100. SECTION X. — PRACTICE. ́

In PRACTICE, questions frequently occur which can be performed more expeditiously than by either of the foregoing methods. Most business operations which contain compound numbers may be abbreviated, by first finding the value for the highest denomination in the question, and then taking aliquot parts of this for lower denominations.

NOTE. The aliquot parts of a number are all the numbers that will measure it. Thus, 1, 2, 3, 4, 6, 8 and 12 are aliquot parts of 24.

1. What cost 18 bushels of wheat, at $1.371⁄2 a bushel?

2. What cost 148 bushels of potatoes, at 12 cts. a bushel? at 183 cts.? at 25 cts.? 31 cts.? 37 cts.? 50 cts.? at 561 cts.? $0.62? $0.75? $0.87 ? $1.12?

3. What will 37 yards of cloth cost, at 15s. 10d. per yard?

4. What will 37 yds. of cloth cost, at £1 12s. Sid. per

yard?

Tables showing the aliquot parts of money, weights and measures, may be prepared by the pupil, if the teacher thinks it best. The following may serve as a model:

5. What cost 3 yd. 3 qr. 2 na. of broadcloth, at $3.25 per yard?

6. At $6.50 per ton, what cost 5 ton 8 cwt. 3 qr. 16 lb. of coal? 1 ton 4 cwt. 2 qr. 21 lb.? 15 cwt. 1 qr. 20 lb.? 7. At $87.50 per acre, what cost 3 A. 3 R. 24 rd. of land? 5 A. 1 R. 15 rd.? 3 A. 75 rd.?

For more examples in Practice, see Art. 94, ques. 27—32.

101. CONTRACTIONS IN MULTIPICATION AND DIVISION.

PROB. I. TO MULTIPLY BY 5. Take of the number to be multiplied, and multiply that by 10. This will give ten halves of the number, which is equal to 5 times the number. Or, Annex a naught, and halve it.

1. How much are 5 times 24? One half of 24 is 12; and ten times 12 are 120. (28, page 30.)

2. How much are 5 times 35? of 35 is 17; and 10 times 173 are 175.

3. How much are 5 times 44? 55? 248? 355-340+ 15? 677-660+17? Say 677 is equal to 660 and 17. of 660 is 330; of 17 is 81, which added to 330 makes 338; 10 times 338-3385.

4. Multiply these numbers by 5. 548; 874; 957; 1684; 25702400+160+10; 2590; 3376; 4087; 5359.

PROB. II. TO MULTIPLY BY 25. Take multiply that by 100. This will give which is equal to 25 times the number. Annex two naughts, and take of it.

of the number, and go of the number,

Or,

5. Multiply the following numbers by 25. 8; 12; 20; 17; 21; 22; 34; 47; 144; 275-240+35; 575-400+160 +15; 576; 648; 185; 668; 754; 3216; 4036.

PROB. III. TO MULTIPLY BY 50. Take of the number, and multiply that by 100. This will give 190, which is equal to 50.

Or,

Annex two naughts, and halve it.

6. Multiply the numbers in Prob. II. by 50.

PROB. IV. TO MULTIPLY BY 121. Take of the number, and multiply that by 100. This will give 180, which is equal to 121.

7. Multiply the numbers in Prob. II. by 121.

PROB. V. TO MULTIPLY BY 125. Take of the number, and multiply that by 1000. Or,

Annex three naughts, and take of it. NOTE. 1900- =

8. Multiply the numbers in Prob. II. by 125.

PROB. VI. TO MULTIPLY BY 331. and multiply that by 100. Or,

125.

Take of the number,

Annex two naughts, and take of it.

NOTE. 180=33}.

9. Multiply the numbers in Prob. II. by 33.

PROB. VII. TO DIVIDE BY 25. Multiply by 4, and divide the product by 100. (39.) This will give four one hundredths, or of the number.

10. Divide these numbers by 25. 150; 225; 400; 500; 675; 270; 576; 384; 990; 1575; 1640; 3248; 4564; 8167.

PROB. VIII. TO DIVIDE BY 125.

Multiply by 8, and divide

the product by 1000. (39.) This will give Tobo, or 125, of the number.

11. Divide the numbers in the last Prob. by 125.

PROB. IX.

TO DIVIDE BY 121. Multiply by 8, and divide

the product by 100.

12. Divide the numbers in Prob. VII. by 121.

PROB. X. TO DIVIDE BY 50. Multiply by 2, and divide the product by 100.

13. Divide the numbers in Prob. VII. by 50.

TO DIVIDE BY 331.

Multiply by 3, and divide

PROB. XI. the product by 100.

14. Divide the numbers in Prob. VII. by 331.

NOTE. The pupil should perform all the exercises in the above problems mentally, as well as on the slate.

PROB. XII. TO MULTIPLY BY ANY NUMBER OF 9's.

Annex

as many naughts to the multiplicand as there are nines in the multiplier, and from this number subtract the given multiplicand; the remainder will be the product.

15. Multiply 841685 by 99999.

SOLUTION. 84168500000

841685

84167658315

16. Multiply 51487 by 9; by 99; by 999; by 99999. 17. Multiply 371548 by 5; by 25; by 50; by 121; by 33; by 125.

18. Divide 419684 by 25; by 125; by 12; by 50; by 334. 19. Multiply 51487 by 50; by 121; by 331; by 25; by 125; by 5.

20. Divide 610840 by 334; by 50; by 12; by 125; by 25.

102. SECTION XI.— PERCENTAGE.

The term percentage is used to express some number of hundredths of a given sum or quantity. Per cent. signifies hundredths, from the Latin per centum, by the hundred. Therefore, any given per cent. of a quantity is so many hundredths of that quantity. Thus, if a man buys goods for $100, and sells them for $108, we say that he gains 8 per cent. of the cost of the goods. So 1 per cent. of any number, or quantity, is .01 of that quantity; 3 per cent., .03; 100 per cent. of a

quantity is 100 hundredths of it; that is, the quantity itself. 500 per cent. of a number is 5 times that number.

1. A man collects bills for a merchant to the amount of $15, and is to receive 3 per cent. for collecting them. How much shall he receive?

NOTE 1. 3 per cent. is 3 cents for every 100 cents, $3 for every $100, &c. If he receives 3 cents for collecting $1, he must receive 3 times as many cents as there are dollars collected. Ans., 45 cents.

2. What is 3 per cent. of $1? Of $20? $40? $25? $55? $75? $517? What is 5 per cent. of $1? Of $8? $28? $87? $185! What is 8 per cent. of $1? $12? $25! $354? $1025 $3108?

3. A man borrowed $20 for a year, and is to pay 6 per cent. for the use of it. How much shall he pay? What is 6 per cent. of $1? $17? $75

$584?

4. What is 12 per cent. of $1? $75? $354?

is $8; and 12

Of $8? $24? $44? $50?

of 100 per cent.

NOTE 2. 12 per cent. is 100 per cent. of $8 per cent. is as much. Ans., $1. Again: 100 per cent. of $44 is $44; 12 per cent. will be as much=$54. See Practice Table.

5. What is 20 per cent. of $40? $35? $54? $48? $357? $485? 20 per cent. is of 100 per cent.

6. What is 25 yards? 100 yards? 7. What is 33 acres?

per cent. of 12 lb.? Of 40 lb.? Of 50 lb.? 60 35 yards?

per cent. of 18? 24? 30? 75? 240 oz.? 270

8. What is 50 per cent. of $1? $10? $25? $87? 84 cwt.? 75 bushels?

RULE FOR COMPUTING PERCENTAGE. Multiply by the rate per cent. expressed decimally; or assume the rate at 100 per cent., and take the aliquot parts as in Practice.

9. What is 23 per cent. of $3? Of $7? $15? $25? $187? 1016? $10.16?