A cake, when put into one of the pans of the scale, appeared to weigh four ounces more than nine-elevenths of its true weight, but when put into the other pan it appeared to weigh three pounds more than in the first pan. What was its true weight? Assume the error is due to unequal lengths of the arms and unequal weights of the pans. Also assume the scale is balanced when the pans are empty.
[Problem submitted by Kevin Windsor, LACC Instructor of Mathematics. Source: 536 Puzzles & Curious Problems by Henry Ernest Dudeney, 1967.]
Solution for Problem 5:
Let w be the true weight of the cake, x the weight of the left pan, a the length of the left arm, y the weight of the right pan, b the length of the right arm.
The scale being balanced when the pans are empty implies ax=by.
Subtract ax from the left hand sides and by from the right hand sides from the above equations, we get
Divide equation (3) by equation (4) we get
, Solve the quadratic equation of w, we get w=143.