Excerpt from Key to the Progressive Practical Arithmetic: Including Analyses of the Miscellaneous Examples in the Progressive Intellectual Arithmetic; For Teachers Only In many examples and problems the same results may be obtained by different processes, but the shortest and most simple method is to be desired; hence the object of a Key should be to give not results only, but the explanation of methods, and a full analysis of such questions as contain a peculiar application of principles involved. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.