quired, are 7 men and 9 days. But $45÷3 men the dollars 1 man earns in 5 days, and this value divided by the 5 days what 1 man earns in 1 day. The 3 and the 5, therefore, become divisors in the natural analysis of the sum. Again, if what 1 man earns in 1 day be multiplied by 7, the result will be what 7 men earn in 1 day, and this multiplied by 9, shows what 7 men will earn in 9 days. In the analysis of the work then, the 7 and 9 become multipliers. RULE. Divide the given value by the agencies which produce it, and multiply the quotient by the agencies, the result of whose combined action is required. Statement of preceding: sum: $45. 7. 9 $189, Ans. 2. If it require 7 horses, 16 days to remove 104 tons of hay a certain distance, how many tons will 14 horses rernove in 21 days? Ans. 273 tons 3. If 16 men earn $192 in 8 days, what will 12 men earn in 27 days? Ans. $486. 4. If 8 men build 18 rods of wall in 12 days, how many rods will 96 men build in 2 days? Ans. 36 rods. 5. If 8 men in 15 days consume 48 pounds of beef, how many pounds will 20 men consume in 25 days? Ans. 200 pounds, 6. If 9 horses eat 128 bushels of oats in 64 days, how many bushels will 24 horses eat in 32 days? Ans. 170 bushels. 7. If 12 men reap 36 acres of wheat in 6 days, how many acres will 18 men reap in 9 days? Ans. 81 acres. Sum 2. What does dividing by 7 and 9 show? What is the rule! Sum 3. Dividing $192 by 16 shows what? What does dividing that result by 8 show?. Multiplying what 1 man earns in 1 day by 12, shows what? That result multiplied by 27 shows what? Similar analytical questions should be asked respecting each sum. 8. If 15 men in 21 days spend $472.50, how much will 27 men spend in 35 days? Ans. $1417.50. 9. If $300 in 5 years gain $90 interest, what will $450 gain in 6 years? Ans. $175.50. 10. If 5 men in 12 days make 180 pairs of shoes, how many pairs will 15 men make in 4 days? Ans. 180 pairs. 11. If 7 men in 9 days cut 84 cords of wood, how many cords will 12 men cut in 21 days? Ans. 336 cords. 12. If 25 persons in 3 years consume 300 bushels of wheat, how many bushels will 27 persons consume in 13 years? Ans. 1404 bushels. 13. If it require 16 men, 12 days to make 64 suits of clothes, how many suits can be made by 24 men in 15 days? Ans. 120 suits. § 151. A result produced by the united action of two given agents; also a result to be produced, and one of the agents employed, are often given, and a second agent is required, which acting unitedly with that given, shall secure the required result. The following example will serve as an illustration : 1. If 8 men build 18 rods of wall in 12 days, how many men will build 36 rods in 2 days? A number of men is required, who, operating 2 days, shall build 36 rods of wall; but 36 rods will require more men than 18 rods. Multiply, therefore, 8 men by 36 and divide the product by 18. Also 2 days, the time in which this greater work is to be performed, is less than the 12 days in which the other work was performed. More men will, therefore, be required. Hence, when one of two agencies is required, an increase in the required result requires a corresponding increase of that agency; but an increase in one of the two agencies necessarily requires a corresponding decrease in the other. Therefore, To obtain an increase in a required agency, we multiply that agency by the result to be attained, and also by the 151. How is a second agent found, when required? Give the analysis of the sum. When two agencies are employed, what is requisite, when an increased result is to be obtained? When one agency is increased, how is the other affected? greater of the two given agencies corresponding in kind, and divide by the attained result and the less of the two given agencies; but to obtain a less agency, we multiply by the less of the given agencies and divide by the greater, the terms expressing the results being used as before. The statement for the previous sum is therefore, 8. 36. 12 =96 men, Ans. Here the number of men required is increased, both by the increase of work and the decrease of the time. OBS. The principle involved in the above is simply, that with a fixed result to be attained, as one of the agencies employed is increased, the other is diminished. 2. If 5 men make 120 pairs of shoes in 12 days, how many men will make 150 pairs in 15 days? Statement: 5. 150. 12 5 men, Ans. 3. If 4 men in 12 days earn $96, in how many days will 8 men earn $288 ? Ans. 18 days. 4. If 12 men reap 120 acres of wheat in 10 days, in how many days will 30 men reap 200 acres? Ans. 63 days. COMPOUND NUMBERS. § 152. Compound numbers increase in no uniform ratio. In Avoirdupois weight, 16 ounces are required for a pound; and in Troy and Apothecaries' weight, 12 ounces make a pound. So in time, the denominations are seconds, minutes, hours, days, weeks, &c. But none of these denominations sustain to each other a uniform relation, since it requires 60 How do we obtain an increased agency? What is the principle involved? 152. How do compound numbers increase? pound numbers? How a less agency? What is said of com seconds to make 1 minute, 60 minutes to make 1 hour, 24 hours to make 1 day, &c. Compound Numbers do not, however, differ in kind, for a lower denomination repeated a required number of times, will produce a higher denomination. Hence the lower are but parts of the higher denominations. They are, however, compound in their relations to each other; that is, their relative numerical value is constantly changing, being expressed at one time by the numbers, 4, 12, and 20, as in English money; at another by 3, 12, 3, 5, 40, 8, &c., as in Long The pound and farthing are denominations of English money not coined. The English sovereign, a common English coin, is of the same value as the pound sterling. By this weight, jewels, precious metals, and distilled liquors are weighed. The pound Troy is the same as the Imperial pound Troy of England, requiring 5760 grains to make an ounce. TABLE III.-APOTHECARIES' WEIGHT. This weight is employed by apothecaries and physicians, in weighing medicines. The pound and ounce are the same as in Troy weight. The drams, scruples, and grains, are minute divisions of weight, to enable the physician to weigh out small doses of medicine. What are such numbers called? In what do they not differ? Why! In what respect are they compound? This is the weight used by grocers, and is employed for weighing all coarse articles: as sugar, tea, coffee, cheese, butter, hay, &c. The pound Avoirdupois is heavier than the pound Troy. The pound Troy=5760 grains, while the pound Avoirdupois 7000 grains. 360° make a circle. That circle, 1 degree of which is 60 geographic or 694 statute miles, is the circumference of the earth. This measure is used for measuring length, and is often called linear measure. The yard in this measure is the same as in Long measure, viz.: 3 ft. or 36 inches. Hence, 36-49 in., the quarter; and 9-4-2 in., the nail. 1 ell Flemish, 1 ell English, 1 ell French, |