2. If a man can do a piece of work in 9ş days, when the days are 12 hours long, in how many days will he do the same when the days are but 9} hours long? Ans. 1253 days. 3. If 55 yards of cloth which is 2; yards wide will make a cloak, how many yards of cloth that is only iyd. wide will make one ? Ans. 1635 yards. 4. If 14 men can do a piece of work in 13 days, how many men will do the same in 612 days? Ans. 28 men. 5. If I lend my friend 250 dollars for of a year, how much ought he to lend me of a year to requite the favour ? Ans. $150. 6. How much baize that is lịyds. wide, will line 163 yards of camblet that is ayd. wide ? 7. How much length that is 8} inches wide, will make Ans. 171 8. If 25 s. will pay for the carriage of a cwt. 145 miles, how far may 6. hundred weight be carried for the same money? Ans. 2226 miles Ans. 9 a square foot ? COMPOUND PROPORTION, OR THE DOUBLE RULE OF THREE, Teaches to solve by one statement such questions as would require two or more statements in simple proportion whether direct or inverse,—having (commonly) five terms given to find a sixth, the first three being a supposition, and the last two a demand. RULE. State the question, that is, place the terms of supposition so that the one which is the principal cause of loss, gain or action, stand in the first place ; that which denotes the space of time, distance of place, &c. in the second place, and the remaining terin of supposition in the third place. Place the other two terms under those of the same kind in the supposition. If the blank place or term sought fall under the third place, the proportion is direct; then multiply the first and second terms together for a divisor, and the other three for a dividend. But if the blank place fall under the first or second term, the proportion is inverse ; then multiply the third and fourth terms together for a divisor, and the other three for a dividend, and the quotient will be the answer. EXAMPLES. 40 : 5 :: 84 acres. 1. If 7 men in 16 days reap 84 acres of grain, how many acres can 40 men reap in 5 days? men. days. acres. {position. In this example, 7:16 :: 84 terms of sup- the blank place falls terms of de under the 3d term; [mand. therefore the pro420 portion is direct; 40 and we multiply the first and 2d terms, 7x16=112)16800(150 Ans. viz.: 7x16=112, 112 for a divisor, and the other three, viz: 560 560 84 X 40 X 5 togeth er for a dividend-16800 ; and 16800-112=150, Ans. By analysis.—If ? men can reap 84 acres in 16 days, 1 man can reap of 84=12 acres in 16 days, which is 13 of an acre a day; and 40 men will reap 40 times 14° =30 acres in 1 day, and in 5 days they will reap times 30=150 acres ; the answer as before. By two statements in simple proportion.-First; taking into consideration the number of men employed, we have the following proportion ;-if 7 men in a certain number of days, reap 84 acres, how many acres will 40 men reap in the same time ? — As 7:84 :: 40 ...... which gives for the fourth term 480 acres. Then, taking into consideration the number of days, it will stand thus : if a certain number of men in 16 days, reap 480 acres, how many acres will they reap in 5 days? days. days. acres. As 16 : 480 :: 5 : 150 Ans. 2. If 100 dollars in 12 months gain 6 dollars interest, what principal in 7 months will gain 14 dollars interest? acres. mo. mo. : : : : $ Here the blank place falls un100 : 12 :: 6 der the first term; therefore the 7::14 proportion is inverse, and we mulAns. $400. tiply the third and fourth terms together (6x7=42) for a divisor, and the other three (100 X 12x14=16800) for a dividend ; and 16800:42=400, Answer. 3. If 100 dollars gain 6 dollars in a year, what will 400 dollars gain in 7 months ? 100 12 : : 6 Ans. 14 dollars. 4. If 100 dollars in 1 year gain 6 dollars, in what time will 400 dollars gain 14 dollars ? mo. :: 14 Ans. 7 inonths. 5. If $247,50 will pay the board of 15 persons eleven weeks, how much will it cost to board 9 persons 5 weeks? 15 11 :: 247,50 Ans. $67,50. 6. If 5 men build 150 rods of wall in 12 days, how many rods will 14 men build in 9 days? Ans, 315 rods. 7. If 5 men build 150 rods of wall in 12 days, in how many days will 14 men build 315 rods? Ans. 9 days. 8. If 5 men build 150 rods of wall in 12 days, how many men will build 315 rods in 9 days ? Ans, 14 9. A usurer put out 125 dollars at interest, and when the same had been out 9 months he received for principal and interest' $132,50; at what rate per cent did he receive interest? Ans. 8 per cent. 10. If the carriage of 6cwt. 140 miles cost $28,50, what will be the cost of carrying 8cwt. 3qrs., 64 miles at the same rate ? Ans. $18,24. 11. If a footman travel 240 miles in 12 days, when the days are 12 hours long, in how many days will he travel 720 miles when the days are 16 hours long? Ans. 27d. 12. If 30 men in 20 days build 300 rods of wall, how many men will build four times as much in a fifth part of the time ? Ans. 600. 13. If 6 men build a wall 20 feet long 6 feet high and 4 24 feet thick in 16 days, in what time will 24 men build one da. ft. long ft. h. ft. thick. :: 200 X 8 X 6 Ans. 80 days. Questions. 1. What is the common measure or higher into lower denominations ? divisor of two or more numbers ? 9. How do you reduce fractions of 2. What is the common multiple of lower into higher denominations ? two or more numbers ? 10. How do you add Vulgar Fractions? 3. When two or more fractions have 11. How do you subtract Vulgar Fracthe same denominator, what is it call- tions ? ed ? 12. How do you multiply Vulgar Frac4. How do you find the greatest com tions ? mon divisor of two numbers ? 13. How do you divide Vulgar Frac5. How do you find the least common tions? multiple of two or more numbers ? 14. IIow do you perform the Rule of 6. How do you reduce a compound Three Direct in Vulgar Fractions ? fraction to a simple one ? 15. How do you perform the Rule of 7. How do you reduce fractions hav- Three Inverse in Vulgar Fractions ? ing different denominators, to a common 16. What is Compound Proportion, or denominator ?-To their least common the Double Rule of Three? denominator ? 17. What is the Rule for Compound 8. How do you reduce fractions of Proportion ? INVOLUTION. Involution, or the raising of powers, is the multiplying any given number into itself, and that product by the former multiplier, and so on; and the products thus produced are called the powers of the given number. The first power or root of any number is that number itself. If the first power be multiplied by itself, it produces the second power or square. If the square be multiplied by the first power, it produces the third power, or cube, &c. The number denoting the height of the power is called the index, or exponent of that power. Thus, 94 denotes that 9 is raised to its 4th power. EXAMPLES. 1. What is the 4th power of 9? 9 the first power or root. 9 9 2. What is the square of 6 ? Ans. 36. 3. What is the square of 12 ? Ans. 144. 4. What is the cube, or 3d power of 6 ? Ans. 216. 5. What is the cube of 12 ? Ans. 1728. 6. What is the square of ,085 ? Ans. ,007225. 7. What is the cube of 12,5 ? Ans. 1953,125. 8. What is the biquadrate of 16? Ans. 65536. 9. What is the square of 62? Thus, 61=2, and 2 x 25 =26=3916, Ans. 10. What is the cube of į? 11. What is the square of 3} ? Ans. Ils 12. What is the cube of 3}? ans. 371 13. What is the cube of 61 ? Ans. 343 512 Ans. 2444 EVOLUTION, OR EXTRACTION OF ROOTS, Is the method of finding the root of any given power, or number. The Root, as we have seen, is that number which, being multiplied continually into itself, produces the given power. The Square Root is a number which, being multiplied by itself or squared, will produce the given number. Thus, the square root of 64 is 8, because 82, that is, 8X8=64; and the cube root of 512 is 8, because 83, that is, 8X8 X 8=512; and so for any other number or power. Although there is no number that will not produce a perfect power by involution, yet there are many numbers of which precise roots cannot be obtained ; but by the help of decimals, we can approximate towards the root to any assigned degree of exactness, Numbers whose precise roots can be obtained are called rational numbers; but those whose precise roots cannot be determined are called surd numbers. This character (V) prefixed to any number shows that the square root of that number is to be extracted. Thus, V 16 shows that the square root of 16 is to be extracted. Other roots are denoted by the same character, with the index of the required root placed before it. Thus, y 27, |