Another method is to group into tens when it can be conveniently done, and still another method in adding up long columns is to add from the bottom to the top, and whenever the numbers make even 10, 20, 30, 40 or 50, write with pencil a small figure opposite, 1, 2, 3, 4 or 5, and then proceed to add as units. The sum of these figures thus set out will be the number of tens to be carried to the next column.

6^{2} 2 8 3 5^{2} 4^{1} 2 8 4 9 6 2 7^{2} 1 8^{2} 8 3^{2} 5 5 2 7 1^{1} 3 2^{1} 5 8 8 _________________ 5 0 2 8

SHORT METHODS OF MULTIPLICATION.

For certain cla.s.ses of examples in multiplication short methods may be employed and the labor of calculation reduced, but of course for the great bulk of multiplications no practical abbreviation remains. A person having much multiplying to do should learn the table up to twenty, which can be done without much labor.

To multiply any number by 10, 100, or 1000, simply annex one, two, or three ciphers, as the case may be. If it is desired to multiply by 20, 300, 5000, or a number greater than one with any number of ciphers annexed, multiply first by the number and then annex as many ciphers as the multiplier contains.

TABLE.

5 cents equal 1/20 of a dollar.

10 cents equal 1/10 of a dollar.

12-1/2 cents equal 1/8 of a dollar.

16-2/3 cents equal 1/6 of a dollar.

20 cents equal 1/5 of a dollar.

25 cents equal 1/4 of a dollar.

33-1/3 cents equal 1/3 of a dollar.

50 cents equal 1/2 of a dollar.

Articles of merchandise are often bought and sold by the pound, yard, or gallon, and whenever the price is an equal part of a dollar, as seen in the above table, the whole cost may be easily found by adding two ciphers to the number of pounds or yards and dividing by the equivalent in the table.

_Example_. What cost 18 dozen eggs at 16-2/3c per dozen?

6)1800 _____ $3.00

_Example_. What cost 10 pounds b.u.t.ter at 25c per pound?

4)1000 ----- $2.50

Or, if the pounds are equal parts of one hundred and the price is not, then the same result may be obtained by dividing the price by the equivalent of the quant.i.ty as seen in the table; thus, in the above case, if the price were 10c and the number of pounds 25, it would be worked just the same.

_Example_. Find the cost of 50 yards of gingham at 14c a yard.

2)1400 ----- $7.00

When the price is one dollar and twenty-five cents, fifty cents, or any number found in the table, the result may be quickly found by finding the price for the extra cents, as in the above examples, and then adding this to the number of pounds or yards and calling the result dollars.

_Example_. Find the cost of 20 bushels potatoes at $1.12-1/2 per bushel.

8)2000 250 ----- $22.50

If the price is $2 or $3 instead of $1, then the number of bushels must first be multiplied by 2 or 3, as the case may be.

_Example_. Find the cost of 6 hats at $4.33-1/3 apiece.

3)600 4 ------ 24.00 2.00 ------ $26

When 125 or 250 are multipliers add three ciphers and divide by 8 and 4 respectively.

To multiply a number consisting of two figures by 11, write the sum of the two figures between them.

_Example_. Multiply 53 by 11. Ans. 583.

If the sum of the two numbers exceeds 10 then the units only must be placed between and the tens figure carried and added to the next figure to the left.

_Example_. Multiply 87 by 11. Ans. 957.

FRACTIONS.

Fractional parts of a cent should never be despised. They often make fortunes, and the counting of all the fractions may const.i.tute the difference between the rich and the poor man. The business man readily understands the value of the fractional part of a bushel, yard, pound, or cent, and calculates them very sharply, for in them lies perhaps his entire profit.

TO REDUCE A FRACTION TO ITS SIMPLEST FORM.

Divide both the numerator and denominator by any number that will leave no remainder and repeat the operation until no number will divide them both.

_Example_. The simplest form of 36/45 is found by dividing by 9 = 4/5.

To reduce a whole number and a fraction, as 4-1/2, to fractional form, multiply the whole number by the denominator, add the numerator and write the result over the denominator. Thus, 4 X 2 = 8 + = 9 placed over 2 is 9/2.

TO ADD FRACTIONS.

Reduce the fractions to like denominators, add their numerators and write the denominator under the result.

_Example_. Add 2/3 to 3/4.

2/3 = 8/12, 3/4 = 9/12, 8/12 + 9/12 = 17/12 = 1-5/12. Ans.

TO SUBTRACT FRACTIONS.

Reduce the fractions to like denominators, subtract the numerators and write the denominators under the result.

_Example_. Find the difference between 4/5 and 3/4.

4/5 = 16/20, 3/4 = 15/20, 16/20-15/20 = 1/20. Ans.

TO MULTIPLY FRACTIONS.

Multiply the numerators together for a new numerator and the denominators together for a new denominator.

_Example_. Multiply 7/8 by 5/6.

7/8 x 5/6 = 35/48. Ans.

TO DIVIDE FRACTIONS.