## Practical Applications

To find the value of any number of articles at 75 cents each, say 248 yards of cloth at 75 cents a yard, deduct one-quarter of 248 from it, and call the remainder dollars. At a dollar a yard the result will be \$248 ; then at 75 cents it must be \$248 - (1/4 of 248)= \$186.

Find the cost of 84 yards of cloth at 121/2 cents a yard.

What will 328 bags of potatoes cost at 75 cents a bag ?

Find the cost of 20 gross of pen-handles at 25 cents each.

What will 216 pounds or raisins cost at 16 2/3 cents a pound ?

A railway charges a cent a mile, for the first 50 miles, for carrying a cord of wood, and then 3 cents for every 4 miles beyond the 50 ; what will it cost to carry 250 cords 90 miles ?

If a clerk receives \$640 a year, and his expenses are \$325 a year, how many years will it take him to pay for a 56-acre farm at \$45 an acre ?

A fruit dealer bought 5 bushels of cherries at \$2.50 a bushel, and sold them at 15 cents a quart; did he gain or lose, and how much ?

6. Other Short Fractional flu Multiplication.

To multiply any number containing 1/2, such as 7 1/2, 19 1/2, 12 1/2, etc., by itself, multiply the whole number by the next higher whole number, and annex 1/4 to the product. Thus, 7 1/2 X 7 1/2 = 7 X 8 + 1/4 = 56 1/4; and 19 1/2 X 19 1/2 = 19 X 20 + 1/4 = 380 1/4. Apply this rule to the first column of exercises below.

To multiply two fractional numbers, such as 7 1/4 and 7 3/4, multiply 7 by 8, and add to the product the product of 3/4 and 1/4, or 3/16, and you have the correct product. Apply this rule to the second column of exercises below.

To multiply two fractional numbers, each containing 1/2, such as 5 1/2 by 7 1/2, add the product of the whole numbers, plus 1/4 to 1/2 of the sum of the whole numbers. Thus,

5 X 7 = 35; 35+1/2 (5+7) = 35 + 6 = 41, and to this add 1/4, making 41 1/4, the product. Apply this rule to the third column of exercises below.

To multiply two fractional numbers each containing 3/4, such as 11 3/4 by 13 3/4 ; to the product of the whole numbers add the product of their sum by 3/4, after which add the product of 3/4 by 3/4. This rule applies in all cases where both fractions are the same. Apply it in working the fourth column of exercises below.

Examples for Practice.

6 1/2 X 6 1/2 4 1/2 X 4 1/2 5 1/2 X5 1/2

8 1/2 X 8 1/2

12 1/2 X 12 1/2 19 1/2 X 19 1/2

5 1/4 X 5 3/4

8 1/4 X 8 3/4

9 1/3 X 9 2/3

6 4/5 X 6 1/5

2 2/5 X2 3/5

9 1/6 X 9 5/6

2 1/2 X 5 1/2 4 1/2 X6 1/2

2 1/2 x 8 1/2

9 1/2 X 8 1/2

2 1/2 X 7 1/2

1 1/2 X 9 1/2

2 3/4 X 4 3/4 3 1/4 X 3 1/4 8 3/4 X 2 3/4

8 3/5 X 4 3/5

9 1/4 X 4 1/4

8 1/3 X2 1/3

7. To Multiply Numbers Ending in 5.

To multiply two small numbers each of which ends in 5, such as 35 and 75, take the product of the 3 and 7, increase this by one-100 half of the sum of these figures, and prefix the result to 25. Thus,

35 5X5 = 25

75 7X3 = 21, 21+1/2 (7+3)=26

This rule will be found to hold good with any two numbers each of which end with 5. Apply it to the examples below :

Examples for Practice. 45 X 85; 95X25; 35X65; 75X95; 85X55.

8. To Multiply Special Large Numbers.

In the multiplication of large numbers, where one part of the multiplier is a multiple of the remainder, the work can always be considerably abbreviated. See the examples below: 2043 427 = 420 + 7 = (7X60)4-7

14301 = 2043X7 858060 ='2043 X 420 = 14301 X 60

872361 = product.

3142 972 = 900+72 = 900 + (9 X 8)

2827800 = 3142 X 900 226224 = 3142 X 72 = 28278X8

3054024 = product.

We first multiply by 7, then by 420, thus taking the number 2043, 427 times. The contraction is made in multiplying by 420. We take its factors, 7 and 60 ; we have already multiplied by 7, so that all that remains to be done is to multiply 14301 by 60 and place it under. The sum of the two partial products gives the whole product. As a test exercise, multiply some number by 14412 so as to have only two lines instead of five to add. See exercises in second and third columns below:

Remember, that any number is divisible by 3 if the sum of its digits is divisible by 3; that any number is divisible by 5 if its right-hand figure is 5 or o; that any number is divisible by 9 if the sum of its digits is divisible by 9.

## Exercises

2013 X 927 ; 1214 X 279 ; 3135 X 728; 2146 X 287; 3210 X 189; 21401 X 729; 31252 X 14412 ; 42001 X 70357; 15421 X 81273; 30012 X 94572.

Lightning Table for Marking Goods Bought by the Dozen.

Retailers buy most of their articles by the dozen, such as boots, shoes, hats, caps, and notions of various kinds. A vast amount of time is employed in marking goods by the old process, and errors are frequent, owing to the unnecessary figures used. If the purchaser will commit to memory the following table, he can instantly find the retail price of a single article with any desired business per cent. added. The following per cents. are those generally used in business:

Divide the cost To make per doz. by

20 per cent. 10

33/3 " 9

50 " 8

100 " 6

40 " 10, and add 1-6 itself.

35 " 10 " 1-8 "

37 1/2 " 10 " 1-7 "

30 " 10 " 1-12 "

25 " 10 " 1-24 "

12 1/2 " 10 and subtract 1-16 "

16 2/3 " 10 " 1-36 "

18 3/4 " 10 " 1-96 "

S. T.Jones purchased one dozen hats for \$27.00, and marked them to sell at a profit of 33 1/3 per cent. What was the retail price of a single hat ?

## Explanation

According to table, 9 ) \$27.00 simply divide the cost per dozen by 9. -------Ans. \$3.00

For how much must I sell books bought at \$28.00 per dozen, to gain 25 per cent ?

## Explanation

Removing the point 24) 2.800 one place to the left, on \$28.00 we get .11 \$2.800. Now add 1-24 itself and we --------have \$2.91 = ans. \$2.91 +

The above table should be committed to memory.