# Dividing values with units

I'm reading about the subject of heat in a basic physics book. If I am not mistaken the formula to work out how much energy is required to increase the temperature of water is

e = M * t * shc


Where

• e is energy in Joules
• M is mass in kg
• t is temperature to increase by in °C
• shc is specific heat capacity in J/kg°C

If I need to solve how much the temperature has varied I rearrange the equation like this...

t = e / M / shc


And this is where I get stuck.

• e = 7.2 * 108 J
• M = 105 kg
• shc = 4.23 J/kg°C

What is the resulting unit of measurement of the following?

e / M = ?
M / shc = ?

-

Just write it out like this:

$t = \frac{e}{\frac{m}{shc}}= \frac{J}{\frac{kg}{\frac{J}{kg\cdot{}^\circ{}C}}} = \frac{J}{J \cdot kg \cdot \frac{1}{kg\cdot^{\circ}C}} = \frac{1}{\frac{1}{^{\circ}C}} = ^{\circ}{\rm{}C}$

-
This was the explanation I found easiest to understand, thanks! – Peter Morris May 22 '13 at 17:13

When multiplying or dividing units, all you need to do is put the units in the numerator or denominator (wherever they appeared) of the answer. So:

$$[e/M]={J\over kg}$$ $$[M/shc]={kg\over{J\over kg^oC}}={kg^2\,^{\circ}\rm C\over J}$$

But this is not the correct way of analyzing your units. You have
$t = e / M / shc = e / (M * shc)$

The units of this are: $$[t]={J\over kg {J\over kg\,^{\circ}\rm C}}=\,^{\circ}\rm C$$

-
Wow, Jim. Apparently your answer was so good that two different people decided to post different halves of your answer afterwards. – Jack Dozer May 21 '13 at 15:40
@Dan Indeed. But everyone is entitled to answer in their own way, so let's not have inflammatory comments like that anymore. Okay? – Jim May 21 '13 at 15:41