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Let the product of two numbers be 288 and one of them 8; I demand the other?

8)288

Answer, 36

PROB. 6. Having the dividend and quotient, to find the divisor
Rule. Divide the dividend by the quotient.

COR. Hence we get another method of proving Division.

Given

{

288 the Dividend.

36 the Quotient.

Required the Divisor.

36)283(8 Divisor.

288

PROB. 7. Having the Divisor and Quotient given, to find the Dividend.

Rule. Multiply them together.

Given

8 the Divisor.

36 the Quotient.

Required the Dividend.

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288 the Dividend.

By a due consideration and application of these Problems only. many questions (of which kind are some of the following) may be resolved in a short and elegant manner, although some of them are generally supposed to belong to higher rules.

3.

APPLICATION OF THE PRECEDING RULES.

1. The least of two numbers is 19418, and the difference between them is 2384: What is the greater, and sum of both? 19418+2384 21802 greater, and 19418+21802=41220 sum. 2. Suppose a man born in the year 1743; when will he be 77 years of age? 1743+77=1820 Answer. What number is that, which, being added to 19418, will make 21802 ? 2384 Ans. 4. Gen. Washington was born in 1732; what was his age in 1799 ? 67 Ans. 5. America was discovered by Columbus in 1492 and its independence declared in 1776: How many years elapsed between

those two eras?

234 Ans. 6. The Massacre at Boston, by the British troops, bappened March 5th, 1770, and the Battle at Lexington, April 19th, 1775; How long between ?

April 19th, 1775-March 5th, 1770=5 y. 1 m. 14 d. Ans. 7. Gen. Burgoyne and his army were captured October 17th, 1777, and Earl Cornwallis and his army, October 19th, 1781 : What space of time between? 4 years and 2 days. Ans. 8. The war between America and England commenced April 19th, 1775, and a general peace took place January 20th, 1783: How long did the war continue? 7 y. 9 m. 1 d. Ans. 9. A, B, C and D purchased a quantity of goods in partnership: A paid £12 10s. a dollar* and a crownt piece; B, 35s. C, 29s. 10d. and D, 79d.: What did the goods cost? Ans. £ 16 14 1.

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viz.

10. A man borrowed, at different times, these several sums, £29 5s. £18 17s. 6d. £45 12s. £98, 3 dollars, one crown piece and an half: Pray how much was he in debt? Ans. £ 193 2 6.

11. There are four numbers; the first 317, the second 912, the third 1229, and the fourth as much as the other three, abating 97 : What is the sum of them all? Ans. 4819. 12. Bonght a quantity of goods for £ 125 10s. paid for truckage 15s. for freight 79s. 6d. for duties 35s. 10d. and my expences were 53s. 9d. What did the goods stand me in? Ans. £ 136 4 1.

13. A Gentleman left his son £ 1725 more than his daughter, whose fortune was 15 thousand, 15 hundred and 15 pounds: What was the son's portion, and what did the whole estate amount to? Ans. The son's fortune, £ 18240, and the whole estate £34755. 14. A merchant had 6 debtors, who together owed him £2917 10s. 6d. A, B, C, D and E, owed him £1675 13s. 9d. of it: What was F's debt? Ans. £ 1241 16 9.

15. What is the difference between £1309 7s. 1d. and the amount of £ 345 13s. 4d. and £ 571 4s. 8d.? Ans. £392 91

16. A merchant, at his first engaging in trade, owed £937 15s. he had in cash £1755 3s. 6d. in goods £ 459 12s. 3d. in good debts £ 197 16s. and he cleared the first year £249 19s. 10d. What was the neat balance at the year's end? Ans. £1724 16 7.

17. What sum of money must be divided between 12 men, so as that each may receive £155?

£ 1860 Ans. 18. What number must I multiply by 9, that the product may be 675?

75 ans. 19. A privateer of 175 men took a prize, which amounted to £59 per man, beside the owner's half: What was the value of the prize? £20650 Ans. 20. What is the difference between thrice five, and thirty, and thrice thirty five?

60 Ans.

21. The sum of two numbers is 750; the less 248: What is their difference and product? diff. 254, 124496 product. 22. What is the difference between six dozen dozen, and half a dozen dozen; and what is their product, and the quotient of the greater by the less?

Ans. 792 difference, 62208 product, and 12 quotient. 23. There are two numbers; the greater of them is 25 times 70, and their difference is 9 times 15; their sum and product are required.

Ans. 1950 the greater, 1815 the less. 3765 the sum, and 3539250 the product.

24. A merchant began trade with £25327; for six years togeth er, he cleared £1253 per annum; the next 5 years, he cleared £1729 per annum; but, the last 4 years, had the misfortune to lose £3013 per annum: What was he worth at the 15 years' end? Ans. £29414.

25. If a man spends £132 in a year: What is that per calendar month?

16 Ans.

26. If the Federal Debt, which is 42 million dollars, be equally divided between the 13 States: What will be the share of each? Ans. 3230769 dollars.

27. If 9000 men march in a column of 750 deep: How many march abreast ? 12 Ans. 28. What number, deducted from the 32d part of 3072, will. leave the 96th part of the same? 64 Ans.

29. What number is that, which, multiplied by 3589, will produce 92050672 ? 25648 Ans. 30. Suppose the quotient arising from the division of two numbers to be 5379, the divisor 37625: What is the dividend, if the remainder came out 9357? 202394232 Ans.

31. There is a certain number, which being divided by 7, the quotient resulting multiplied by 3, that product divided by 5, from the quotient 20 being subtracted, and 30 added to the remainder, the half sum shall make 35: Can you tell me the number?

700 Ans. 32. A sheepfold was robbed three nights successively; the first night, half the sheep were stolen, and half a sheep more; the second half the remainder were lost, and half a sheep more; the last night they took half what were left and half a sheep more ; by which time they were reduced to 30: How many were there at first?

Begin with 30, and, reckoning back from the last night to the first, you will find that 31 were stolen the 3d night, 62 the 2d, and 124 the first. Ans. 247.

33. Two boys, A and B, had 850 chesnuts between them; but A had 150 more than B: How many had each. 850-2=425 half sum, and 150-2=75 half diff.; then 425+75 =500 A's, and 425-75=350 B's.

34. What number added to the 27th part of 6615, will make 570 ?

325 Ans.

REDUCTION

TEACHES to bring numbers of one denomination to others of different denominations, retaining the same valuë.

It is of two sorts, viz. Descending and Ascending.

REDUCTION DESCENDING

Teaches to change numbers from a higher to a lower denomiIt is performed by multiplication.

nation.

RULE.*

Multiply the highest denomination given, by so many of the next less as make one of that greater, and thus continue until you have brought it down as low as your question requires.

PROOF. Change the order of the question, and divide your last product by the last multiplier, and so on.

Note. From this rule and Case VI. of Simple Multiplication, it appears, that Federal Money is reduced from higher to lower denominations by annexing as many cyphers as there are places from the denomination given, to that required; or, if the given sum be of different denominations, by annexing the several figures of all the denominations in their order, and continuing with cyphers, (if necessary,) to the denomination required; or, what amounts to the same thing, by reading the whole number from the left to the required denomination, as one number in the required denomination.

EXAMPLES.

1. In 3 eagles 2 dollars, how many mills? 2. In 91 dollars 75 cents, how many cents? 3. In 50 eagles, how many dollars?

Ans. 32000 m.

Ans. 9175 c.

4. In 44 dollars, 1 cent, 4 mills, how many mills? 5. In 9 dollars, 31 cents, 7 mills, how many mills? 6. How many cents in 39 dollars 5 cents?

7. In 28 dollars 17 cents, 5 mills, how many mills? 8. In £27 15s. 9d. 2qrs. how many farthings?

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Ans. 500 D

Note. In multiplying by 20, I added in the 15s. by 12, the 2. and by 4, the 2qrs. which must always be done in like cases.

To prove the above question, change the order of it, and it will stand thus: In 26673 farthings how many pounds?

The reason of this Rule is exceedingly obvious; for pounds are brought into shillings by multiplying them by 20; shillings into pence by multiplying them by 12; and pence into farthings by multiplying them by 4; and the contrary by division; and this will be true in the reduction of numbers consisting of any denomination whatever. The rule for Reduction ascending is simply the reverse of this, and equally evident.

4)26673

12)6669 2qrs.

2/0)55/5 9 d.

+

Answer, £27 15 9 2

Ans. 35177. Ans. 91751. Ans. 345358. Ans. 9744.

9. In £36 12s. 10d. 1qr. how many farthings? 10. In £95 11s. 5d. 3qrs. how many farthings? 11. In £719 9s. 11d. how many half pence? 12. In 29 guineas, at 28s. how many pence? 13. In 37 pistoles, at 22s. how many shillings, pence, and farthings? Ans. 814s. 9768d. 39072qrs. 14. In 49 half johannes, at 48s. how many sixpences? Ans. 4704. 15. In 473 French crowns, at 6s. 8d. how many threepences? Ans. 126131. 16. In 53 moidores, at 36s. how many shillings, pence and farthings? Ans. 1908s. 22896d. 91581qrs. 17. In £29 how many groats, threepences, pence, and farthings? Ans. 1740 groats, 2320 threepences, 6960d. 27840yrs. 18 Reduce 47 guineas and one fourth of a guinea into shillings, sixpences, groats, threepences, twopences, pence and farthings. Ans. 1323 shillings, 2046 sixpences, 3969 groats, 5292 threepences, 7938 twopences, 15876 pence, and 63504 qrs.

REDUCTION ASCENDING

Teaches to change numbers from a lower to a higher denomination. It is performed by division.

RULE.

Divide the lowest denomination given, by so many of that name, as make one of the next higher, and thus continue till you have brought it into that denomination which your question requires.

Note. From this rule and the note under Case II. of Simple Division, it appears, that Federal Money is reduced from lower to higher denominations by cutting off as many places as the given denomination stands to the right of that required; the figures cut off belonging to their respective denominations.

EXAMPLES.

1. How many eagles in 42000 mills?
2. In 3175 cents, how many dollars?
3. In 500 dollars how many Eagles?
4. In 4414 mills, how many dimes?
s. In 9317 mills, how many dollars?
6. How many dollars in 28175 mills?

Ans. 4 E. $2 Ans. $31 75 c. Ans. 50

7. In 547325 farthings, how many pence, shillings, and pounds?

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