### Metric conversion where two units (numerator and denominator) are convertedProblems #1 - 10

Problem #1: A sign gives the speed limit at 50.0 km/hr. What is this speed in centimeters per second?

Solution:

1) Convert km to cm:

50.0 km/hr times (105 cm / km) = 5.00 x 106 cm / hr

Convert hour to second:

5.00 x 106 cm / hr times (1 hr / 3600 s) = 1388.9 cm/s

Three sig figs gives 1390 cm/s as the answer.

Problem #2: An object is traveling at a speed of 7500. centimeters per second.  Convert the value to kilometers per minute.

Solution:

Convert cm to km. Convert seconds to minutes. Here it is set up in one line:

7500. cm/s times (1 km / 105 cm) times (60 s / min) = 4.500 km/min

Problem #3: 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:

Consider that 1 m3 is a cube 1 meter on each side. Change each meter to its equivalent in dm:
1 m3 = 10 dm x 10 dm x 10 dm = 1000 dm3

Since 1 dm3 = 1 L, we arrive at this many liters for 1 m3:

1000 L

Therefore:

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

Comment: notice that I converted from cubic meter to L, then from L to μL. The direct conversion is this:

(1 m3 / 1000 L) times (1 L / 106 μL) = 1 m3 / 109 μL

and a one-line solution looks like this:

10.6 kg/m3 times (106 mg/kg) times (1 m3 / 109 μL) = 10.6 x 10-3 mg/μL = 0.0106 mg/μL

Problem #4: Convert 9.98 x 1012 g/mL to kg/m3

Solution:

1) I propose to create a one-line solution in a step-by-step way. First, change g/mL to kg/mL:

9.98 x 1012 g/mL x (1 kg / 1000 g)

2) Next, I want to change mL to cubic meters, but suppose I really don't know (off the top of my head) what the conversion is. However, I do know this:

9.98 x 1012 g/mL x (1 kg / 1000 g) x (1000 mL / 1 dm3)

That cancels the mL and replaces it with dm3.

3) I have memorized the conversion from dm3 to m3 (see the discussion in the solution to Problem #3, just above):

9.98 x 1012 g/mL x (1 kg / 1000 g) x (1000 mL / 1 dm3) x (1000 dm3 / 1 m3)

4) The above calculation gives this answer:

9.98 x 1015 kg/m3

Suppose I did know the direct conversion from mL to m3. My conversion might then look like this:

9.98 x 1012 g/mL x (1 kg / 1000 g) x (106 mL / 1 m3)

The direct mL to m3 conversion depends on you knowing that 1 mL equals 1 cm3 and that 1 m3 is a cube 100 cm on each of its three sides (100 x 100 x 100 = 106)

Problem #5: Convert 1.17 x 10-8 kg/cm3 to g/mL

Solution: Problem #6: A light-year is the distance that light can travel in one year. If the sun is 150,000,000 km away, how many light years is the Sun from Earth? Assume light travels at a speed of 3.0 x 1010 cm/s

Solution:

1) Determine the kilometers in one light-year:

(3.0 x 1010 cm/s) (1 km / 105 cm) (3600 s / 1 hr) (24 hr / 1 day) (365.25 day) = 9.46728 x 1012 km <--- this is one light-year

In order, these are the above conversions:

cm to km
per second to per hour
per hour to per day
per day to one year

Note that 365.25 days was used. Some teachers will use 365 days. Note also that a rounded-off value for the speed of light was used. It is actually slight less than the value used.

Note that I did not use the unit year in the denominator at the end. This is to emphasize that a light-year is a measure of distance. There are 9.46728 x 1012 km in one light-year.

You could write the unit as km/ly.

2) Determine light-years that the Sun is from Earth:

150,000,000 km / 9.46728 x 1012 km/ly = 0.000015844 ly

Problem #7: Convert 0.153 mol/L to micromole/nm3.

Solution:

1) Convert the numerator:

0.153 mol/L times (106 μmol / mol) = 1.53 x 105 μmol/L

2) Convert the denominator:

Replace L with dm3 to obtain 1.53 x 105 μmol/dm3 (remember, 1 L = 1 dm3)

deci- is 10-1 and nano- is 10-9, so we have an absolute exponential distance of 108

1.53 x 105 μmol/dm3 times (1 dm / 108 nm)3 = 1.53 x 10-19 μmol / nm3

Problem #8: Convert 3.75 μm/s to km/hr

Solution:

micro- = 10-6 and kilo- = 103. The absolute exponential distance is 109.

3.75 μm/s times (1 km / 109 μm) x (_______)

In the empty spot, put a conversion that changes seconds to hours. There are 3600 second in one hour.

The rest is left to the reader.

Problem #9: 4.18 x 104 kg/L to dg/mL

Solution: Notice that, in there numerator, there is a total of 108 and, in the denominator is 103. That makes 105.

The answer is 4.18 x 105 dg/mL.

Problem #10: Convert 5.00 x 105 cm/s to m/day

Solution:

1) Convert the numerator:

5.00 x 105 cm/s times (1 m / 1000 cm) <--- cancels cm, replaces it with m

Based on this one conversion alone, I will use 5.00 x 102 m/s in the next step.

2) Convert the denominator:

The starting unit is now m/s and we have to take seconds up to day. The first conversion is seconds to minutes:
5.00 x 102 m/s times (60 s / min) <--- the seconds will cancel

The next conversion will replace minutes with the next time unit up, which is hours:

5.00 x 102 m/s times (60 s / min) times (60 min / hr) <--- notice how minutes will cancel

The last conversion replaces hours with day, which is what we want:

5.00 x 102 m/s times (60 s / min) times (60 min / hr) times (24 hr / day) <--- hour cancels and day stays in the denominator

3) Everything put together in one line (I start with the original cm/s value):

(5.00 x 105 cm/s) x (1 m / 1000 cm) x (60 s / min) x (60 min / hr) x (24 hr / day) = 4.32 x 107 m/day