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3. 1 half-dollar, 5 half-dimes, and 7 three-cent pieces?
4. 2 dimes, 9 three-cent pieces, and 7 cents ?

5. 1 half-dollar, 1 quarter-dollar, 3 dimes, 3 half-dimes, 3 three-cent pieces, and 3 cents ?

6. 1 half-eagle, 4 dollars, 7 quarter-dollars, 8 three-cent pieces ?

7. 1 gold dollar, 1 silver dollar, 1 half-dollar, 1 quarterdollar, 1 dime, 1 half-dime, 1 three-cent piece, and 1 cent?

8. 3 quarter-eagles, 5 dollars, 5 half-dollars, 9 dimes, 7 halfdimes, and 4 three-cent pieces ?

9. 7 quarter-eagles, 5 quarter-dollars, and 5 half-dimes? 10. 3 half-eagles, 3 quarter-eagles, 9 half-dollars, 9 quarterdollars, 7 half-dimes, and 7 three-cent pieces ?

11. 2 double eagles, 3 eagles, 5 quarter-eagles, 4 dollars, 4 half-dollars, 9 quarter-dollars, 7 dimes, 9 half-dimes, 8 threecent pieces, and 6 cents ?

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Note. Many of the following examples admit of more than one

For instance, a debt of 6 cents can be paid with two coins, by using a half-dime and a cent, or by using 2 three-cent pieces. The pupil should find all the answers each question admits. The kind of work here required is very valuable to ensure quickness and accuracy, and to fit for the business transactions of real life.

1. If you should have coins of each kind coined in the United States, and should wish to use the smallest number of coins in the payment, how (i. e., with what coins) would you pay a debt of 6 cents ?

The same suppositions continuing, 2. How would you pay 25 cents ? 15? 13? 93? 76 ? 3. How would you pay 26 cents ? 16 ? 10 ? 50 ? 75 ? 4. How would you pay 99 cents ? 48? 36 ? 79? 84 ? 5. How would you pay 3 cents ? 8? 19 ? 91 ? 73 ? 6. How would you pay 67 cents ? 14? 43? 37? 23 ? 7. How would you pay $1.25 ? $2.50 ? $7.50 ? $1.36 ? 8. How would you pay $2.63 ? $3.50 ? $4.25 ? $15? 9. How would you pay $3.00 ? $7.60 ? $9.81 ? $6.37 ? 10. How would you pay $2.33 ? $1.42? $3.96 ? $4.17? 11. How would you pay $1.23 ?. $5.75? $25? $22.75 ? 12. How would you pay $12.60 ? $4.07? $1.08? $3.16 ?

Note. Bank bills of various denominations above a dollar are often used as a substitute for gold and silver coins. These bills most com monly represent some one of the following values, viz.: 1 dollar, 2 dollars, 3 dollars, 5 dollars, 10 dollars, 20 dollars, 50 dollars, and 100 dollars; but they sometimes represent 500 dollars, and 1000 dollars.

13. If you had coins of each kind coined in the United States, and bank bills of each denomination mentioned above, and should wish to pay as much as possible of each debt in bills, and at the same time should wish to use as small a number of bills and of coins as possible, how would you pay a debt of $3.75 ?

The same suppositions continuing, 14. How would you pay $4.25 ? $8.37 ? $6.00 ? 15. How would you pay $7.16 ? $27.49 ? $60. ? 16. How would you pay $16.84 ? $30.06 ? $95.29 ? 17. How would you pay $47.31 ? $83.24 ? $47.97 ? 18. How would you pay $76.83 ? $13.08? $99.86 ? 19. How would you pay $47.38? $27.04 ? $2.96 ? 20. How would you pay $35.31 ? $73.21 ? $99.99 ?

21. What would have been the answer to each of the above questions under this letter if the only bills used had been three, five, ten, and twenty dollar bills ?

I. It often happens that a person wishes to pay a debt when he does not have such coins or bank bills as will enable him to pay the exact sum he owes. In such cases he usually gives coins or bank bills to make a value greater than the debt he owes, receiving in return a sum equal in value to the excess of what he has paid over what he owed. This is called making change.

1. If each of two persons have several coins of each kind coined in the United States, and wish to use in each transaction as few coins as possible, how can a debt of 19 cents be paid by one to the other?

Ans. By giving a quarter-dollar, and receiving in return either 2 three-cent pieces,

a half-dime and a cent; or, by giving 2 dimes, receiving in return 1 cent.

Subject to the same conditions as in the last example: 2. How can a debt of 38 cents be paid ? of 63 ? of 49 ? 3. How can a debt of 22 cents be paid ? of 8? of 18? 4. How can a debt of 24 cents be paid ? of 78? of 99 ?

5. How can a debt of 90 cents be paid ? of 40 ? of 70 ? 6. How can a debt of 64 cents be paid ? of 44 ? of 43 ? 7. How can a debt of 86 cents be paid ? of 84 ? of 74? 8. How can a debt of $1.14 be paid ? of $2.25? of $3.50 ? 9. How can a debt of $7.50 be paid ? of $1.47 ? of $1.86 ? 10. How can a debt of $3.14 be paid ? of $13.42? of $1.67?

11. Miss Ireson bought at a store 3 yards of ribbon at 8 cents per yard, 4 papers of pins at 9 cents per paper, and a yard of muslin for 23 cents. She gave in payment a gold dollar. How much change ought she to receive, and in what coins can it best be paid ?

12. Mr. Jones has 3 half-dollars, 3 half-dimes, and 4 cents. Mr. French has 2 dollars, 1 half-dollar, 1 quarter-dollar, 1 halfdime, and 4 three-cent pieces. What coins must be exchanged that Mr. Jones may pay Mr. French a debt of 15 cents ? That Mr. French may pay Mr. Jones 15 cents ?

13. What coins must be exchanged that Mr. Jones may pay Mr. French $.20 ? $.11 ? $1.16 ? $.31 ? $.23 ? $1.37 ? $.85 ?

14. What coins must be exchanged that Mr. French may pay Mr. Jones $.20 ? $.11 ? $1.16 ? $.31 ? $.23 ? $1.37 ? $.85 ?

15. Mr. Brooks has 2 quarter-eagles, 1 three-dollar bill, 4 silver dollars and 9 three-cent pieces. Mr. Upham has 1 halfeagle, 3 two-dollar bills, 1 gold dollar, 7 half-dimes and 2 cents. In what ways can change be so made that Mr. Brooks shall pay Mr. Upham $.20 ? $.40 ? $4.50 ? $.50 ? $5.25 ? $.31 ? $.47? $.89 ? $1.10 ? $5.58 ? $9.37 ? $12.23 ?

16. In what way can change be so made that Mr. Upham shall рау.

Mr. Brooks the sums mentioned in the last example ? 17. Alfred has 10 three-cent pieces, and Edmund has 8 halfdimes. In what ways can change be so made that Alfred shall pay Edmund 1 cent ? 2 cents ? 11 ? 4? 25 ? 19 ? 5? 13? 16 ? 8? 17 ?

18. In what ways can change be so made that Edmund shall pay Alfred 11 cents ? 3? 1 ? 24? 28 ? 2? 4 ? 13 ? 19 ?

19. Mr. Allen bought of Mr. Mason 2 gallons of molasses at $.30 per gallon, 5 lb. of sugar at $.08 per pound, 4 oz. of nutmegs at $.124 per ounce, and 9 yards of calico at $.11 per yard. He gave in payment 2 bushels of potatoes at $.75 per bushel, 2 doz. eggs at 12 cents per doz., 1 melon worth $.13, and the balance in money. What was the balance ? On coming to pay the balance he found that the only coins he had were i

half-dollar, 1 quarter-dollar, 1 half-dime, and 2 three-cent pieces; while Mr. Mason had 4 half-dollars, 5 dimes and 2 cents. What coins must be exchanged that the debt may be cancelled ?

20. Two persons have each a half-dollar, a quarter-dollar, a dime, a three-cent piece, and a cent. How can change be so made that one shall pay the other 8 cents ? 16 cents ? 17 cents ? 18 cents ? 23 cents ? 61 cents ? 87 cents ? 69 cents ?

SECTION XIX.

INTRODUCTORY Note. The numbers employed in arithmetical problems are sometimes so large that it is difficult to remember them, and the results of our operations on them. When this is the case we use what is called Written Arithmetic; that is, we write the numbers we use and the results we obtain, instead of retaining them in our memory as in Mental Arithmetic. Aside from those connected with the writing of numbers, written arithmetic involves no principles, mental operations, or reasoning processes, that are not involved in mental arithmetic, and a person who fully understands the latter will have no difficulty with the former. In both, the work must be done in the mind. The only use of the slate is to relieve the memory, or to exhibit to the eye the numbers used and operations performed.

The subsequent sections are intimately connected with the preceding, and indeed contain much that has before been stated in another form. The importance of the principles seems to demand the repetition.

A. Any number whatever may be represented by the decimal point, and the ten characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

These characters are usually called figures, but sometimes digits, or digit figures. Their names, (one, two, three, four, five, six, seven, eight, nine, and zero, or, as it is sometimes called, nothing, cipher or naught,) are always the same, but the value expressed by any one of them depends upon its place with reference to the decimal point.

A figure written in the first place at the left of the point, represents as many units or ones as its name indicates; written in the second place, it represents as many tens as its name indicates ; in the third place, as many hundreds ; in the fourth, as inany thousands; and so the value would change with each change of place in reference to the point.

Hence each place has its peculiar name; the first second,

third, and fourth places, being called, respectively, the units' place, the tens' place, the hundreds' place, and the thousands' place. The following will illustrate this.

Thousands.
Hundreds.
Tens.
Units.
Point.

0000. In this example, each zero marks a place, and shows that there are none of the denomination of the place it occupies expressed in that place.

As another illustration, take the expression 2503. Here each figure marks a place, and denotes as many of the denomination of that place as its name implies; i. e., they represent 2 thou. sands, 5 hundreds, 0 tens, and 3 units, which, as we have before learned, is read, two thousand five hundred and three.

The decimal point is often omitted in writing numbers, but in all such cases it is understood to belong at the right of the given number, thus making the right hand figure represent units.

The largest number we can express by a single figure is 9, by two figures is 99, and by three figures is 999; for if we add 1 to either of these numbers, it will require one more figure and place to represent the sum than it did to represent the original number.

B. By extending these principles, we can take as many places as we please, giving to the figure in each, ten tiines the value it would have if written one place further to the right. The names of the places as far as the twenty-fourth are given in the following example.

000,000,000,00 0,000,000, By inspecting the above example it will be seen that the first three places are occupied by units, tens, and hundreds, - the second three by thousands, tens of thousands, and hundreds of thousands, the third three by millions, tens of millions, and hundreds of millions, and so on. If the first three places were called, as they might be with perfect propriety, units, tens of

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