is as warm as 4 of Z 's: and one of L 's as warm as 3 of M 's. Which is best, giving equal weight in the result to rapidity of work, lightness, and warmth?
Answer.—The order is M, L, Z.
Solution.—As to rapidity (other things being constant) L's merit is to M's in the ratio of 5 to 2: Z's to L's in the ratio of 4 to 3. In order to get one set of 3 numbers fulfilling these conditions, it is perhaps simplest to take the one that occurs twice as unity, and reduce the others to fractions: this gives, for L, M, and Z, the marks 1, 2⁄5, 2⁄3. In estimating for lightness, we observe that the greater the weight, the less the merit, so that Z's merit is to L's as 5 to 1. Thus the marks for lightness are 1⁄5, 2⁄3, 1. And similarly, the marks for warmth are 3, 1, ¼. To get the total result, we must multiply L's 3 marks together, and do the same for M and for Z. The final numbers are 1 × 1⁄5 × 3, 2⁄5 × 2⁄3 × 1, 2⁄3 × 1 × ¼; i.e. 3⁄5, 2⁄3, 1⁄3; i.e. multiplying throughout by 15 (which will not alter the proportion), 9, 10, 5; showing the order of merit to be M, L, Z.
Twenty-nine answers have been received, of which five are right,