Book-keepers, bank clerks and others who have constant calculations to make need to be both expert and rapid. There are many short cuts practiced which every person should know, and a few of the most important are here explained.

The art of adding quickly is acquired by learning to read a column of figures as you would a sentence of words, and those words composed of letters.

By Practice we may become so familiar with figures that when we see a group of them, we can tell at a glance what the sum of them reads, without spelling the figures at all. In practicing the reading of a column of figures in this way, we do not let the brain work at all, but simply pass the eye over the figures slowly at first, but increase the speed as proficiency is acquired.

A few minutes' daily practice will produce astonishing results in a very short time; beginning with two figures, then three, four, and so on until finally we become able to write the Sum total of long columns. For example, when we see the figures 9, 8, 6, 4, we know at a glance that the sum is 27 without reading the figures themselves or spelling them out.

Reading a column of figures is done by dividing a large group of figures into smaller ones and from group to group through the column, just as from word to word we read through a sentence.

The most important qualities of an accountant are accuracy and speed. The most speedy calculators are usually the most correct.

No labor should be regarded too great to master this, the key to all numerical as well as business transactions.

3. To multiply by numbers ending or beginning with 1; as 21, or 31, or 13, or 17, or 51 or 501, or 103, or any number of two figures where one of them is 1, or of three figures where two of them are o and 1, a good deal of time can be saved by abbreviating the ordinary process, as below :

Rule I. When the I stands at the right, multiply by the figure at its left, and place the product under the place of tens under the number to be multiplied. Add the two together, and their sum is the product.

If the multiplier is 201, or 301, or 401, or any number of hundreds, place the product two places to the left. If the multiplier is 2001, or any number of thousands, place the product three places to the left. In other words, the product of hundreds under hundreds' place ; of thousands under thousands' place, etc.

Rule II. When the 1 stands at the left of the multiplier, multiply by the number at the right, and place the product under the multiplicand one place to the right of the units' place. The second figure of the product under units' place of the multiplicand.

Examples. 231423X21=4859883.

231423=231423X1

4628460=231423X20

4859883=product. 20213 X13 20213X201

60639 40426

262769=product. 406281i3=product. Examples for Practice. 2134X11; 6215X11; 2143X11; 3212X11; 4215X11 ; 2153X21; 1024X31; 8461X41; 2222X14; 3120X19; 2132X201; 2146X102; 9842X 301; 8002X402; 4621X105.

3. Short Cuts in General Multiplication.

Example i. Multiply 96 by 97.

96 . . .4 (Complement.)

97 3 (Complement.)

The complement of a number is the difference between the number and the unit of the next higher order, thus the complement of 96 is 4 (100-96); of 97 is 3 ; of 987 is 13, etc. To multiply these two numbers, multiply the complements 4 and 3, and place the product, 12, in the answer. For the remaining two figures subtract across, either the 4 from the 97, leaving 93, or the 3 from the 96, leav-es 93. Apply this rule to the first and second lines of exercises below.

Example 2. Multiply 37 by 43.

The mean number - that is, the number which s as much greater than 37 as it is less than 43 - is 40. Forty squared, or multiplied by itself, gives 1600. The square of 3, the difference between the mean number and one of the numbers is 9. 1600 - 9=1591=the product of 37 and 43. Apply this rule to the exercises below.

Example 3. Multiply 76 by 46. 46 6X6=36, carry 3. 76 6X (7+4)=6X 11=66, and 3

------ carry 69.

3496 4X7=28, and 6 to carry, 34. Multiply units by units for the first figure of the product, the sum of the tens by units for the second figure, and tens by tens for the third figure, carrying when necessary. A similar rule applies to numbers having the left-hand figures the same. Work the following exercises.

Exercises. 97X98; 95X94; 97X96; 5X93; 93X97; 97X94; 99X89; 994X

995 ; 993X994; 989X998; 992X995 ; 988 X997; 976X999; 954X998; 87x73; 63

X57; 42X38; 45X35; 116X124; 1012X 988; 1025x975; 56X56; 72X32; 87x37; 61X63; 114X114; 137X177; 125x112.

4. Short Cuts in Multiplication and Division by .Special Numbers.

To multiply any number by 25, add two ciphers, and divide the number by 4.

To multiply any number by 125, add three ciphers, and divide the number by 8.

To multiply a number by any number of nines, add as many ciphers to the number as there are nines, and from this subtract the original number.

5. Short Multiplication and Division in Fractional Numbers.

To multiply any number by 21/2, add one cipher, and divide by 4.

To multiply any number by 31/3, add one cipher, and divide by 3.

To multiply by 331/3, add two ciphers and divide by 3.

To multiply any number by 13/7, add one cipher, and divide by 7.

To multiply by 162/3, add two ciphers, and divide by 6.

To multiply by 142/7, add two ciphers, and divide by 7.

To multiply by 875, add three ciphers, multiply by 7 and divide by 8.

To divide by 25, multiply by 4, and cut off two figures.

To divide by 125, multiply by 8, and cut off three figures.

To multiply by 121/2, add two ciphers, and divide by 8.

To divide by 121/2, multiply by 8, and cut off two figures at the right.

To divide by 331/3 multiply by 3, and cut off two figures at the right.