Rule. Divide the difference of their squares by the difference of the numbers, and the quotient will be their sum; then proceed by Problem 4.

What are those two numbers, whose difference is 20, and the difference of whose squares is 2000 ?

20)2000(100 sum. 50+10=60, the greater; and

50-10 40 the less.

PROB. 7. Having the product of two numbers, and one of them given, to find the other.

Rule. Divide the product by the given number, and the quotient will be the number required.

Let the product of two numbers be 288, and one of them 8 ; we demand the other. 8)288

Ans. 36

PROB. 8. Having the dividend and quotient, to find the divisor.
Rule. Divide the dividend by the quotient.

Hence we get another method of proving division.

PROB. 9. Having the divisor and quotient given, to find the dividend.

REMARK. The scholar, having now surveyed the groundwork of his studies, begins to see their application to the common concerns of life ;--and it is important, while proceeding in the higher rules, that his memory be strengthened by repeated examinations in the previous studies. The instructor is advised, therefore, to state questions of his own, promiscuously under the several rules, that the good scholar may have an opportunity of proving to his teacher and friends, by prompt and ready answers to difficult questions, that he thoroughly understands the subject before him. This hint, improved now, may be of essential service hereafter.

REDUCTION.

REDUCTION is the method of changing numbers of one denomination into others of different denominations, without altering their value.

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

REDUCTION DESCENDING

Teaches to change numbers from a higher denomination to a lower; and is performed by multiplication.

RULE.

Multiply the highest denomination given, by so many of the next less, as make one of that greater, and thus continue, till 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 II, of Simple Multiplication, it appears, that FEDERAL MONEY is reduced from higher to lower denominations, by annexing as many ciphers 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 ciphers, 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?

Ans. 32000m.

*The reason of this rule is obvious; for pounds are brought into shillings by multiplying them by 20; shillings into pence by multiplying them by 12; and pence into far things by multiplying them by 4; and this will be true in the reduction of numbers consisting of any denomination whatever. The reason of the rule in Reduction Ascending, is equally evident; for farthings are brought into pence by dividing them by 4; pence into shillings by dividing them by 12; and shillings into pounds by dividing them by 20.

126. What is Reduction? 127. Of how many parts does it consist? What is Reduction Descending ?-129. from a greater to a less denomination?

128.

What is the rule for changing any number 130. What is the reason of this rule?

2. In 91 dollars 75 cents, how many cents?

3. In 50 eagles, how many dollars ?

Ans. 9175c.

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 £27 15s. 9d. 2qrs. how many farthings?

Ans. $500.

In multiplying by 20, we add in the 15s; by 12,

Multiplied by 20=shillings in a pound. the 9d. and by 4, the 2qrs.

which must always be done in like cases.

This is in fact, in the present

case, simply changing the order of the question, thus: In 26678 farthings, how many pounds?

8. In £36 12s. 10d. 1q. how many farthings?

9. In £719 9s. 11d. how many half pence ? 10. In 37 pistoles, at 22s. how many shillings, things?

Ans. 35177.

Ans. 345358.

pence and farAns. 814 shill, 9768 pence. 39072 farth

11. In 53 moidores, at 36s., how many shillings, pence and far

things?

12. Reduce 47 guineas, (at 28s. each,) and one fourth of a guinea into shillings, six-pences, groats, (4d.) three-pences, two-pences, pences, and farthings.

Ans.

1323 shillings, 2646 six-pences, 3969 groats, 5292 three-pences, 7938 two-pences, 15876de and 63504qrs.

REDUCTION ASCENDING,

Teaches how to change numbers from a lower to a higher de nomination; and 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.

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

131. How do you change numbers of a lower denomination into a higher; or, in other words, how will you reduce farthings to pounds, or mills to eagles?

NOTE.-The remainder is always of the same denomination with the dividend.

8. Bring 35177 farthings into pounds.

9. Bring 365358 half pence into pence, shillings and pounds.

10. In 39072 farthings, how many pistoles, at 22s.?

11. In 63504 farthings, how many pence, twopences, threepen ces, groats, sixpences, shillings, and guineas?

NOTE.-These questions may serve as proofs to some of those in Deduction Descending.

REDUCTION ASCENDING AND DESCENDING.

1. MONEY.

1. In £97, how many pence, and English or French Crowns, at 6s. 8d.? Ans. 23280d. & 291 crowns.

2. In 735 French crowns, how many shillings and French guineas, at 26s. Ed. ? Ans. 4900s.,& 183 guin. 24s.

NOTE.-When it is required to know how many sorts of coin, of dif ferent values, and of equal numbers, are contained in any number of another kind; reduce the several sorts of coin into the lowest denomination mentioned, and add them together for a divisor; then reduce the money given into the same denomination for a dividend, and the quotient arising from the division will be the number required.

Observe the same direction in weights and measures.

132. I wish to ascertain how many moidores, guineas, pistoles, dollars, shillings and sixpences, each of the like number, are contained in 275 half joes: how shall I proceed?