Metric Unit Conversion Problems

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Problem #1: How many palladium atoms would it take to encircle the Earth at its equator? (The atomic diameter of Pd is 140 pm; the equatorial diameter of the Earth is 40,075.02 km.)

Problem #2: In America, a car's gasoline efficiency is measured in miles/gallon. In Europe, it is measured in km/L. If your car's gas mileage is 40.0 mi/gal, how many liters of gasoline would you need to buy to complete a 142 km trip? Use the following conversions: 1 km = 0.6214 mi and 1 gal = 3.7854 L

Answers to 1 and 2


Problem #3: Convert 2.1 Å to nanometers

Solution:

1) Convert Å to meter:

1 Å = 10-8 cm (by definition)

10-8 cm x (1 m / 100 cm) = 10-10 m

1 Å = 10-10 m

2) Convert 2.1 x 10-10 m to nm:

2.1 x 10-10 m times (109 nm / 1 m) = 0.21 nm

Problem #4: Convert 0.0030 mm to nm

Solution:

3.0 x 10-3 mm x (106 nm / 1 mm) = 3.0 x 103 nm

Problem #5: Convert 6.25 x 10¯4 s to ms

The answer is 0.625 ms

Problem #6: Suppose a football, diameter 22 cm, is scaled up so that it becomes as big as the earth, diameter 13000 km. Calculate whether an atom of diameter 0.32 nm will become as big as:

a) a pin head, diameter 1 mm
b) a coin, diameter 1.9 cm
c) a football, diameter 22 cm
d) a weather balloon, diameter 1.8 m

Solution:

1) Convert all measurements to the same value. I used meters:

football: 0.22 m
earth: 13000000 m
atom: 3.2 x 10¯10 m

2) Solve via a ratio and proportion technique:

football diameter : earth diameter = atom diameter : an unknown value

0.22 m ÷ 13000000 m = 3.2 x 10¯10 ÷ x

Solving for x gives 0.0189 m

3) Answer b (1.9 cm) is the correct answer, since 0.0189 m = 1.9 cm.


Problem #7: Convert 10.6 kg/m3 to mg/μL

Solution:

1) Convert kg to mg:

10.6 kg/m3 times (106 mg/kg) = 1.06 x 107 mg/m3

2) Convert m3 to L:

I have made a video on this topic: How to Convert between Cubic Meters and Liters.

1 m3 = 1000 dm3 = 1000 L

1.06 x 107 mg/m3 times (1 m3 / 1000 L) = 1.06 x 104 mg/L

3) Convert L to μL:

1.06 x 104 mg/L x (1 L / 106 μL) = 1.06 x 10-2 mg/μL = 0.0106 mg/μL

Suppose the problem had been to convert kg/m3 to mg/mL? Then, the last part of the solution would look like this:

1.06 x 104 mg/L x (1 L / 103 mL) = 1.06 x 101 mg/mL = 10.6 mg/mL

10.6 kg/m3 = 10.6 mg/mL


Problem #8: Light travels at a speed of 3.00 x 1010 cm/s. What is the speed of light in km/h?

Solution:

1) Convert cm/s to km/s:

answer = 3.00 x 105 km/s

2) Convert seconds to hours:

answer = 1.08 x 109 km/hr

Problem #9: (a) Convert 38.0 centiliters to kiloliters; (b) How many Mg are there in 100,000,000 mg? (c) Convert 0.050 kg to mg.

Solution:

(a) 38.0 cL x (1 kL / 105 cL) = 3.80 x 10¯4 kL
(b) 1 x 108 mg x (1 Mg / 109 mg) = 0.1 Mg
0.050 kg x (106 mg / 1 kg) = 5.0 x 104 mg

Note that, in each case, a '1' is associated with the larger of the two units in the conversion factor. Those units are, in turn, kL, Mg, and kg.


Problem #10: If an oil tanker carrying 2.0 x 105 metric tons of crude oil of density 0.70 g/cm3 spills its entire load at sea, what area of ocean in square kilometers will be polluted if the oil spreads out to a thickness of 0.10 mm? (A metric ton = 103 kg.)

Solution:

1) Determine kg of oil:

2.0 x 105 times 1000 = 2.0 x 108 kg of oil

2) Change kg to g:

2.0 x 108 kg times 1000 = 2.0 x 1011 g

3) Determine volume of oil:

2.0 x 1011 g divided by 0.70 g/cm3 = 2.857 x 1011 cm3

4) Change 0.10 mm to cm:

0.10 mm times (1 cm / 10 mm) = 0.010 cm

5) Determine area of oil:

2.857 x 1011 cm3 divided by 0.010 cm = 2.857 x 1013 cm2

6) Change cm2 to km2:

2.857 x 1013 cm2 times (1 km2 / 1010 cm2) = 2857 km2

7) Comment: 1 km2 is 1 km on each side. 1 km = 105 cm, so the 1 km2 is 105 cm on each side, so:

105 cm times 105 = 1010 cm2 (which equals 1 km2)

Problem #11: A water bed has the following dimensions: 2.00 m x 1.40 m x 15.0 cm. Calculate the mass of water in (kg, please) in the bed ? Assume the density of the water to be 1.00 g/mL.

Solution:

1) Change water bed dimensions to cm:

200 cm x 140 cm x 15 cm = 420000 cm3

2) Change volume of water to mass of water:

1cm3 = 1 mL, so 420000 cm3 = 420000 mL

420000 mL x 1.00 g/mL = 420000 g

3) Change g to kg:

420. kg

Problem #12: (a) Convert 722 μg to mg. (b) How many pg in a ng? (c) How many mm in a km? (d) How many cm in a km? (e) How many mg in 722 micrograms?

Solutions:

(a) 722 μg times (1 mg / 1000 μg) = 0.722 mg
(b) 1 ng times (1000 pg / 1 ng) = 1000 pg (nano- means 10-9 and pico- means 10-12)
(c) 1 km times (106 mm / 1 km) = 1,000,000 mm
(d) 1 km times (105 cm / 1 km) = 100,000 cm
(e) 722 μg times (1 mg / 1000 μg) = 0.722 mg

Note the reversal of the order of the units in parts (a) and (e) of this problem. They are, of course, asking for the exact same thing.


Problem #13: Convert 3000 mL to daL.

Solution: 1) milli- means 10-3 and deca- means 101, therefore:

3000 mL times (1 daL / 104 mL) = 0.3 daL

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