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What is the value of 961b. of tea, at 38. 103d. per lb. The price at £1 per lb. would be £96 at 3s. 4d.= { £

16
at
i 3s. 4d.

1 " 12
at
2d.=1
4d.

16
at id.= 12d.

4 at jd.= lofd.

2 at 3s. 10 d.

£19 66 4s. 1. What is the price of 894yds. of broadcloth, at $4.75 per yd.?

2. What is the price of 127 pounds of sugar, at 12 cts.

per 16.?

3. What is the value of 49 A. 3R. 15r. of land, at $125

per acre ?

4. What is the value of 15T. 13cwt. of hay, at 87cts.

per cwt. ?

5. What is the value of 96yds. 3qr. 3na. of broadcloth, at £1 2s. 6d. per yard ?

6. What will 17T. 11cwt. 2qr. 21lb. of iron cost, at $19.75 per ton ?

7. What will 7cut. 2qr. 11lb. of sugar cost, at $7.621

per cwt. ?

8. What is the cost of 389bu. 2pk. 4qt. of wheat, at $1.12į per bushel ?

9. What is the cost of 163 A. 2R. 25r. of land, at $15.75

per acre ?

10. If i lb. of beef costs 61 cents, what will 16cwt. 2qr. 71b. cost, at the same rate ?

11. If an ounce of indigo costs 37cents, what must I give for lcwt. lqr. 141b. 60%., at the same rate ?

12. What is the value of 364yd. 3qr. Ina. of sheeting, at 12cts. a yard ?

13. What is the value of 37921b. of coffee, at 10cts. per pound? At 12 cts.?

14. Bought 76bu. 3pk. of potatoes, at 37 cts. a bushel;

19bu. 2pk. of wheat, at $1.10 a bushel ; 37bu. 1pk. of barley, at 62 cts. a bushel; and 10 T. 15cwt. of ħay, at $16.00 a ton. What was the amount of the whole ?

15. What is the value of 11cwt. 3qr. 9lb. of sugar, at £2 3s. 6 d. per hundred weight ?

16. What is the price of 361b. 50%, of tea, at 87}cts. per pound?

17. What is the value of 191bu. 31pk. of apples, at $1.25 per bushel ?

18. What is the value of 187yd. lqr. 2na. of silk, at 93 cents a yard ?

19. What is the interest of $174.50 for 4yr. 6mo. 27d., at .05 per year ?

20. What is the interest of $2375 for 5yr. 11mo. 23d., at .055 per year?

21. What is the interest of $4814.25 for 3yr. 7mo. 14d., at .06 per year? At.075 per year?

There are a variety of other contractions that may frequently be adopted in practice. A few are given below, which will often be found useful.

(1.) When the multiplier consists of any number of 9's, increase it by 1, and subtract the multiplicand from the product. Thus, 18473 x 9999=184730000—18473=1847281527.

(2.) To multiply by 5, divide the multiplicand by .2. Thus, 187x5=187+2=935. To divide by 5, multiply the dividend by.2.

(3.) To multiply by 25, divide the multiplicand by .04. Thus, 1289 X 25=12897.04=32225. To divide by 25, multiply the dividend by .04.

(4.) To multiply by 75, multiply by 100 and subtract of the product. Thus, 18645x75=1864500—466125=1398375. To divide by 75, divide by 100, and add į of the quotient.

(5.) To multiply by 125, divide the multiplicand by .008. Thus, 1641 X 125=1641-.008=205125. To divide by 125, multiply the dividend by .008.

(6.) To multiply by 375, divide by .008, and multiply the quotient by 3. Thus, 294 x375=294---.008 X3=110250. To divide by 375, multiply by .008 and divide by 3.

(7.) To multiply by 625, divide the multiplicand by.0016. Thus, 4812 X 625=4812-2016.=300750. To divide by 625, multiply the dividend by,2016.

(8.) To multiply by 875, multiply by 1000 and subtract of

the product. Thus, 7539 x 875=735000—91875=643125. To divide by 875, divide by 1000 and add of the quotient.

(9.) To multiply by any number within 12 of 100, 1000, &c., annex to the multiplicand as many zeroes as there are figures in the multiplier, and subtract as many times the multiplicand as are equivalent to the excess of 100, 1000, &c., over the multiplicand. Thus, 24796 X 99989=2479600000—(11X 24796)=2479327244.

(10.) To square* a number ending in 5, multiply the number of tens by one more than itself, and place 25 at the right of the product. Thus, 3X4=12, and 35X35=1225; 12x13=156, and 125 X 125=15625; 6X7=42, and 65 X 65=4225.

(11.) When the tens in two numbers are alike, and the sum of the units is 10, to obtain the product multiply the number of tens by one more than itself for the hundreds, and place the product of the units at the right of this product, for the tens and units. Thus, 4 X5520, and 43x47=2021; 42 X 48=2016; 44 X 46=2024; 7 X 8=56, and 72 X 78=5616;71 x 79=5609, &c.

(12.) The sum of two numbers multiplied by their difference, is equal to the difference of their squares. Hence we may readily find the product of two numbers, one of which is as much above as the other is below, a certain number of tens. Thus, 87X73=(80+7)*(80—7)=802–72=6400—49=6351.

(13.) To square any number between 50 and 60, add the units of the given number to 25 for the hundreds, and annex the square of the units for the tens and units. Thus, for the square of 51; 25+1=26 hundreds, and ixl=l; hence 51 X 51=2601. In like manner 53 x 53=2809; 59 x 59=3481.

(14.) When one figure of the multiplier is an aliquot part of one or more of the remaining figures, the work may be abbreviated as in the following example: Multiply 489.137 by 7261.8. 489.137

We see at 7261.8

once that 18

is a multiple 2934822 =prod. by 6.

of 6, and 72 8804466 =prod. by 3x6=prod. by 18. is a multiple 35217864 =prod. by 4x 18=prod. by 72.

of 18. There3552015.0666

fore, multi

plying first by 6, we take 3 times the product for the product by 18, and 4 times the product by 18, for the product by 72.

* The product of any number multiplied by itself, is called the square of the number.

mon measure.

(15.) In the ordinary mode of determining the greatest common divisor of two numbers, any prime factor or square number, contained in one number but not in the other, or any prime factor or square number in a remainder that is not in the preceding divisor, may be rejected, and the work thus abbreviated. For example, let the greatest common measure of 689 and 901 be required. 689 ) 2279 (3

Here the square number 4 is a 2067

factor of 212, and not of 689. We

therefore divide 212 by 4, and im4) 212

mediately obtain the greatest comG. C. Meas. 53 ) 689 (13 of this principle to the reduction of

In the application 53

fractions, we observe that 53 divides 159

689, 13 times, and it divides 212, 4 159

times. It therefore divides 3 X 689

+212 or 2279,3x13+4 or 43 times. Therefore 689

2210=13. Reduce 45 to its lowest terms. 457) 563 (1 457

Neither 2 nor 53 being factors of 457, the

fraction is already in its lowest terms. 106=2 x 53 (16.) We have already seen the application of cancelling, in the reduction of fractions to their lowest terms. The principle is applicable in all cases in which the product of one set of numbers is to be divided by the product of another set. If either multiplier contains a factor of either divisor, the factor may be removed from both without altering the result. Upon this truth is founded a ready mode of solving many intricate questions, which will be explained more fully in the chapter on Proportion.

22. Multiply 576.3 by 99; by 999000.
23. Multiply 7.894 by 5; by 25; by 7500.
24. Multiply 48.302 by 1250 ; by 375000.
25. Divide 1879.4 by 5; by 250 ; by 75.
26. Divide 4449.17 by 125 ; by 375.

27. Multiply 3.0872 by 525125, by adding three partial products.

28. Multiply 41909 by 999625125, in the most expeditious manner.

29. Multiply 89443 by 625; by 875. 30. Divide 141.982 by 625; by 875.

31. Multiply 89443 by 625875. 32. Multiply 283172 by 9992; by 991. 33. What is the square of 15 ? of 85? of 115? 34. What is the product of 73% 77 ? 12x 18 ? 44 x 46 ? 35. What is the product of 81 x 89 ? 75 x 75 ? 34 X 36 ?

36. What is the product of 16 x 24? 1921 ? 35x 45 ? 89 x 71 ? 67 x 53 ? 78 x 82 ? 96 X 84 ? 113 x 107 ? 112 x 128?

37. What is the square of 58? of 56 ? 52 ? 55 ? 57 ? 54 ?

38. Find the greatest common divisor of 804 and 938 ; of 741 and 1083; of 1343 and 1817.

39. Reduce each of the following fractions to its lowest terms: 391, 1980, 897, 1143, 9:41.

40. Multiply 476384 by 9995125625.

CHAPTER IX.

PERCENTAGE.

It is customary to estimate all sums paid as commission or brokerage for the sale or purchase of property, insurance against loss by fire or otherwise, interest for the use of money, &c., at a given number of hundredths, called the rate per cent. (from the Latin per centum, signifying by the hundred). Thus, 6 per cent. is .06 ; 5$ per cent. is .05, or .0525; 7} per cent. is .07} or .073. To compute any percentage, Multiply by the rate expressed deci. mally.

COMMISSION is an allowance of a certain percentage to a Factor, Correspondent, or Broker, for buying or selling property.

1. A broker sold goods at a commission of 41 per cent., for which he received $1963.50. What was the amount of his commission, and how much did he pay to his employer?

2. How much shall I receive for my real estate, by sell.

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