4. Divide £150 between A, B, and C, so that A may have 301 more than B, and B 157 more than C? 5. Suppose 8s. 8d. was distributed between some poor people, consisting of men, women, and children; 3d to each man, 2d to each woman, and Id to each child; now if there were twice as many women as there were men, and three times as many children as there were women; how many were there of each ? 6. A captain after an engagement, finds of his company either killed or missing; + are sent wounded to hospital, ib are on duty, and 51 are on parade; what was the strength of his company before the engagement? 7. If I wish to divide 60d between some poor people, consisting of an equal number of men, women, and children, and gave to each man 6d, each woman 4d, and each child 3d, and had 8d to spare; how many were there of each? 8. Suppose I lent at interest a certain sum, at 6 per cent, per an. simple interest, and at the end of 8 years both prin cipal and interest amounted to 512/ what was the sum lent? 9. What number is that, whose third exceeds its fourth part by 24? ᄒ 10. What sum of money is that whose +,, and parts added together, shall make 28/ 10s,? 11. Required a number, which being multiplied by 8, the product divided by 5, the quotient increased by twice the required number, and by 6, will make 60? 12. Required a number, to which if of itself be added, half the sum shall be 15? 13. A person after spending his yearly income, and 451, had left of it, together with 57; what was his income? 14. A sum of money which had been put out to interest, is increased by of the original sum, which is now 1441; what was the sum lent? 15. What number is that, to which if its and its be added together with half the sum thus produced, the whole shall be 484 ? DOUBLE POSITION Is the method of resolving certain questions by means of two suppositions. Assume any two convenient; numbers, and proceed with each of them separately, according to the conditions of the question, as in Single Position, and find how much each result differs from that mentioned in the question, calling these differences the errors, noting also whether they are too great or too little; if too much, mark with the sign+; and if too, little, with the sign Then multiply each of the said errors into, or by the contrary position, and if the errors are alike, or both marked with the same sign, divide the difference of the products by the difference of the errors; but, if the errors are unlike, divide the sum of the products by the sum of the errors, for the answer. OR, Having assumed two different numbers, perform on them separately the operations indicated in the question, and find the errors of the results. Then, as the difference of the errors if alike, or sum if unlike, is to the difference of the supposi tions; so is either of the errors to a fourth number, which added to or subtracted from the number which produced that error, will give the number sought. If at any time it should happen that the errors are the same in quantity, and unlike in quality, half the sum of the suppo sitions will be the number sought.. Examples. 1: What number is that, to which if 75 be added, the sum shall be four times the said number? Or thus by the second rule. As 30+ 45 = 75: 25 :: 30: 10. And 15 +10=25 Ansr. 2. What number is that, which being multiplied by 4, and 6 subtracted from the product, the remainder divided by 3, shall make 10? 3. What number is that, which being increased by 5, and multiplied by 3, the sums and products shall be équal? 4. Two persons, A and B, engage at play. When they began, A had 30s. and B had 20s. After a certain number of games, A's money is three times B's; what sum did A wir from B? 5. Divide 18 into two such parts, that 8 times the less part, less 1, may be three times the greater? 6. A person being asked what o'clock it was, answered, if the hour was multiplied by 3, and 12 subtracted therefrom, the remainder would be just of the hour; pray what was that?. 7. What number is that, which being multiplied by 5, the product increased by 15, and the sum divided by the number itself more 1, shall have for a quotient 64 ? 8. What number is that, which being multiplied by 5, the product increased by 15, and the sum divided by the number itself less 1, shall have for a quotient 64 ? 9. Two persons travelling between Cork and Limerick, distant, suppose, 48 miles. After eight hours they meet, when it appeared that one left Limerick at the same hour the other left Cork, and had travelled 11⁄2 miles per hour faster than he from Cork; at what rate per hour did they travel? 10. Suppose 60 was to be divided into two parts, so that the greater divided by 2, may be equal to the less multiplied by 2; what are the parts? 11. A gentleman riding from his own house to a certain town, weht at the rate of 6 miles an hour; returning, he rode at the rate of 5 miles per hour, and was 7 the road; what was the distance? minutes longer on 12. Divide 100 into two parts, so that of the greater, with 5 added thereto, may be of the less part? 13. What number is that, which being multiplied by 6, and divided by 3, the product less 24, may be 10 times greater than the quotient? 14. A workman was hired for 28 days, at 3s. 4d. per day, and his diet, for every day he worked; but for every day he was idle, he was to pay 1s. 8d. for his diet. When he came to settle, there was 27 13s. 4d. coming to him; how many days did he work, and how many days was he idle ? 15. A gentleman on mounting his horse, was asked what o'clock it was, and answered, I must be at a friend's house at 5 o'clock. Now, if I ride at the rate of 6 miles an hour, I will have 15 minutes to spare; but if I ride at the rate of 5miles an hour, I shall be 21 minutes too late what was the hour? 16. A man being asked his age, answered, ten years ago I was 6 times the age of my son; but if each of us live 10 years, I shall be only twice as old as he; pray what were their ages? 17. A company wanting to make 30s. for a charitable purpose, find they have 9d more to pay than if there were two more in company; how many were there? 18. A gentleman's horses are + of his black cattle, which are of his sheep. Now, if his sheep are lessened by 162, they are just one half his black cattle; how many of each had he? L DUODECIMALS. DUODECIMALS, or as it is commonly called, Cross Multiplication, is a rule made use of by artificers in computing the contents of their several works. It is so called, because every superior place is twelve times its next inferior, decreasing from the place of feet towards the right hand, as in the following TABLE. 12- Inches = 1 Foot. 12 Fourths = 1 Third. Inches or primes, seconds or parts, &c. are marked thus ; inches or primes (), seconds ("), thirds ("), and fourths (""), &c... Dimensions are generally taken in feet, inches, and quarters; any smaller than these being neglected as of no consequence. One quarter being three seconds or parts, two quarters six, and three quarters nine. GENERAL RULE for multiplying duodecimally: Arrange the terms of the multiplier under the corresponding names of the multiplicand: beginning at the lowest, multiply each term of the multiplicand by the feet in the multiplier, setting down the result of each product under its respective term, carrying one for every twelve, from each lower denomination to the product of the next higher. Multiply each term of the multiplicand, as before, by the inches in the multiplier, and write the result of each product, one place towards the right hand of those names in the multiplicand. Proceed in the same manner with the parts or seconds in the multiplier, writing the result of each product two places removed to the right hand, and for the thirds three places, &c. Lastly, add all the products, and the sum will be the answer required. Instead of multiplying by the inches, &c. if we take such parts of the multiplicand, or of the first product as there are of a foot, or of the first multiplier, and add the lines together, the same result will be obtained. The work may be proved by making the multiplicand multiplier, and multiplier multiplicand, and again performing the operation, or by multiplying half the multiplicand by double the multiplier. Examples. |