The suppositions and errors will then stand as follows: For A. 5 For B. X For C. For D. 1 1018101 1018-1333 10481638 1013 2A 8101 A 8-134 A 82 -161 A 83 X As the same errors are used in finding every share, it will be most convenient to reduce them to 15ths, before multiplying. 3. It now remains to show how to find B's and C's proportions of the first and second supposed sums, $399, and $398, for after being found, the most difficult part of the work is performed, A's being assumed, and D's being found by subtracting A's, B's and C's from each whole supposed number, viz., $399, and $398. 4. To find B's part of the $399. Suppose $51; from 399-51,-3-116, of A, C and D's, which plus 51=167; then 200-167=33, first error. Suppose $90; from 399-90,-÷3=103, of A, C and D's, which plus 90-193; then, 200--193-7, second error. These suppositions and errors give $1014 for B. 5. To find C's part of the $399. Suppose $83. Then, from 399-83,÷4-79, of A, B and D's, which plus 83-162; then, 200-162-38, first error. Suppose $103; from 399-103;÷4-74, of A, B and D's, which plus 103 =177; then, 200-177-23, second error. These suppositions and errors give $1333 for C. 6. To find D's share of the $399. Subtract A, B ́and C's from the whole. Proceed as in the last two cases to find all the proportions of the second supposed number, $398. Ans. A had $54, B $10239, C $135, and D $15137. 7. At a certain time between two and three o'clock, the minute hand of the clock was between three and four. Within an hour after, the hour hand and minute hand had exactly changed places with each other. What was the precise time, when the hands were in the first position? * 1. Suppose the time to be 16 minutes past 2 o'clock, the hour hand must have passed 16 of the distance from 2 o'clock to 3 o'clock; and if the minute hand was in the place of the hour hand, it would be 11m. 20 sec. from 12 o'clock; and if the hour hand was in the place of the minute hand, it would be 12 minutes past 3 o'clock. The difference between 12 m. and 11 m. 20 sec, is 40 seconds. Therefore, let 40 sec. be the first error. 2. Suppose the time to be 18 minutes past two o'clock; the hour hand, at that time, has passed 8 of the distance from 2 o'clock to 3 o'clock; and if the minute hand was in the place of the hour hand, it would be 11 m. 20 sec. from 12 o'clock. And if the hour hand was in the place of the minute hand, the time would be 36 minutes past 3 o'clock. The difference between 36 m. and 11 m. 30 sec. is 24 m. 30 sec., being 1470 seconds. Let this be the second error. 3. 1470×16,-40X18,÷1470-40-15 m. 56 seconds past 2 o'clock, Ans. MISCELLANIES. REMARKS.-Besides the remarks on Proportionals and Percentage Proportionals, it may not be amiss to observe, that, the same said of them is also applicable to this division and the Promiscuous Questions in Mensuration, and it may be said of these four divisions, that, the operations of their questions, though the common method of solving such, are to be considered but the outlines or notes of their thorough analysis, of which the questions are the texts acting as exercises; consequently, pupils and the teacher should go hand in hand in giving the analysis; and from this course, one may judge the result. EXAMPLES. 1. A lady purchased a piece of silk frocking, at 80 cents per yard, and lining for it, at 30 cents per yard; the frock and lining contained 15 yards, and the price of the whole was $7,00. How many yards were there of each? * $715-464 cents, average price per yard. 80-463-331; 463—30—163; hence, it is evident, that the quantity of lining will be to that of the silk as 33 to 163, that is, the quantity of lining will be double the quantity of silk. Hence, 10 yards of lining, and 5 yards of silk. 2. A and B bought a quantity of calico for $30,59. A paid 15 cents per yard for his, and the price per yard of B's was equal to of the whole number of yards. Required, the price of B's per yard. *(3059X2÷7+15÷14)=61; then, 61-15,÷2=23 cents, Ans. Questions of this nature admit of an infinite number of answers, one as correct as the other. 3. A party of lively young gentlemen and ladies, going into the country on a tour of pleasure, had a bill of $24,99 to pay, part of which the funny females insisted on discharging; hence, it was agreed that each gentleman should pay $1,17 cents of this expense, and each lady 34 cents. What number of each sex was there? * RULE.-Divide the amount to be paid by one of the respective shares, or some multiple of it, till the remainder becomes divisible by the other. Ans., 17 gentlemen, and 15 ladies. 4. A man was hired 50 days on these conditions that; for every day he wrought he was to have 75 cents, and for every day he was idle he was to forfeit 25 cents. How many days was he idle, if he received $27.50 at the end of the time? *Had he worked every day, his wages would have been $37.50, $10 more than he received; but every day he was idle lessened the $37.50 just 75 cents plus 25 cents, $1.00; therefore, he was idle 10 days. 5. A and B have the same income. A saves an eighth of his; B spending $30 yearly more than A, finds himself $40 in debt at the end of 8 years. Required their income and what each spends per annum. *$40-8 $5 yearly, more than his income, $30—5—$25, what A saves yearly; hence, $25 of either's income; consequently $200 their income. 6. The head of a fish weighs 4lbs., its tail weighs as much as its head and half of its body, and its body weighs as much as head and tail. Required the weight of the fish. *The head and tail weigh 8lbs. and of the body, and as the body weighs as much as the head and tail, it is evident that 8lbs. is of the required weight. Hence, its tail weighs 4 lbs. plus 8lbs. equal 12lbs. 7. A man when he was married was 3 times as old as his wife; 15 years after, he was but twice the age of his lady. At what age was each married? *When they were married her age was 1 year to his 3 years; in 15 years his age was 2 to her 1, that is, 15 years doubled her age, and his was what it was; hence, he was 45 and the lady 15 years of age. 8. In a mixture of wine and cider, of the whole plus 25 gallons is wine, and part less 5 gallons is cider. How many gallons of each kind in this mixture? *+3=12 in all. Therefore, 25—5—20— of the required quantity; hence, 120 gallons in all. Consequently, 85 gallons is wine, and 35 gallons is cider. as 9. A man bought some lemons at 2 cents each, and many at 3 cents each, and then sold them all at the rate of 5 cents for 2, and thus gained 25 cents. How many lemons did he buy? *He bought 4 at 2 cents each, as often as he bought 3 at 3 cents each, therefore he gave 17 cents for every 7 lemons, at 23 cents each, but sold them at 2 cents each. Then, 22-23 of a cent gained on one, consequently 25 cents gained on 350 lemons, Ans. 10. The stock of a cotton manufactory is divided into 32 shares, and owned equally by 8 persons, A, B, C, &c. A sells 3 of his shares to a ninth person, who thus becomes a member of the company, and B sells 2 of his shares to the company, who pay for them from the common stock. After this, what proportion of the whole stock does A own? *Each one owns of the whole, and A reserved but of his, and 2 of B's shares being taken up by the company, hence only 30 shares in all, consequently A owns of the whole. 11. A lady has two silver cups of unequal weight with but one cover. The first cup weighs 12oz. If the first cup be covered, it will weigh twice as much as the second; but if the second cup be covered, it will weigh three times as much as the first. What is the weight of the cover and of the second cup ? *First cup, oz. 12×3, second cup and cover, viz. 26oz.; hence, both cups and cover 48oz., its 32oz., first cup and cover, for either cup covered are to each other as 2 to 3; then, 48-32-16oz., second cup, and 48,-16+12-28 12. A general disposing his army into a square, found he had 231 over and above; but increasing each side with one soldier, he wanted 44 to fill up the square. Of how did his army consist? many men *231+44+1,÷2=138; then, 138X138,-44-19000 men, Ans. The 1 is added because the soldier standing in the corner of the square is counted twice. 13. A military officer drew up his soldiers in rank and file, having the number in rank and file equal; on being reinforced with three times his first number of men, he placed them all in the same form, and then the number in rank and file was double what it was at first; he was again reinforced with three times his whole number of men, and after placing the whole in the same form as before, his number in rank and file was 40 men each. How many men had he at first? *His first number call 1; his first reinforcement is 3, and second 12; therefore, 40x40,÷1+3+12=100 men, number at first, Ans. Or, 3x3,+, X3,+8=32. Then, 40X40X2,32-100 men, Ans. 14. A fellow said that when he counted his plums two by two, three by three, four by four, five by five, and six by six, there was still an odd one; but when he counted them seven by seven they came out even. How many had he? *Thus, 2×3×4×5×6+1=721, Ans. 15. A paid $100 for 100 animals, consisting of oxen, sheep, and geese, paying $10 for an ox, $1 for a sheep, and a shilling for a goose, respectively. How many of each did he buy? *First, 6 | Then, 100—54+5=41 sheep. { oxen, 60- 5 16. How much wine at 4s. 6d. and at 5s. per gallon must be mixed with 6 gallons at 4s. and 6 gallons at 3s. per gallon, that the mixture may be worth 4s. 6d. per gallon? *Limited, 6gls. at 4s.-24 Simples, 6" at 3s.-18) } )42=3s. 6d., rate of the limited 12gls. at 4s. 6d. per gallon. } ::12:{12gls. at 4s. 6d. at 5s. |