MISCELLANEOUS QUESTIONS ON DECIMALS AND FEDERAL MONEY.-What does the word decimal come from? How do decimals arise? How can you find the denominator of a decimal? Does annexing a naught to a decimal increase or diminish its value? When adding decimals, where do you place the decimal point in the result? When subtracting? When multiplying? When dividing? In division of decimals, when will the quotient be a whole number? How do we multiply a decimal by 10, 100, 1000, &c.? How do we divide a decimal by 10, 100, 1000, &c.?

What is a circulating decimal? What is a repetend? How do you reduce a decimal to a common fraction? How do you reduce a repetend to a common fraction? How do you reduce a common fraction to a

decimal?

What is federal money? How do you write federal money? Why are two places appropriated to cents? How do you read federal money? How do you add, subtract, multiply, and divide federal money? What is a bill? When are credits to be entered in a bill? When must a bill be receipted? What are the forms to be used by a clerk in receipting a bill?

216. How many cents in five dollars?

100 cents make $1; in $5, therefore, there are 5 times 100 cents, or 500 cents. Ans. 500 cents.

We have here changed the denomination from dollars to cents, without changing the value. This process is called Reduction. We have reduced dollars to cents.

217. Reduction is the process of changing the denomination of a number without changing its value.

218. There are two kinds of Reduction :—

Here we

1. Reduction Descending, in which we change a higher denomination to a lower, as dollars to cents. must multiply.

216. How many cents in $5? What have we here done? What is this process called?-217. What is Reduction?-218. How many kinds of Reduction are there? What are they called? What is Reduction Descending?

2. Reduction Ascending, in which we change a lower denomination to a higher, as cents to dollars. Here we must divide.

219. Reduction Descending.

EXAMPLE 1.-Reduce $41 to mills. 100 cents make $1; in $41, therefore, there are 100 times 41 cents, or 4100 cents.

10 mills make 1 cent; in 4100 cents, therefore, there are 10 times 4100 mills, or 41000 mills.

EXAMPLE 2.-Reduce $41.375 to mills.
Reduce $41 to cents:
Add in 37 cents:

Reduce 4137 cents to mills:
Add in 5 mills:

$41

100 4100 c. 10

Ans. 41000 m.

41 x 100 = 4100 c.
4100 + 37 = 4137 c.

4137 x 10 = 41370 m.
41370541375 m. Ans.

220. RULE FOR REDUCTION DESCENDING.-Multiply the highest given denomination by the number that it takes of the next lower to make one of this higher, and add in the number belonging to such lower denomination, if any be given. Go on thus with each denomination in turn,

till the one required is reached.

221. Reduction Ascending.

EXAMPLE 3.-Reduce 41375 mills to dollars.

10 mills make 1 cent; therefore in 41375 mills there are as many cents as 10 is contained times in 41375, or 4137 cents, and 5 mills over.

10) 41375 m.
100) 4137 c., 5 m

Ans. $41.375

100 cents make 1 dollar; therefore in 4137 cents there are as many dollars as 100 is contained times in 4137, or $41, and 37 cents over. The last quotient and the two remainders form the answer-$41, 37 cents, 5 mills, or $41.375.

222. RULE FOR REDUCTION ASCENDING.-Divide the given denomination by the number that it takes of it to

What is Reduction Ascending?-219. Solve the given examples, explaining the several steps.-220. What is the rule for Reduction Descending ?-221. Reduce 41375 mills to dollars.-222. What is the rule for Reduction Ascending?

make one of the next higher. Divide the quotient in the same way, and go on thus till the required denomination is reached. The last quotient and the several remainders form the answer.

223. In Example 2 we reduced $41.375, and obtained 41375 mills. In Example 3, we reduced 41375 mills, and obtained $41.375. Thus it will be seen that Reduction Descending and Reduction Ascending prove each other.

224. Reduction of Federal Money.

In Example 1, § 219, we reduced dollars to cents by annexing two naughts, cents to mills by annexing one naught.

In Example 2, § 219, comparing the result, 41375 mills, with $41.375, the amount to be reduced, we find it is the same, with the dollar-mark and decimal point omitted.

In Example 3, § 221, comparing the result, $41.375, with 41375 mills, the amount to be reduced, we find that we have simply pointed off three figures from the right, and inserted the dollar-mark. Hence the following rules:

RULES FOR THE REDUCTION OF FEDERAL MONEY.1. To reduce dollars to mills, annex three naughts; to reduce dollars to cents, two; to reduce cents to mills, one.

2. To reduce dollars and cents to cents, or dollars, cents, and mills, to mills, simply remove the dollar-mark and the decimal point.

3. To reduce mills to dollars, point off three figures from the right; to reduce cents to dollars, two; to reduce mills to cents,

one.

223. How is Reduction Descending proved? Reduction Ascending ?—224. Recite the rules for the reduction of federal money.

11. 8620 cents to dollars.

12. 490000 mills to cents.

13. 56000 cents to dollars.

14. 56000 cents to mills.

15. 8705 cents to dollars. 16. $87.05 to mills.

17. How many cents in 4 eagles? (4 eagles = $40) Ans. 4000 c. 18. How many cents is a double eagle worth? Ans. 2000 c. 19. How many eagles are 8000 cents worth? 2000 cents? 20. Reduce 423756890 mills to dollars.

21. How many cents in $891? In $1024? In $41?

22. How many mills in 373 cents? In $5.621?

23. How many quarter-dollars equal a double eagle? 24. How many dimes in $1? In $15? In $30? In $49? 25. How many cents in 1 dime? In 5 dimes? In 20 dimes? 26. How many dimes are equal to 10 cents? To 150 cents? 27. How many half-dollars ought I to receive in change for an eagle? For two double eagles?

28. How many cents is a quarter-eagle worth? A half-eagle? A three-dollar piece? A half-dollar? Five dimes?

29. Reduce each of the following to cents, and add the results: 2 eagles; 5 half-dollars; 15 dollars; 1 double eagle; 3 quarterdollars; 12 dimes; 120 mills. Ans. 5957 cents.

Compound Numbers.

225. A Compound Number is one consisting of different denominations; as, 3 dollars, 19 cents.

226. .Compound numbers may be reduced, added, subtracted, multiplied, and divided.

227. To show the relations that different denominations bear to each other, Tables are constructed. These are now presented in turn, with examples in Reduction under each; they should be thoroughly committed to memory. For convenience of reference, these Tables are reproduced together on the last page of the book.

225. What is a Compound Number?-226. What operations may be performed on Compound Numbers?-227. For what purpose have Tables been constructed in connection with Compound Numbers?

The pound mark £ is a capital 1, standing for the Latin word libra, a pound; it always precedes the number, as £2. S. stands for the Latin solidus, a shilling; d. for denarius, a penny; qr. for quadrans, a farthing.

Shillings are sometimes written at the left of an inclined line, and pence at the right: 2/- 2s. -/6 = 6d. 2% = 2s. 6d. Farthings are sometimes written as the fraction of a penny, 1 far. as id., 2 far. as id., 3 far. as 3d.

The pound is simply a denomination; a gold coin called the Sovereign represents it. The Sovereign is worth $4.84. The English shilling is worth 24 cents, and the English penny about 2 cents.

Guineas, originally made of gold brought from Guinea, are no longer coined. The Crown is a silver coin, worth 5 shillings.

229. In the twelfth century, some traders from the Baltic coasts, called by the people Easterlings because coming from regions farther east, were employed to regulate the coinage of England. From these Easterlings the currency took the name of Sterling Money.

230. Recite the rules for Reduction, § 220, 222. EXAMPLE 1.-Reduce £5 19s. 3 far. to farthings.

228. What is English or Sterling Money? Recite the Table of Sterling Money. What is the pound mark, and where does it stand? What do 8., d., and gr., stand for? How are shillings sometimes written? How are farthings sometimes written? Is the pound a denomination or a coin? What represents it? What is the sovereign worth? The English shilling? The English penny? Why were guineas so called i What is the Crown?-229. From whom did sterling money receive its name ?~230. Go through and explain the given examples in Reduction.