}\).

\(V=2\,\text{L}=2000\,\text{cm}^3\)

- How are milliliters and cubic centimeters related? - How many grams are in one kilogram? - Use the given mass of one cubic centimeter of water.

1. Since \(1\,\text{mL}=1\,\text{cm}^3\), the bottle holds \(750\,\text{cm}^3\) of water. 2. At \(1\,\text{g}\) per cubic centimeter, \(750\,\text{cm}^3\) of water has a mass of \(750\,\text{g}\). 3. Since \(1000\,\text{g}=1\,\text{kg}\), \(750\,\text{g}=0.75\,\text{kg}\).

a) \(750\,\text{cm}^3\) b) \(0.75\,\text{kg}\)

- Convert both amounts to the same unit before comparing. - Align decimal points when adding decimal numbers. - Convert the decimal part of the total liters to milliliters.

1. Container B holds \(1\,\text{L}+800\,\text{mL}=1.8\,\text{L}\). Since \(2.05>1.8\), Container A holds more. 2. The total is \(2.05\,\text{L}+1.8\,\text{L}=3.85\,\text{L}\). 3. Since \(0.85\,\text{L}=850\,\text{mL}\), \(3.85\,\text{L}=3\,\text{L}+850\,\text{mL}\).

a) Container A b) \(3.85\,\text{L}\) c) \(3\,\text{L}+850\,\text{mL}\)

- Adjacent cubic metric units differ by a factor of \(1000\). - One liter equals one cubic decimeter. - Convert each mixed number to a decimal or improper fraction before multiplying.

1. For part a, \(4\frac{3}{4}=4.75\), and \(4.75\times1000=4750\,\text{dm}^3\). 2. For part b, \(0.2\times1000=200\,\text{cm}^3\). 3. For part c, \(1\frac{1}{2}=1.5\), and \(1.5\times1000=1500\,\text{mm}^3\). 4. For part d, \(2\frac{1}{4}=2.25\), and \(2.25\times1000=2250\,\text{mL}\).

a) \(4750\,\text{dm}^3\) b) \(200\,\text{cm}^3\) c) \(1500\,\text{mm}^3\) d) \(2250\,\text{mL}\)

- How are liters and cubic decimeters related? - Find three-fourths of \(120\) before converting units. - How many cubic centimeters are in one cubic decimeter?

1. Since \(1\,\text{L}=1\,\text{dm}^3\), \(120\,\text{L}=120\,\text{dm}^3\). 2. The water volume is \(\frac{3}{4}\) of \(120\,\text{L}\): \(120\div4\times3=90\,\text{L}\). 3. Since \(1\,\text{L}=1000\,\text{cm}^3\), \(90\times1000=90{,}000\,\text{cm}^3\).

a) \(120\,\text{dm}^3\) b) \(90{,}000\,\text{cm}^3\)

- Convert both volumes in each pair to the same unit. - Convert the fractions or mixed numbers to equivalent decimals if helpful. - Check whether the numerical value should increase or decrease when the unit changes.

1. For part a, \(\frac{1}{4}\,\text{m}^3=0.25\,\text{m}^3=250\,\text{L}\), so the volumes are equal. 2. For part b, \(3\frac{1}{5}\,\text{dm}^3=3.2\,\text{dm}^3=3200\,\text{cm}^3\). Since \(3200>320\), the left volume is greater. 3. For part c, \(0.8\,\text{L}=800\,\text{mL}\). Since \(800<850\), the left volume is less. 4. For part d, \(\frac{1}{20}\,\text{cm}^3=0.05\,\text{cm}^3=50\,\text{mm}^3\), so the volumes are equal.

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

- How are liters and cubic decimeters related? - How many liters are in one cubic meter? - Convert every amount to the same unit before comparing.

1. Container A: \(0.05\,\text{m}^3=50\,\text{L}\). 2. Container B: \(48\,\text{L}\). 3. Container C: \(52{,}000\,\text{cm}^3=52\,\text{L}\). 4. Since \(48\,\text{L}<50\,\text{L}<52\,\text{L}\), the order is Container B, Container A, Container C.

Container B \(<\) Container A \(<\) Container C, because \(48\,\text{L}<50\,\text{L}<52\,\text{L}\).

- Express the tank's capacity in the requested unit. - Add the volumes already in the tank. - Subtract the filled volume from the total capacity.

1. Convert the total capacity: \(1\,\text{m}^3=1000\,\text{dm}^3\). 2. Convert the oil volume: \(120\,\text{L}=120\,\text{dm}^3\). 3. The filled volume is \(650\,\text{dm}^3+120\,\text{dm}^3=770\,\text{dm}^3\). 4. The remaining volume is \(1000\,\text{dm}^3-770\,\text{dm}^3=230\,\text{dm}^3\).

\(230\,\text{dm}^3\) of space remains.

- Write the conversion as a fraction with the larger unit's conversion factor in the denominator. - Most adjacent cubic metric units and the milliliter-liter pair use a factor of \(1000\). - Reduce each fraction using a common factor.

1. For part a, \(15\,\text{cm}^3=\frac{15}{1000}\,\text{dm}^3=\frac{3}{200}\,\text{dm}^3\). 2. For part b, \(800\,\text{mL}=\frac{800}{1000}\,\text{L}=\frac{4}{5}\,\text{L}\). 3. For part c, \(125\,\text{dm}^3=\frac{125}{1000}\,\text{m}^3=\frac{1}{8}\,\text{m}^3\). 4. For part d, \(600\,\text{mm}^3=\frac{600}{1000}\,\text{cm}^3=\frac{3}{5}\,\text{cm}^3\). 5. For part e, \(350\,\text{mL}=\frac{350}{1000}\,\text{L}=\frac{7}{20}\,\text{L}\).

a) \(\frac{3}{200}\,\text{dm}^3\) b) \(\frac{4}{5}\,\text{L}\) c) \(\frac{1}{8}\,\text{m}^3\) d) \(\frac{3}{5}\,\text{cm}^3\) e) \(\frac{7}{20}\,\text{L}\)

- Convert all values in an expression to one common unit before calculating. - Give the final result in the unit requested.

1. For part a, \(300\,\text{mL}=0.3\,\text{L}\) and \(500\,\text{mL}=0.5\,\text{L}\). Then \(1.2+0.3+0.5=2\,\text{L}\). 2. For part b, \(20\,\text{L}-15\,\text{L}=5\,\text{L}\). Since \(1\,\text{L}=1000\,\text{mL}\), \(5\,\text{L}=5000\,\text{mL}\).

a) \(2\,\text{L}\) b) \(5000\,\text{mL}\)

- Put both starting amounts in one cubic unit before adding. - How many cubic centimeters are in \(1\,\text{dm}^3\)? - After finding the total in cubic centimeters, use the relationship between cubic centimeters and milliliters.

1. Convert \(1.35\,\text{dm}^3\) to cubic centimeters. Since \(1\,\text{dm}^3=1000\,\text{cm}^3\), \(1.35\times1000=1350\), so the starting volume is \(1350\,\text{cm}^3\). 2. Add the new liquid: \(1350+275=1625\,\text{cm}^3\). 3. Since \(1\,\text{cm}^3=1\,\text{mL}\), \(1625\,\text{cm}^3=1625\,\text{mL}\). 4. The numerical values match because one cubic centimeter and one milliliter represent equal volumes.

a) \(1625\,\text{cm}^3\) b) \(1625\,\text{mL}\) c) The numbers are the same because \(1\,\text{cm}^3=1\,\text{mL}\).

- Convert every amount to the same unit. - Use \(1000\,\text{mL}=1\,\text{L}\). - Compare the total with the bowl’s capacity.

1. Convert every amount to liters: \(1\frac{1}{4}\,\text{L}=1.25\,\text{L}\), \(750\,\text{mL}=0.75\,\text{L}\), \(\frac{1}{5}\,\text{L}=0.2\,\text{L}\), and \(50\,\text{mL}=0.05\,\text{L}\). 2. Add: \(1.25+0.8+0.75+0.2+0.05=3.05\). The total volume is \(3.05\,\text{L}\). 3. Since \(3.05\,\text{L}>3\,\text{L}\), the bowl is too small.

The total volume is \(3.05\,\text{L}\). It will not all fit in a \(3\,\text{L}\) bowl.

- First find the total amount of water removed. - Keep track of the unit requested for the answer. - How are liters and milliliters related?

1. The total amount removed is \(60\times250\,\text{mL}=15{,}000\,\text{mL}\). 2. Convert the amount removed to liters: \(15{,}000\,\text{mL}=15\,\text{L}\). 3. The amount remaining is \(18.5\,\text{L}-15\,\text{L}=3.5\,\text{L}\). 4. Convert to milliliters: \(3.5\,\text{L}=3500\,\text{mL}\).

\(3500\,\text{mL}\) of water remains.

- How much space remains in Pitcher B before any water is added? - Convert the amounts to the same unit. - Compare the new total with the pitcher's capacity.

1. Convert the added amount to liters: \(125\,\text{mL}=0.125\,\text{L}\). 2. The new amount in Pitcher B is \(2.45\,\text{L}+0.125\,\text{L}=2.575\,\text{L}\). 3. Since \(2.575\,\text{L}<2.6\,\text{L}\), Pitcher B does not overflow.

No. Pitcher B will contain \(2.575\,\text{L}\), which is less than its \(2.6\,\text{L}\) capacity.

- Convert the bottle capacity from milliliters to liters first. - Find the total capacity of the large containers and the small bottles. - Compare the total amount with the amount the festival needs.

1. Convert each small bottle to liters: \(2000\,\text{mL}=2\,\text{L}\). 2. The four large containers hold \(4\times10=40\,\text{L}\). The three small bottles hold \(3\times2=6\,\text{L}\). 3. The total is \(40+6=46\,\text{L}\). Since \(46\ge45\), the school has enough juice.

Yes. The school has \(46\,\text{L}\) of juice altogether.

- Put both amounts in milliliters. - Recall the number of milliliters in one liter. - After comparing, subtract to find how much more.

1. Convert the first amount to milliliters: \(0.4\times1000=400\), so \(0.4\,\text{L}=400\,\text{mL}\). 2. Compare \(400\,\text{mL}\) with \(350\,\text{mL}\). The first cup contains more. 3. Find the difference: \(400-350=50\,\text{mL}\).

The first cup contains \(50\,\text{mL}\) more water.

- Convert the added liters and milliliters to one decimal number of liters. - Add the new water to the amount already in the barrel. - Compare the total with the barrel’s capacity.

1. Convert the added water to liters: \(45\,\text{L}\,350\,\text{mL} = 45.35\,\text{L}\). 2. Add the amounts: \(155.75\,\text{L} + 45.35\,\text{L} = 201.10\,\text{L}\). 3. Since \(201.10\,\text{L} > 200\,\text{L}\), the barrel will overflow.

Yes. The barrel would contain \(201.10\,\text{L}\), which is \(1.10\,\text{L}\) more than its capacity.

- Begin with the innermost parentheses. - Express liters and milliliters in one unit. - Use \(1000\,\text{mL} = 1\,\text{L}\). - Follow the grouping symbols from inside to outside.

1. Evaluate the inner parentheses: \(1\,\text{L}\,250\,\text{mL} - 800\,\text{mL} = 1250\,\text{mL} - 800\,\text{mL} = 450\,\text{mL}\). 2. Convert: \(450\,\text{mL} = 0.45\,\text{L}\). 3. Evaluate the brackets: \(3.4\,\text{L} + 0.45\,\text{L} = 3.85\,\text{L}\). 4. Subtract: \(10\,\text{L} - 3.85\,\text{L} = 6.15\,\text{L}\).

\(6.15\,\text{L}\)

- Find the amount in each container separately. - Convert liters to milliliters before calculating. - One-third means divide by \(3\). - Add all amounts and convert back to liters.

1. Convert the bucket amount: \(4.8\,\text{L} = 4800\,\text{mL}\). 2. The watering can contains \(4800 + 1500 = 6300\,\text{mL}\). 3. The pitcher contains \(4800 \div 3 = 1600\,\text{mL}\). 4. Add: \(4800 + 6300 + 1600 = 12{,}700\,\text{mL}\). 5. Convert: \(12{,}700\,\text{mL} = 12.7\,\text{L}\).

The three containers contain \(12.7\,\text{L}\) of water altogether.

- Convert every volume to the requested unit first. - How many cubic decimeters are in one cubic meter? - How many cubic centimeters are in one cubic decimeter? - Then calculate from left to right.

1. Convert each volume to cubic decimeters: \(1.2\,\text{m}^3=1200\,\text{dm}^3\) and \(12{,}000\,\text{cm}^3=12\,\text{dm}^3\). 2. Substitute the converted values: \(1200\,\text{dm}^3-450\,\text{dm}^3+12\,\text{dm}^3\). 3. Calculate from left to right: \(1200-450=750\), and \(750+12=762\). 4. The result is \(762\,\text{dm}^3\).

\(762\,\text{dm}^3\)

- Convert the known volumes to one common unit. - How are cubic centimeters, cubic decimeters, and cubic meters related? - Think of the equation as a balance: both sides must have the same total volume.

1. Convert the known volumes to cubic decimeters. 2. Since \(3500\,\text{cm}^3=3.5\,\text{dm}^3\) and \(0.02\,\text{m}^3=20\,\text{dm}^3\), the equation becomes \(3.5+\square=20\). 3. Subtract to find the missing number: \(20-3.5=16.5\). 4. The missing number is \(16.5\).

\(16.5\)

- Determine how many smaller units make one larger unit. - Recall the relationship between liters and cubic decimeters. - When converting from a larger unit to a smaller unit, the numerical value increases. - In part d), convert both quantities to the same unit before adding.

1. Since \(1\,\text{L}=1000\,\text{mL}\), \(4.5\times1000=4500\,\text{mL}\). 2. Since \(1000\,\text{cm}^3=1\,\text{dm}^3\), \(250\div1000=0.25\,\text{dm}^3\). 3. Since \(1\,\text{m}^3=1000\,\text{L}\), \(0.03\times1000=30\,\text{L}\). 4. Convert \(12\,\text{dm}^3\) to \(12{,}000\,\text{cm}^3\). Then \(12{,}000+5=12{,}005\,\text{cm}^3\).

a) \(4500\,\text{mL}\) b) \(0.25\,\text{dm}^3\) c) \(30\,\text{L}\) d) \(12{,}005\,\text{cm}^3\)

- Quantities with different units cannot be added directly. - Convert each term to liters first. - One liter equals one cubic decimeter.

1. Convert the first term: \(0.4\,\text{m}^3=400\,\text{L}\). 2. Convert the second term: \(120\,\text{dm}^3=120\,\text{L}\). 3. Convert the third term: \(5000\,\text{cm}^3=5\,\text{L}\). 4. Add the volumes: \(400\,\text{L}+120\,\text{L}+5\,\text{L}=525\,\text{L}\).

\(525\,\text{L}\)

- Convert both quantities in each pair to the same unit. - One liter equals one cubic decimeter. - One milliliter equals one cubic centimeter. - Keep track of the order \(\text{m}^3\), \(\text{dm}^3\), \(\text{cm}^3\), and \(\text{mm}^3\).

1. For part a, \(0.45\,\text{dm}^3=450\,\text{cm}^3=450\,\text{mL}\), so the volumes are equal. 2. For part b, \(0.02\,\text{m}^3=20{,}000\,\text{cm}^3\), and \(20{,}000>2000\). 3. For part c, \(1500\,\text{mm}^3=1.5\,\text{cm}^3\), so the volumes are equal. 4. For part d, \(12\,\text{L}=12{,}000\,\text{cm}^3\), and \(12{,}000>1200\).

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

- Recall the conversion factor between adjacent cubic metric units. - How many smaller cubes fit inside one cube of the next larger unit? - Convert step by step between cubic meters, cubic decimeters, and cubic centimeters.

1. First row: \(1.2\,\text{m}^3=1200\,\text{dm}^3=1{,}200{,}000\,\text{cm}^3\). 2. Second row: \(3000\,\text{cm}^3=3\,\text{dm}^3=0.003\,\text{m}^3\). 3. Third row: \(40\,\text{dm}^3=0.04\,\text{m}^3=40{,}000\,\text{cm}^3\).

a) \(1200\) b) \(1{,}200{,}000\) c) \(0.003\) d) \(3\) e) \(0.04\) f) \(40{,}000\)

- Picture a cube with an edge length of \(1\,\text{cm}\). How many \(1\,\text{mm}\) cubes fit along one edge? - How many layers of the smaller cubes are there? - What happens to volume when each of three edge lengths is multiplied by \(10\)?

1. A cube with volume \(1\,\text{cm}^3\) has dimensions \(1\,\text{cm}\times1\,\text{cm}\times1\,\text{cm}\). 2. In millimeters, those dimensions are \(10\,\text{mm}\times10\,\text{mm}\times10\,\text{mm}\), so \(1\,\text{cm}^3=1000\,\text{mm}^3\). 3. The volume conversion factor is \(10\times10\times10=1000\), not \(100\). 4. Therefore, \(100\,\text{mm}^3\div1000=0.1\,\text{cm}^3\).

Lucas applied a two-dimensional conversion factor to a three-dimensional measurement. Since \(1\,\text{cm}^3=1000\,\text{mm}^3\), the correct conversion is \(100\,\text{mm}^3=0.1\,\text{cm}^3\).

- Decide how many smaller units make one larger unit. - Adjacent cubic metric units differ by a factor of \(1000\). - One cubic decimeter equals one liter. - Decide whether the numerical value should increase or decrease.

1. For part a, multiply by \(1000\): \(1.5\times1000=1500\,\text{dm}^3\). 2. For part b, divide by \(1000\): \(450\div1000=0.45\,\text{dm}^3\). 3. From cubic decimeters to cubic millimeters, the factor is \(1000\times1000=1{,}000{,}000\). Therefore, \(0.02\times1{,}000{,}000=20{,}000\,\text{mm}^3\). 4. Since \(1\,\text{L}=1000\,\text{cm}^3\) and \(1\,\text{mL}=1\,\text{cm}^3\), \(2\times1000+5=2005\,\text{cm}^3\).

a) \(1500\,\text{dm}^3\) b) \(0.45\,\text{dm}^3\) c) \(20{,}000\,\text{mm}^3\) d) \(2005\,\text{cm}^3\)

- Convert all measurements to the same unit, such as cubic decimeters. - How many liters are in one cubic meter? - Adjacent cubic metric units differ by a factor of \(1000\).

1. Convert all measurements to cubic decimeters or liters. 2. \(2.5\,\text{dm}^3\) remains \(2.5\,\text{dm}^3\). 3. \(0.025\,\text{m}^3=25\,\text{dm}^3\). 4. \(2500\,\text{cm}^3=2.5\,\text{dm}^3\). 5. \(25\,\text{L}=25\,\text{dm}^3\). 6. Therefore, \(2.5\,\text{dm}^3=2500\,\text{cm}^3<0.025\,\text{m}^3=25\,\text{L}\).

\(2.5\,\text{dm}^3=2500\,\text{cm}^3<0.025\,\text{m}^3=25\,\text{L}\)

- Decide whether each target unit is larger or smaller than the starting unit. - For cubic units, think about how a length-unit change affects length, width, and height. - If two metric unit steps separate the units, handle one step at a time and check the direction of each change.

1. For part a, \(3.5\times1000=3500\), so \(3.5\,\text{m}^3=3500\,\text{dm}^3\). 2. For part b, \(7500\div1000=7.5\), so \(7500\,\text{mL}=7.5\,\text{L}\). 3. For part c, \(1\,\text{dm}^3=1{,}000{,}000\,\text{mm}^3\). Therefore, \(0.2\times1{,}000{,}000=200{,}000\,\text{mm}^3\). 4. For part d, \(450{,}000\,\text{cm}^3\div1{,}000{,}000=0.45\,\text{m}^3\), so the missing unit is \(\text{m}^3\).

a) \(3500\,\text{dm}^3\) b) \(7.5\,\text{L}\) c) \(200{,}000\,\text{mm}^3\) d) \(0.45\,\text{m}^3\)

- Evaluate the parentheses first. - Convert each term to liters before calculating. - How many liters are in one cubic meter?

1. Convert each volume to liters: \(1.5\,\text{m}^3=1500\,\text{L}\), \(400\,\text{dm}^3=400\,\text{L}\), and \(250{,}000\,\text{cm}^3=250\,\text{L}\). 2. Evaluate the parentheses: \(400\,\text{L}+250\,\text{L}=650\,\text{L}\). 3. Subtract: \(1500\,\text{L}-650\,\text{L}=850\,\text{L}\).

\(850\,\text{L}\)

- Convert all quantities in each equation to the unit shown next to the blank. - Recall the factor of \(1000\) between adjacent cubic metric units. - How many cubic centimeters are in one-quarter liter?

1. For part a, \(1\,\text{dm}^3=1000\,\text{cm}^3\) and \(0.25\,\text{L}=250\,\text{cm}^3\). The missing amount is \(1000-250=750\,\text{cm}^3\). 2. For part b, \(0.002\,\text{m}^3=2\,\text{L}\) and \(1500\,\text{cm}^3=1.5\,\text{L}\). The difference is \(2-1.5=0.5\,\text{L}\).

a) \(750\) b) \(0.5\)

- Convert all amounts to the same unit before comparing them. - How do you convert cubic decimeters to cubic meters? - How many unit steps separate cubic centimeters from cubic meters? - How are liters related to cubic meters?

1. Site A: \(11{,}500\,\text{dm}^3=11.5\,\text{m}^3\). Since \(11.5<12\), one truckload is enough. 2. Site B: \(12{,}500{,}000\,\text{cm}^3=12.5\,\text{m}^3\). Since \(12.5>12\), one truckload is not enough. 3. Site C: \(11{,}000\,\text{L}=11{,}000\,\text{dm}^3=11\,\text{m}^3\). Since \(11<12\), one truckload is enough. 4. Therefore, one truckload is enough for Sites A and C.

One truckload is enough for Site A, which ordered \(11.5\,\text{m}^3\), and Site C, which ordered \(11\,\text{m}^3\). It is not enough for Site B, which ordered \(12.5\,\text{m}^3\).

- Decide how many smaller units make one larger unit. - It may be easier to calculate first in the smaller unit. - One cubic decimeter equals one liter.

1. For part a, \(1.5\,\text{dm}^3=1500\,\text{cm}^3\). The sum is \(1500+800=2300\,\text{cm}^3=2.3\,\text{dm}^3\). 2. For part b, \(0.04\,\text{m}^3=40\,\text{dm}^3\). The difference is \(40-15=25\,\text{dm}^3=0.025\,\text{m}^3\). 3. For part c, \(2.4\div6=0.4\), so the result is \(0.4\,\text{L}\).

a) \(2.3\,\text{dm}^3\) b) \(0.025\,\text{m}^3\) c) \(0.4\,\text{L}\)

- Convert the measurements in each part to compatible units first. - Think about the missing addend, missing subtrahend, or missing factor in each statement. - Check each value in the original equation after the units match.

1. For part a, \(2\,\text{dm}^3=2000\,\text{cm}^3\). The missing amount is \(2000-450=1550\,\text{cm}^3\). 2. For part b, \(0.5\,\text{m}^3=500\,\text{L}\). The missing amount is \(500-320=180\,\text{L}\). 3. For part c, \(1.5\,\text{L}=1500\,\text{mL}\). The missing amount is \(1500\div5=300\,\text{mL}\).

a) \(1550\,\text{cm}^3\) b) \(180\,\text{L}\) c) \(300\,\text{mL}\)

- Keep the whole-number part in the original unit. - Convert the decimal part to the listed smaller unit. - Check that recombining the whole-unit and smaller-unit parts gives the original decimal measurement.

1. For part a, \(0.007\,\text{m}^3=7\,\text{dm}^3\), so \(4.007\,\text{m}^3=4\,\text{m}^3+7\,\text{dm}^3\). 2. For part b, \(0.65\,\text{dm}^3=650\,\text{cm}^3\), so \(12.65\,\text{dm}^3=12\,\text{dm}^3+650\,\text{cm}^3\). 3. For part c, \(0.304\,\text{m}^3=304\,\text{dm}^3\), so \(2.304\,\text{m}^3=2\,\text{m}^3+304\,\text{dm}^3\). 4. For part d, \(0.08\,\text{L}=80\,\text{mL}\), so \(5.08\,\text{L}=5\,\text{L}+80\,\text{mL}\).

a) \(4\,\text{m}^3+7\,\text{dm}^3\) b) \(12\,\text{dm}^3+650\,\text{cm}^3\) c) \(2\,\text{m}^3+304\,\text{dm}^3\) d) \(5\,\text{L}+80\,\text{mL}\)

- Convert every smaller-unit amount to the largest unit shown. - A factor of \(1000\) means that three decimal places represent the next smaller cubic unit. - Keep placeholder zeros when a smaller-unit amount does not fill all three decimal places.

1. For part a, \(50\,\text{dm}^3=0.05\,\text{m}^3\), so the total is \(7.05\,\text{m}^3\). 2. For part b, \(4\,\text{cm}^3=0.004\,\text{dm}^3\), so the total is \(2.004\,\text{dm}^3\). 3. For part c, \(5\,\text{mL}=0.005\,\text{L}\), so the total is \(10.005\,\text{L}\). 4. For part d, \(125\,\text{dm}^3=0.125\,\text{m}^3\), so the total is \(1.125\,\text{m}^3\).

a) \(7.05\,\text{m}^3\) b) \(2.004\,\text{dm}^3\) c) \(10.005\,\text{L}\) d) \(1.125\,\text{m}^3\)

- Convert both quantities in each pair to the same unit. - Add the parts of each sum after converting them. - Adjacent cubic metric units differ by a factor of \(1000\).

1. For part a, \(2.05\,\text{m}^3=2050\,\text{dm}^3\), while \(2\,\text{m}^3+500\,\text{dm}^3=2500\,\text{dm}^3\). Therefore, \(2050<2500\). 2. For part b, \(3.4\,\text{dm}^3=3400\,\text{cm}^3\), while \(3\,\text{dm}^3+40\,\text{cm}^3=3040\,\text{cm}^3\). Therefore, \(3400>3040\). 3. For part c, \(0.008\,\text{L}=8\,\text{mL}\), and \(8<80\). 4. For part d, \(2\,\text{mm}^3=0.002\,\text{cm}^3\), so \(1\,\text{cm}^3+2\,\text{mm}^3=1.002\,\text{cm}^3\).

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

- How are milliliters and cubic centimeters related? - How many milliliters are in one liter? - Compare the converted capacity with the stated range for a typical backpack.

1. Since \(1\,\text{mL}=1\,\text{cm}^3\), \(150{,}000\,\text{mL}=150{,}000\,\text{cm}^3\). 2. Since \(1000\,\text{cm}^3=1\,\text{L}\), \(150{,}000\div1000=150\,\text{L}\). 3. A typical school backpack holds about \(20\) to \(30\) liters. A capacity of \(150\) liters is about \(5\) to \(7.5\) times as large, so Jonas's claim is not realistic.

No. \(150{,}000\,\text{mL}=150{,}000\,\text{cm}^3=150\,\text{L}\), which is far greater than the typical \(20\)- to \(30\)-liter capacity of a school backpack.

- Convert values to a common unit before comparing or combining them. - Evaluate the parentheses before dividing. - One cubic decimeter equals one liter.

1. For part a, convert to liters: \(0.2\,\text{m}^3=200\,\text{L}\) and \(21{,}000\,\text{cm}^3=21\,\text{L}\). Therefore, \(21{,}000\,\text{cm}^3<0.2\,\text{m}^3<220\,\text{L}\). 2. For part b, \(4.8\,\text{L}=4800\,\text{cm}^3\). Then \((4800-800)\,\text{cm}^3\div2=4000\,\text{cm}^3\div2=2000\,\text{cm}^3=2\,\text{L}\). 3. For part c, \(0.05\,\text{m}^3=50\,\text{dm}^3\) and \(100\,\text{L}=100\,\text{dm}^3\). The missing value is \(100-50=50\,\text{dm}^3\).

a) \(21{,}000\,\text{cm}^3<0.2\,\text{m}^3<220\,\text{L}\) b) \(2\,\text{L}\) c) \(50\,\text{dm}^3\)

- Use the cube-volume formula. - Convert the large cube's edge length to centimeters. - Compare the large cube's volume with the small cube's volume. - Remember that all three dimensions change when converting cubic units.

1. The small cube has volume \(10\times10\times10=1000\,\text{cm}^3\). 2. The large cube has an edge length of \(1\,\text{m}=100\,\text{cm}\). Its volume is \(100\times100\times100=1{,}000{,}000\,\text{cm}^3\). 3. The number of small cubes is \(1{,}000{,}000\div1000=1000\). 4. Therefore, \(1\,\text{m}^3=1{,}000{,}000\,\text{cm}^3\).

a) \(1000\,\text{cm}^3\) b) \(1000\) small cubes c) \(1\,\text{m}^3=1{,}000{,}000\,\text{cm}^3\)

- Convert quantities to the same unit before comparing them. - How many milliliters are in one liter, and how many liters are in one cubic meter? - Convert \(\frac{3}{4}\) of a cubic meter directly to liters.

1. For part a, \(0.2\,\text{m}^3=200{,}000\,\text{cm}^3\). Since \(200{,}000>20{,}000\), the correct symbol is \(>\). 2. For part b, \(800\,\text{mL}=0.8\,\text{L}\). Then \(4.5-0.8=3.7\,\text{L}=3.7\,\text{dm}^3\). 3. For part c, \(\frac{3}{4}\,\text{m}^3=750\,\text{L}\) and \(1\,\text{m}^3=1000\,\text{L}\). The missing amount is \(1000-750=250\,\text{L}\). 4. For part d, \(12\times250\,\text{mL}=3000\,\text{mL}=3\,\text{L}\), and \(0.003\,\text{m}^3=3\,\text{L}\). The volumes are equal.

a) \(>\) b) \(3.7\,\text{dm}^3\) c) \(250\,\text{L}\) d) \(=\)

- One liter equals one cubic decimeter. - How many cubic centimeters equal one milliliter? - Be careful when converting cubic meters to cubic centimeters. - Convert all quantities before performing the operations.

1. Convert each volume to cubic centimeters: \(0.5\,\text{L}=500\,\text{cm}^3\), \(450\,\text{mL}=450\,\text{cm}^3\), and \(0.001\,\text{m}^3=1000\,\text{cm}^3\). 2. Substitute the converted values: \(500\,\text{cm}^3+450\,\text{cm}^3+250\,\text{cm}^3-1000\,\text{cm}^3\). 3. Add the positive terms: \(500+450+250=1200\). 4. Subtract: \(1200-1000=200\). 5. The result is \(200\,\text{cm}^3\).

\(200\,\text{cm}^3\)

- In parts a and b, convert all quantities to the unit requested in the answer. - Cubic metric units change by a factor of \(1000\) at each adjacent unit step. - In part c, first convert \(0.1\,\text{dm}^3\) to cubic millimeters.

1. For part a, \(0.8\,\text{m}^3=800\,\text{dm}^3\). Then \(800+150=950\,\text{dm}^3\). 2. For part b, \(4200\,\text{cm}^3=4.2\,\text{L}\) and \(1.2\,\text{dm}^3=1.2\,\text{L}\). Then \(4.2-1.2=3\,\text{L}\). 3. For part c, \(0.1\,\text{dm}^3=100{,}000\,\text{mm}^3\). Then \(100{,}000\div500=200\).

a) \(950\,\text{dm}^3\) b) \(3\,\text{L}\) c) \(200\) times

- Identify the requested unit for each part. - Convert every term to that unit before calculating. - One liter has the same volume as one cubic decimeter. - Write the mixed number as an equivalent decimal or improper fraction if helpful.

1. For part a, \(0.75\,\text{m}^3=750\,\text{dm}^3\) and \(400\,\text{L}=400\,\text{dm}^3\). Then \(750+250-400=600\,\text{dm}^3\). 2. For part b, \(1\frac{3}{4}\,\text{L}=1750\,\text{cm}^3\) and \(0.1\,\text{dm}^3=100\,\text{cm}^3\). Then \(1750-850+100=1000\,\text{cm}^3\).

a) \(600\,\text{dm}^3\) b) \(1000\,\text{cm}^3\)

- Convert both sides of each comparison to the same unit. - What decimal is equivalent to \(\frac{1}{5}\)? - Recall the relationships among liters, milliliters, and cubic decimeters.

1. For part a, the left side is \(12.5\,\text{L}+0.5\,\text{L}=13\,\text{L}\). The right side is \(0.014\,\text{m}^3=14\,\text{L}\). Therefore, \(13\,\text{L}<14\,\text{L}\). 2. For part b, \(2\frac{1}{5}\,\text{dm}^3=2.2\,\text{dm}^3\) and \(200\,\text{cm}^3=0.2\,\text{dm}^3\), so the left side is \(2\,\text{dm}^3\). The right side is \(2000\,\text{mL}=2\,\text{L}=2\,\text{dm}^3\). Therefore, the two sides are equal.

a) \(<\) b) \(=\)

- Evaluate the parentheses first. - Choose a common unit before adding or subtracting. - Convert the final value to liters if needed. - In part b, divide the remaining volume into \(4\) equal parts.

1. For part a, \(200\,\text{mL}=0.2\,\text{L}\), so the amount in parentheses is \(0.2\,\text{L}+0.3\,\text{L}=0.5\,\text{L}\). Then \(15\times0.5\,\text{L}=7.5\,\text{L}\). 2. For part b, \(0.008\,\text{m}^3=8\,\text{dm}^3\). The amount in parentheses is \(8\,\text{dm}^3-3\,\text{dm}^3=5\,\text{dm}^3\). Then \(5\,\text{dm}^3\div4=1.25\,\text{dm}^3=1.25\,\text{L}\).

a) \(7.5\,\text{L}\) b) \(1.25\,\text{L}\)

- Before adding, subtracting, or dividing measurements, decide which units must match. - Convert the result when the problem requests a different unit. - For the quotient in part c, express both volumes in the same unit before dividing.

1. For part a, \(6\times750\,\text{mL}=4500\,\text{mL}=4.5\,\text{L}\). 2. For part b, \(3.6\,\text{dm}^3\div9=0.4\,\text{dm}^3\). Since \(1\,\text{dm}^3=1000\,\text{cm}^3\), the result is \(400\,\text{cm}^3\). 3. For part c, \(500\,\text{dm}^3=0.5\,\text{m}^3\) and \(2\frac{1}{2}\,\text{m}^3=2.5\,\text{m}^3\). Then \(2.5\div0.5=5\). 4. For part d, \(\frac{1}{4}\,\text{L}=0.25\,\text{L}\), so \(1.2\,\text{L}-0.25\,\text{L}=0.95\,\text{L}=950\,\text{mL}\).

a) \(4.5\,\text{L}\) b) \(400\,\text{cm}^3\) c) \(5\) d) \(950\,\text{mL}\)

- Choose one unit and convert every measurement to it. - Pay careful attention to factors of \(1000\) between cubic units. - One liter equals one cubic decimeter.

1. Convert every volume to cubic centimeters. 2. \(4000\,\text{mm}^3=4\,\text{cm}^3\). 3. \(45\,\text{cm}^3\) remains \(45\,\text{cm}^3\). 4. \(0.05\,\text{dm}^3=50\,\text{cm}^3\). 5. \(0.4\,\text{L}=400\,\text{cm}^3\). 6. \(0.003\,\text{m}^3=3000\,\text{cm}^3\). 7. Therefore, \(4000\,\text{mm}^3<45\,\text{cm}^3<0.05\,\text{dm}^3<0.4\,\text{L}<0.003\,\text{m}^3\).

\(4000\,\text{mm}^3<45\,\text{cm}^3<0.05\,\text{dm}^3<0.4\,\text{L}<0.003\,\text{m}^3\)

- Convert quantities to compatible units before adding, subtracting, or dividing. - Use the target unit written after each expression to decide whether another conversion is needed. - For a quotient of two measurements of the same kind, consider whether the units cancel.

1. For part a, \(0.75\,\text{L}=750\,\text{mL}\). Then \(450+750=1200\,\text{mL}=1.2\,\text{L}\). 2. For part b, \(0.004\,\text{m}^3=4000\,\text{cm}^3\). Then \(4000-1200=2800\,\text{cm}^3=2.8\,\text{dm}^3\). 3. For part c, \(1.2\times5=6\), so the result is \(6\,\text{dm}^3\). Since \(1\,\text{dm}^3=1000\,\text{cm}^3\), this is \(6000\,\text{cm}^3\). 4. For part d, \(6\,\text{L}=6000\,\text{mL}\). Then \(6000\div15=400\,\text{mL}\). 5. For part e, \(2\,\text{m}^3=2000\,\text{L}\). Then \(2000\,\text{L}\div400\,\text{L}=5\). The units cancel.

a) \(1.2\,\text{L}\) b) \(2.8\,\text{dm}^3\) c) \(6000\,\text{cm}^3\) d) \(400\,\text{mL}\) e) \(5\)

- Follow the order of operations: parentheses first, then multiplication or division before addition or subtraction. - Convert intermediate values to the requested unit. - Milliliters and cubic centimeters represent equal volumes.

1. For part a, \(\frac{1}{4}\,\text{m}^3=250\,\text{L}\). The parentheses equal \(250\,\text{L}+150\,\text{L}=400\,\text{L}\). Then \(400\times2=800\,\text{L}\), and \(0.3\,\text{m}^3=300\,\text{L}\). The result is \(800-300=500\,\text{L}\). 2. For part b, \(0.08\,\text{dm}^3=80\,\text{cm}^3\). Then \(80\div4=20\,\text{cm}^3\). Since \(15\,\text{mL}=15\,\text{cm}^3\), the result is \(20+15=35\,\text{cm}^3\). 3. For part c, \(\frac{2}{5}\,\text{L}=0.4\,\text{L}\). Then \(0.4\times10=4\,\text{L}\), and \(3500\,\text{mL}=3.5\,\text{L}\). The result is \(4-3.5=0.5\,\text{L}\).

a) \(500\,\text{L}\) b) \(35\,\text{cm}^3\) c) \(0.5\,\text{L}\)