462. Registered bonds are recorded, with the names of their owners, and can not be transferred from one party to another without a change of the record.

463. Coupon bonds are bonds to which certificates are attached calling for the payment of certain interest at specified times.

These certificates called coupons are cut off and presented for payment when they become due. No record is made of the holders of coupon bonds, hence they may be transferred from one person to another by delivery as bank-notes.

464. Bonds issued by the United States Government (sometimes called Government securities), and State bonds (called State securities), are distinguished by their rates of interest, dates at which they are made payable, etc.

Thus, in the daily papers of May 20, 1886, we find mentioned "U. S. 41/2's, '91, reg.," that is, United States registered bonds bearing 41/2 % interest and payable in 1891. (See quotations, Art. 465 )

465. Bonds, like stocks, are bought and sold at the Stock Exchanges of all the principal cities, the brokerage on the purchase and sale of both being the same. (See Art. 291, page 291.)

The following are the quotations for the United States securities, in the market May 22, 1886:

Note.-Currency 6's are bonds issued by the United States Government in aid of the trans-continental railways, bearing 6% interest, and payable in currency at the times specified in the quotations.

In the following problems, $100 bonds are referred to, and brokerage is reckoned at 1% on par values, unless otherwise stated.

1. Find the cost to the buyer of 38 bonds at 971/2.

2. If a person sells 38 bonds at 9712, what will he receive from his broker?

3. What amount in bonds at 112 can be bought for $10,586.75 ?

The brokerage being added to the price of the bond, the cost of 1 bond is found to be 1125/. At this rate, how many can be bought for the given sum?

Note.--No principles are involved in the solution of problems like the foregoing but such as are applicable to the purchase and sale of stocks; but when questions arise as to the advantage of investing in one kind of bonds rather than another, such problems as the following occur:

4. If a person buys 5% stock at 125, what rate of interest does he receive on the money invested?

Whatever he pays for the bond, the interest he receives is always the interest specified in the bond. Hence he receives $5 interest on his investment. What per cent. of $125 is $5?

Or, if stock at 100 yields 5%, at 125 it must yield 100/125, or 4/5 of 5 % = 4 %.

interest will be realized on money invested

What

13. In 4%

How much money must be invested

bonds at 923/, to produce $352 income.

8

17. That I may receive 6% on the money invested, what price may I pay for 8% bonds?

What would be the difference between the income from an investment

18. Of $8400 in 41/2's at 120, and in 31/2's at 112.

"41/2's 90.

"3's

21. If a person invest $4488 in 3% stocks at 70, and $5505 in currency 6's at 1372, paying the usual brokerage, what per cent. will his income be on the sum invested?

APPENDIX.

Testing the Accuracy of Addition, Subtraction, Multiplication, and Division.

466. There is no better way of making sure of the correctness of an addition than adding "both ways"; and in subtraction the best test of accuracy is that the sum of the subtrahend and remainder is equal to the minuend.

In multiplication, the most thorough proof of accuracy is found if the product of the multiplier by the multiplicand is equal to the first product. In division, if the sum of the remainder added to the product of the quotient by the divisor is equal to the dividend, the work may be relied upon as correct. But these methods of proof require as much time as the original operation, hence the common use of the method called

Casting out Nines.

To cast out the nines of a number we may add all the terms of the number, and divide the sum by 9; the remainder will be the result sought. Thus, the sum of the terms of 4787763 is 42. Dividing this by 9, we obtain 6 as the excess of nines. But, since we wish to know only what the remainder is, we may drop the nines from the results as we proceed.

Thus, in the operation of casting out nines from 4787763, we may think the process indicated in light-faced italics, and speak the numbers printed in full-faced type, as follows:

In this process we skip the 9's, for there is no use of adding a nine and at once dropping it.

Explanation.-Casting out the nines from the divisor and quotient separately, we find the remainder written over the last figure of each. We then multiply the one remainder by the other, cast out the nines of the product, and carry the excess to the remainder found by the division 562, and think 3 + 5 + 6 = 14, 1495, and 5 + 2 = 7. We finally cast out the nines from the dividend, and since we obtain 7 from this also we judge the work to be correct.

467. The principle on which this method is based is, that The remainder arising from dividiny any number by 9 is always the same as the remainder that arises from dividing the sum of all its terms by 9.

That this must be so is evident from the fact that, on being divided by 9, there is a remainder of 1 for every ten, hundred, or thousand that go to make up a number, thus:

Dividing 10, 100, or 1000 by 9, the remainder is always 1. Dividing 20, 200, or 2000 by 9, the remainder is always 2, etc. Hence, if we divide separately the parts of a number represented by its digits by 9, the remainders will be expressed by those digits: e. g., if we divide the 2000, 400, 70, and 8, in 2478, separately by 9, the remainders will be 2, 4, 7, and 8, the excess of 9's in the sum of which is evidently the same as in the sum of the digits or in the number itself.

468. Another excellent test of the correctness of an operation in division is that the remainder after division added to all the subtrahends produces a sum equal to the dividend.

Greatest Common Divisor. (See Art. 141.) Example. Let it be required to find the greatest common divisor of 91 and 224.

The process as given in Art. 141, page 142, is as follows: We divide the greater number by the less, and the divisor by the remainder, and so on till we find that 7 will exactly divide the preceding divisor or remainder, as we choose to regard it. By trial, we find that 7 is a common divisor of 91 and 224. But, by reasoning, we might conclude that it must always be the case that the last remainder in such a succession of divisors will be a common divisor of the given numbers. The reasoning would be based on two principles:

91)224(2

182

42)91(2
84

7)42(6

42

1. That an exact divisor of any number must be an exact divisor of any multiple (number of times) that number.

If there is an exact whole number of times 7 apples in one heap, there would be an exact whole number of times 7 apples of the same size in any number of equal heaps.

2. That an exact divisor of two numbers will be an exact divisor of their sum or difference.

If there are 5 times 12 buttons on one string, and 2 times 12 buttons on another, there will be an exact whole number of times 12 buttons on one string more than the other, and an exact whole number of times 12 buttons on both.

With these principles in view, to show that the last divisor is

a common divisor, we would reason thus:

Seven being a divisor of 42, it is, according to principle 1, a divisor of 84; and being a divisor of 84, it is (principle 2) a divisor of 91 (84 +7); and being a divisor of 91, it is a divisor (principle 1) of 182 (2 × 91); and being a divisor of 182 and 42, it is (principle 2) a divisor of their sum, 224. Hence, 7 must be an exact divisor of 91 and 224.

And to show that the last divisor is the greatest common divisor, we would reason as follows:

According to principle 1, the common divisor of 91 and 224 must be an exact divisor of 182 (2 × 91); and hence, being a common divisor of 182 and 224, it must be, according to principle 2, an exact divisor of their difference, 42; and, reasoning in a similar manner, being a divisor of 42, it must be a divisor of 91-84, or 7. Hence, the greatest common divisor can not be greater than 7.

Thus, having found that 7 is a common divisor, and that the common divisor can not be greater than 7, we conclude that 7 is the greatest common divisor.