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An easy and uniform method of computing interest D Fish's method, is to place the principal, the rate, and the me in months, on the right of a vertical line, and 12 on the its or, if the time is short and contains days, reduce to days and place 360 on the left. After canceling equal actors on sides of the line, the product of the remaining factors. right, divided by the factor, if any, on the lett will g required interest.

To find the interest of $184.80 for 1 yr. 5 mo, at 5%.

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See interest tables in our Lightning Calculator. An Aliquot Part or Even Part of a number is such a part as will exactly divide that number. Thus, 2, 21, 31, and 5, are aliquot parts of 10.

An aliquot part may either be an integer or mixed num ber, while a component factor must be an integer.

ALIQUOT PARTS OF ONE DOLLAR

plying the

Rate per cent. 2 $12.25 Int. for 1 yr.

principal by

the rate %, we

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Int. for 6 mo.

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.510 Int. for 15 da.

of 1 year's int.,

$19.905=Int. for 1 yr. 7 mo. 15 da. for I mo. take

$175 =Principal.

$194.905 Amt. for 1 yr. 7 mo. 15 da.

of 6 months'

int., and for

15 da. take

of 1 month's int. The sum of the several results is the int. for the whole time.

Adding the principal to the interest gives the amount.

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TO TELL ANY NUMBER THOUGHT OF.

ESIRE any person to think of a number, say a certain number of shillings, tell him to borrow that sum of some one in the company, and add the number borrowed to the amount thought of 1: wil here be proper to name the person who lends him the shillings and to beg the one who makes the calculation to do it with great care, as he may readily tail into an error especially the first time Then, say to the person- I do not lend you, but give you TO, add them to the former sum Continue in this manner "Give the half to the poor and retain in your memory the other half." Then add Return to the gentleman, or lady, what you borrowed, and remember that the sum ient you was exactly equal to the number thought of " Ask the person if he knows exactly what remains. He will answer "Yes." You must then say "And I know, also, the number that remains; it is equal to what I am going to conceal in my hand." Put into one of your hands five pieces of money, and desire the person to tell how many you have got. Ile will answer five; upon which open your hand, and show him the five pieces. You may then say "I well knew that your result was five, but if you had thought of a very large number, for example, two or three millions, the result would have been much greater, but my hand would not have held a number of pieces equal to the remainder." The person then supposing that the result of the calculation must be different, according to the difference of the number thought of, will imagine that it is necessary to know the last number in order to guess the result. but this idea is false; for, in the case which we have here supposed whatever be the number thought of, the remainder must always be five. The reason of this is as follows:-The sum, the half of which is given to the poor, is nothing else than twice the number thought of, plus 10, and when the poor have received their part, there remains only the number thought of. plus 5; but the number thought of is cut off when the sum borrowed is returned, and, consequently, there remain only 5

It may be hence seen that the result may be easily known, since it will be the half of the number given in the third part of the operation; for example, whatever be the number thought of, the remainder will be 36 or 25, according as 72 or

50 have been given. It this trick be performed several times successively the number given in the third part of the operation must be always different, for if the result were several times the same the deception might be discovered. When the first five parts of the calculation for obtaining a resuit are finished, it will be best not to name it at first, but to continue the operation, to render it more complex, by saying, for example:Double the remainder deduct 2, add 3. take the fourth part, etc. and the different steps of the calculation may be kept in mind in order to know how much the first result has been increased or diminished This irregular process never fails to confound those who attempt to follow it.

A Second Method. Bid the person take : from the number thought of, and then double the remainder desire hin. to take 1 from the doubie, and to add to it the number thought of, in the last place, ask him the number arising from this addition, and, if you add 3 to it, the third of the sum will be the num ber thought of The application of this rule is so easy. that it is needless to illustrate it by an example.

A Third Method. - Desire the person to add 1 to the triple of the number thought of and to multiply the sum by 3 then bid him add to this product the number thought of and the result will be a sum, from which, if 3 be subtracted, the remainder will be ten times the number required, and if the cipher on the right be cut off from the remainder, the other figure will indicate the number sought.

Example:-Let the number thought of be 6. the triple of which is 18; and if I be added, it makes 19; the triple of this last number is 57. and if 6 be added, it makes 63, from which, if 3 be subtracted, the remainder will be 60; now, if the cipher on the right be cut off, the remaining figure, 6, will be the number required.

A Fourth Method.-Bid the person multiply the number thought of by itself; then desire him to add 1 to the number thought of, and to multiply it also by itself, in the last place, ask him to tell the difference of these two products, which will certainly be an odd number, and the least half of it will be the number required. Let the number thought of, for example, be to, which, multiplied by itself. give 100; in the next place, to increased by 1 is 11, which, multiplied by itself. makes 121; and the difference of these two squares is 21, the least half of which, being 10, is the number thought of. This operation might be varied by desiring the person to multiply the second number by itself, after it has been diminished by

In this case, the number thought of will be equal to the greater half of the difference of the two squares. Thus, in the preceding example, the square of the number thought of is 100, and that of the same number less 1, is 81; the difference of these is 19; the greater half of which, or 10, is the number thought of.

TO TELL TWO OR MORE NUMBERS
THOUGHT OF.

.If one or more of the numbers thought of be greater than 9, we must distinguish two cases; that in which the number or the numbers thought of is odd, and that in which it is even. In the first case, ask the sum of the first and second, of the second and third, the third and fourth, and so on to the last, and then the sum of the first and the last. Having written down all these sums in order, add together all those, the places of which are odd, as the first, the third, the fifth, etc., make another sum of all those, the places of wrich are even, as the second, the fourth, the sixth. etc. subtract this sum from the former, and the remainder will be the double of the first number. Let us suppose, for example, that the five fol. lowing numbers are thought of, 3, 7. 13, 17. 20, which, when added two and two as above, give 10, 20, 30, 37, 23 the sum of the first, third, and fifth. is 63, and that of the second and fourth is 57. if 57 be subtracted from 63 the remainder, 6 will be the double of the first number 3 Now if; be taken. from 10, the first of the sums the remainder 7 will be the second number, and by proceeding in this manner we may find all the rest.

In the second case, that is to say, if the number or the numbers thought of be even you must ask and write down as above the sum of the first and the second that of the second and third and so on. as before but instead of the sum of the first and last, you must take that of the second and last; then add together those which stand in the even places and form them into a new sum apart, add also those in the odd places, the first excepted, and subtract this sum from the former, the remainder will be the double of the second number; and if the second number, thus found, be subtracted from the sum of the first and second, you will have the first number; if it be taken from that of the second and third, it will give the third; and so of the rest. Let the numbers thought of be, for example, 3, 7, 13, 17; the sums formed as above are 10, 20, 30, 24, the sum of the second and fourth is 44, from which, if 30, the third, be subtracted, the remainder will be 14, the double of 7. the second number. The first, therefore, is 3, the third 13, and the fourth 17.

When each of the numbers thought of does not exceed 9, they may be easily found in the following manner :

Having made the person add 1 to the double of the first number thought of, desire him to multiply the whole by 5, and to add to the product the second number. If there be a third, make him double this first sum, and add 1 to it; after which, desire him to multiply the new sum by 5, and to add to it the third number. If there be a fourth, proceed in the same manner, desiring him to double the preceding sum, to add to it I, to multiply by 5, to add the fourth number, and

so on.

Then ask the number arising from the addition of the last number thought of, and if there were two numbers, subtrac! 5 from it; if there were three, 55; if there were four, 555, and so on, for the remainder will be composed of rigures which the first on the left will be the first number thought the next the second, and so on.

Suppose the number thought of to be 3. 4. 6 by addin to 6, the double of the first, we shall have 7, which being tiplied by 5. will give 35, if 4, the second number though. be then added, we shall have 39, which. doubled. gives 38 and, if we add 1, and multiply 79, the sum, by 5. the re will be 395. In the last place, if we add 6, the num thought of, the sum will be 401, and if 55 be deducted it, we shall have, for remainder, 346, the figures of whi 4, 6, indicate in order the three numbers thought of

THE MONEY GAME.

A person having in one hand a piece of gold, and in the other a piece of silver. you may tell in which hand he has the gold, and in which the silver by the following method -Some value, represented by an even number, such as 8 must be assigned to the gold and a value represented by an odd num ber, such as 3. must be assigned to the silver after which, desire the person to multiply the number in the right hand, by any even number whatever such as 2 and that in the left hand by an odd number as 3 then bid him add together the two products, and if the whole sum be odd the gold will be in the right hand. and the silver in the left if the sum be even, the contrary will be the case

To conceal the artifice better it will be sufficient to ask whether the sum of the two products can be halved without a remainder for in that case the total will be even and in the contrary case odd

It may be readily seen that the pieces instead of being in the two hands of the same person may be supposed to be in the hands of two persons one of whom has the even number, or piece of gold, and the other the odd number or piece of silver. The same operations may then be performed in regard to these two persons, as are performed in regard to the two hands of the same persons, calling the one privately the right, and the other the left.

THE GAME OF THE RING.

This game is an application of one of the methods emploved to tell several numbers thought of, and ought to be performed in a company not exceeding nine, in order that it may be less complex. Desire any one of the company to take a ring and put it on any joint of whatever finger he may think proper The question then is to tell what person has the ring, and on what hand. what finger, and on what joint.

For this purpose, you must call the first person I, the second 2, the third 3, and so on. You must also denote the ten fingers of the two hands by the following numbers of the natural progression, 1, 2, 3. 4, 5, etc., beginning at the thumb of the right hand, and ending at that of the left, that this order of the number of the finger may, at the same time, indicate the hand. In the last place, the joints must be denoted by 1, 2 3, beginning at the points of the fingers.

To render the solution of this problem more explicit, let us suppose that the fourth person in the company has the ring on the sixth finger, that is to say, on the little finger of the left hand, and on the second joint of that finger.

Desire some one to double the number expressing the perSon which, in this case, will give 8; bid him add 6 to this doble, and multiply the sum by 5, which will make 65; then te him to add to this product the number denoting the finger, this to say 6, by which means you will have 71; and, in the place, desire him to multiply the last number by 10, and to to the product the number of the joint, 2; the last result be 712; if from this number you deduct 250, the remainwill be 462 the first figure of which, on the left, will te the person; the next, the finger, and, consequently, hand, and the last, the joint.

It must here be observed, that when the last result contains cipher, which would have happened in the present example had the number of the figure been 10, you must privately subtract from the figure preceding the cipher, and assign the value of 10 to the cipher itself.

THE GAME OF THE BAG.

To let a person select several numbers out of a bag, and to tell him the number which shall exactly divide the sum of those he had chosen :-Provide a small bag, divided into two parts, into one of which put several tickets, numbered 6, 9. 15, 36, 63, 120, 213, 309, etc., and in the other part put as many other tickets, marked No. 3 only. Draw a handful of tickets from the first part, and after showing them to the company, put them into the bag again, and having opened it a second time, desire any one to take out as many tickets as he thinks proper; when he has done that, you open privately the other part of the bag. and tell him to take out of it one ticket only. You may safely pronounce that the ticket shall contain the number by which the amount of the other numbers is divisible; for, as each of these numbers can be multiplied by 3, their sum total must. evidently, be divisible by that number. An ingenious mind may easily diversify this exercise, by marking the tickets in one part of the bag with any numbers that are divisible by 9 only, the properties of both 9 and 3 being the same; and it should never be exhibited to the same company twice without being varied.

THE CERTAIN GAME.

ve persons agree to take, alternately, numbers less than er number, for example, 11, and to add them together

of them has reached a certain sum, such as 100. By means can one of them infallibly attain to that number →the other?

whole artifice in this consists in immediately making

the numbers 1, 12, 23, 34, and so on, or of a series + continually increases by 11, up to 100. Let us suppose he first person, who knows the game, makes choice of 1; evident that his adversary, as he must count less than II, Can at most reach 11. by adding 10 to it. The first will then fase which will make 12. and whatever number the second may add the first will certainly win, provided he continually

add the number which forms the complement of that of his adversary to 11; that is to say, if the latter take 8, he must take 3 if 9, he must take 2; and so on. By following this method he will infallibly attain to 89: and it will then be im possible for the second to prevent him from getting first to 100; for whatever number the second takes he can attain only to 99; after which the first may say "and I makes 100.' If the second take 1 after 89, it would make 90, and his ad. versary would finish by saying—“' and 10 make 100." Between two persons who are equally acquainted with the game, he who begins must necessarily win.

If your opponent have no knowledge of numbers, you may take any other number first, under 10, provided you subse quently take care to secure one of the last terms, 56, 67, 78, etc., or you may even let him begin, if you take care afterward to secure one of these numbers.

This exercise may be performed with other numbers; but, in order to succeed, you must divide the number to be attained by a number which is a unit greater than what you can take each time, and the remainder will then be the number you must first take Suppose. for example, the number to be attained be 52, and that you are never to add more than 6; then, dividing 52 by 7. the remainder, which is 3, will be the number which you must first take; and whenever your opponent adds a number you must add as much to it as will make it equal to 7, the number by which you divided, and so in continua. tion.

ODD OR EVEN.

Every odd number multiplied by an odd number produces an odd number; every odd number multiplied by an even number produces an even number; and every even number multiplied by an even number also produces an even number. So, again, an even number added to an even number, and an odd number added to an odd number, produce an even number; while an odd and even number added together produce an odd number.

If any one holds an odd number of counters in one hand, and an even number in the other, it is not difficult to discover in which hand the odd or even number is. Desire the party to multiply the number in the right hand by an even number, and that in the left hand by an odd number, then to add the two sums together, and tell you the last figure of the product; if it is even, the odd number will be in the right hand; and if odd, in the left hand; thus, supposing there are 5 counters in the right hand, and 4 in the left hand, multiply 5 by 2, and 4 by 3, thus:-5 × 2 = 10, 4 × 3 = 12, and then adding 10 to 12, you have 10+12= 22, the last figure of which, 2, is even, and the odd number will consequently be in the right hand.

PROPERTIES OF CERTAIN NUMBERS.

OF ODD NUMBERS.-All the odd numbers above 3, that can only be divided by 1, can be divided by 6, by the addition or subtraction of a unit. For instance, 13 can only be divided by I; but after deducting I, the remainder can be divided by 6; for example, 5 + 1 = 6; 7—1=6; 17+ = 18; IQ I 18 35 I = 24 and so on

OF NUMBER THREE.-Select any two numbers you please, and you will find that either one of the two, or their amount when added together, or their difference, is always 3, or a number divisible by 3. Thus, if the numbers are 3 and 8, the first number is 3; let the numbers be 1 and 2, their sum is 3; let them be 4 and 7, the difference is 3. Again, 15 and 22, the first number is divisible by 3; 17 and 26, their difference is divisible by 3, etc.

OF NUMBER FIVE.-If you multiply 5 by itself, and the quotient again by itself, and the second quotient by itself, the last figure of each quotient will always be 5. Thus, 5 × 5 = 25; 25 × 25 = 625; 125 × 125 = 15,625, etc. proceed in the same manner with the figure 6, will constantly be 6.

Again, if you the last figure

To divide any number by 5, or any multiplicand of that number, by means of simple addition :-To divide by 5, double the number given, and mark off the last figure, which will represent tenths. Thus, to divide 261 by 5-261 + 261 = 522, or 5 22-10ths. Again, to divide the same number by 25, you must take four times the number to be divided, and mark off the last two figures, which will be hundredths, thus, 261 + 261 +261 +261 : = 1044, or 10 44-100ths.

=

OF NUMBER NINE.-The following remarkable properties of the number 9 are not generally known :-Thus, 9 × 1 = 9; 9 × 2 = 18, 1 + 8 = 9; 9 × 3 = 27, 2 + 7 = 9, 9 × 4 == 36, 3+6=9:9 × 5= 45, 4 + 5 F9; 9×6 = 54, 549; 9 × 7=63, 6 + 3: 9; 9 × 872, 7+ 2 = 9; 9 x9= 81, 8 + 1 = 9. It will be seen by the above that-1. The component figures of the product made by the multiplication of every digit into the number 9, when added together, make NINE. 2. The order of these component figures is reversed, after the said number has been multiplied by 5. 3. The component figures of the amount of the multipliers (viz. 45), when added together, make NINE. 4. The amount of the several products, or multiples of 9 (viz. 405), when divided by 9, gives, for a quotient, 45, that is, 4 + 5 = NINE.

It is also observable, that the number of changes that may be rung on nine bells is 362,880; which figures, added together, make 27; that is, 2 + 7 = NINE.

And the quotient of 362,880, divided by 9, will be 40,320; that is, 4 + 0 + 3 + 2 + 0 = NINE.

To add a figure to any given number, which shall render it divisible by Nine :-Add the figures together in your mind, which compose the number named; and the figure which must be added to the sum produced, in order to render it divisible by 9, is the one required. Thus, suppose the given number to be 7521 :

Add those together, and 15 will be produced; now 15 requires 3 to render it divisible by 9; and that number 3, being added to 7521, causes the same divisibility:

7521 3

9)7524(836

This exercise may be diversified by your specifying, before the sum is named, the particular place where the figure shall

be inserted, to make the number divisible by 9; for it is exactly the same thing whether the figure be put at the head of the number, or between any two of its digits.

To multiply by Nine by Simple Subtraction.—Supposing you wish to multiply 67583 by 9, add a cipher to the end of the sum, then place the sum to be divided underneath the amount, and subtract it from the same; the quotient will be the pro duct of 67583 multiplied by 9; thus:

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TO DISCOVER A SQUARE NUMBER. A square number is a number produced by the multiplica tion of any number into itself; thus, 4 multiplied by 4 is equal to 16, and 16 is consequently a square number, 4 being the square root from which it springs. The extraction of the square root of any number takes some time; and after all your labor you may perhaps find that the number is not a square number. To save this trouble, it is worth knowing that every square number ends cither with a 1, 4, 5, 6, or 9, on with two ciphers, preceded by one of these numbers.

Another property of a square number is, that if it be divided by 4, the remainder, if any, will be 1-thus, the square of 5 is 25, and 25 divided by 4 leaves a remainder of I and again, 16, being a square number, can be divided by 4 without leave ing a remainder.

A MAGIC SQUARE.

The following arrangement of figures, from 1 to 36, in the form of a square, will amount to the same sum if the numbers are cast up perpendicularly, horizontally, or from

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