MATHEMATICS. 27 



by 4 ; the product is the same as if you had multi- 

 plied by 25, and is quicker done. 



Any sum to be multiplied by 125 requires three 

 cyphers added, and then to be divided by 8. Instead 

 of multiplying by 5, add a cypher, and divide by 2. 



In the same manner, instead of multiplying by 9, 

 or 99, or 999, add as many cyphers to the multipli- 

 cand as there are figures in the multiplier, then sub- 

 tract the multiplicand from the total. The product 

 will be found quicker. 



Multiplication may thus be easier effected by 

 figures that are not exact dividends of whole num- 

 bers. 



Ratios. 



The sum of two or more ratios is obtained by 

 multiplying the antecedents together, and the conse- 

 quents together ; the products will form a ratio, 

 which is called the sum of the given ratios. Thus 

 the ratio of 2 to 3 ; plus the ratio of 2 to 5 ; plus 

 the ratio of 2 to 7, is equal to the ratio of 16 to 

 105; viz. 2x2x4=16 



3x5x7=105. 



Binary Arithmetic. 



Binary division in arithmetic, consisting of halves, 

 quarters, eighths, sixteenths, &c., has always been 

 found much more convenient and more in unison 

 with the habits of the people, than the decimal divi- 

 sion, which in France was found to produce great 

 frauds, in consequence of uneducated persons being 

 incapable of accurately distinguishing such divisions. 



