How to do Weight & Balance for Cessna 172?
The Calculation section of “Center of gravity of an aircraft” on Wikipedia gives an overview of the steps.


  
  Determine the weights and arms of all mass within the aircraft.
  Multiply weights by arms for all mass to calculate moments.
  Add the moments of all mass together.
  Divide the total moment by the total weight of the aircraft to give an overall arm.
  


The Basic Empty Weight (BEW) of the aircraft will come from the Weight &amp; Balance information that is unique to each aircraft. The computed arm for the BEW may have some rounding error, so treat it as informational. Use the precise figures for Weight and Moment from your aircraft’s W&amp;B.

The arms, distances from some reference datum, will also come from your aircraft’s W&amp;B info: fuel, front seats, rear seats, and baggage areas on a Cessna 172. Copy those into your table as well.

Avgas (100LL) weighs 6 pounds per gallon, so the weight is 6 times the number of gallons.

I like to work top-to-bottom and left to right. For the grand finale, divide the sum of the Moment column by the sum of the Weight column to get the overall center of gravity.

The example below is from my PP-ASEL checkride taken in a Cessna 172. The items in boldface are computed values. The digits after the decimal are there for typographical purposes and do not necessarily indicate significant figures.

$$\begin{array}{c|r|r|r|}
                     &amp; \textit{Weight} &amp; \textit{Arm} &amp; \textit{Moment} \\ \hline
\text{Airplane}      &amp; 1{,}665.6     &amp; 39.2       &amp; 65{,}928.93      \\
                     &amp;               &amp;            &amp;                  \\
\text{Fuel (50 gal)} &amp; \bf{300.0}    &amp; 48.0       &amp; \bf{14{,}400.00} \\
                     &amp;               &amp;            &amp;                  \\
\text{Bacon}         &amp; 180.0         &amp; 37.0       &amp; \bf{6{,}660.00}  \\
\text{Shelton}       &amp; 172.0         &amp; 37.0       &amp; \bf{6{,}364.00}  \\
                     &amp;               &amp;            &amp;                  \\ \text{Duffel}        &amp; 10.0          &amp; 73.0       &amp; \bf{730.00}      \\
\text{Shelton bag}   &amp; 10.0          &amp; 73.0       &amp; \bf{730.00}      \\
                     &amp;               &amp;            &amp;                  \\ 
\text{Backpack}      &amp; 15.0          &amp; 93.3       &amp; \bf{1{,}399.50}  \\ \hline
                     &amp; \bf{2{,}352.6}&amp; \bf{40.9}  &amp; \bf{96{,}212.43}
\end{array}$$

In this particular case—given for information only—the center of gravity (40.9 ≈ 96,212.43 / 2,352.6) at the total weight is within limits for the Normal category as given in the aircraft’s Pilot Operating Handbook (POH). Your case may vary. The Pilot-in-Command is responsible for computing Weight &amp; Balance for each flight.

As a point of interest, my DPE was Clyde Shelton who recently flew his ten thousandth checkride.