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Definitions. 1. An annuity is a rent or sum payable or receivable yearly for a certain number of years, or forever.
2. A deferred annuity is one which is not to be entered upon immediately, but after a certain number of years.
3. A reversionary annuity is one which is not to be entered upon until after the death of some person or persons now living.
4. The present worth of an annuity is such a sum of money as will, at compound interest, produce an amount equal to the amount of the annuity.
To find the amount of an annuity, which has remained unpaid (or been
forborne), for a given time : in other words, to find the sum of a series, when the first term, the number of terms, and the ratio, are given. [See Case III., Progression by Ratios.]
1. What is the amount of an annuity of $100, forborne 6 years, at 5 per cent. compound interest?
Ans. $680-19. 2. What is the amount of an annuity of $150, payable half yearly, forborne 2 years, allowing half-yearly compound interest at 6 per cent. per annum?
Ans. $313-772. 3. What would a salary of $700 a year amount to, if left unpaid for 4 years, allowing quarterly compound interest at 12 per cent. per annum?
Ans. $3527-454. 4. If the annual rent of a house be $200 per annum, payable quarterly, and it remains unpaid 4 years, how much will be due, allowing compound interest at the rate of 12 per cent. per annum?
Ans. $10076844. 5. What would be the amount of a pension at $65 per annum, which had been unpaid for 14 years, if compound interest were reckoned at the rate of 6 per cent. per annum?
To find the present worth of an annuity which is to terminate in a given
number of years. 1. What is the present worth of an annuity of $150, to continue 6 years; reckoning compound interest at 6 per cent. per annum ? Ans. Amount of $150 for 6 years, $1046-297. Present worth of
$1046*297, 6 years hence, at 6 per cent., $7374598.
* In most works on Arithmetic, which treat of compound interest and annuities, the student is furnished with tables of the amount of $1, at various rates per cent., for from 1 to 30 or 40 years, and also of the present worth of annuities, &c., in like manner. Such tables are exceedingly convenient in offices where this kind of business is transacted, as the object there is to ascertain values as quickly as may be, and with as little calculation as possible. But they are entirely out of place in a school, where the main object is mental discipline, and expertness in calculation. Where the teacher, however, thinks such calculations of ratios occupy too much time, each student can be directed to form the requisite tables for himself, as separate exercises.
2. How much ready money will purchase an annuity of $250, to continue 20 years, at 6 per cent. compound interest ? Ans. $2867647.
3. What is the present worth of an annuity of $150 for 30 years, at 5 per cent. compound interest?
Ans. $2315-96. 4. What is the present worth of an annuity of $500, to continue 10 years, at 6 per cent. per annum, compound interest ? Ans. $3680-045.
5. A gentleman subscribed $1000 for a college, payable in 10 annual instalments, of $100 each. What is the present worth of such a subscription ; that is, what sum of money placed at interest would exactly pay each instalment as it fell due, reckoning the interest at 6 per cent. ?
Ans. $736. CASE III.
To find the present worth of an annuity in reversion.
a. Where the term of the annuity is limited. 1. What is the present worth of an annuity of $100, to be continued 6 years, but not to commence till 3 years hence, allowing 6 per cent. compound interest ? Present worth of an annuity of $100 for 9 years, at 6 per cent.,
$680-169. Worth of do. for 3 years, $267,301. Difference,
$412-868. Ans. 2. What is the present worth of an annuity of $320, to continue 7 years, but not to commence till after the expiration of 5 years, reckoning interest at 5 per cent. per annum?
Ans. $1450*728. 3. What is the present worth of an annuity of $200, to be continued 5 years, but not to commence till after the expiration of 2 years, reckoning interest at 6 per cent. ?
Ans. $749-798. 4. A father, dying, left a will, by which, among other arrangements for his family, he directed that his only son should have $500 a year as soon as he attained his majority, the annuity to be continued for 4 years, to assist him at the commencement of his profession. The son was exactly 18 on the day of his father's death. What sum would be necessary for the payment of this bequest, allowing compound interest at the rate of 6 per cent. per annum ?
Ans. $14546684. 5. What is the present worth of a reversion of $400 a year, to continue 7 years, but not to commence until the end of 8 years, allowing compound interest at 4 per cent. per annum ?
Ans. $1754°356. b. Where the term of the reversionary annuity is unlimited. [Where an annuity is to continue forever from the present time, it is
manifest that its present worth will be that sum whose interest for 1 year is equal to the annuity. All that is necessary, therefore, where the annuity is reversionary, is to find what sum of money will produce such a principal at the time when the annuity is to commence ]
1. What is the present worth of a reversionary annuity of $200 a year, to commence in 6 years, and to continue forever, interest being 6 Ans. $, .
Present worth, if entered on immediately, $3333}. 2. Find the present worth of a reversionary annuity of $500 a year, to commence in 4 years, and to continue forever, interest at 6 per cent.
per cent. ?
3. A man left his son an annuity of $400, to commence on his majority 3 years thereafter, and to continue forever to him, and his nearest heir. What sum would secure such an annuity at the father's death, allowing interest at 5 per cent. ?
Ans. #6910670. 4. What is the present worth of an annuity of $1000, to commence in 6 years, and to continue forever, interest at 6 per cent. ?
Ans. $11,749-35. 5. What is the present worth of a reversion in perpetuity of $100, to commence in 4 years, interest at 5 per cent. ? Ans. $1645-40.
OR, CHANGE OF ARRANGEMENT.
Definitions. PERMUTATION is the method of finding in how many different ways the order of things may be varied.
Exercises for the Slate or Black-board. 1. In how many different positions can 5 blocks be placed in succession ?
1st. If the blocks be letters in the order of the alphabet, it is obvious that a (the first) alone can only be placed in one position.
2d. If now 6 (the second block) be added, the two can be placed in two positions, namely, ab and ba· or 1X2=2.
3d. Adding c (the third) we have (commencing with c) cab, cbu; (with c in the middle) acb, bca ; (with c last) abc, bac: or 1X2X3=6.
4th. Adding d (the fourth), it is plain that the 6 changes are increased four fold, since d may be placed as the first letter of the series, and also as the second, third, and fourth, making the number 1X2X3X4=24.
5th. In the same manner the 5th will increase the last product five fold ; and, in general, every number added in the natural arrangement of numbers (1,2,3,4, &c.) will increase the number so many times.
All such questions, then, are evidently solved by taking the products of the natural numbers from 1 to the given number inclusive, the last product furnishing the answer.
Ans. to Exercise 1, 1X2X3X4X5=120. 2. How many ways may the 6 vowels, a, e, i, o, u, y, be placed, one after another?
Ans. 720. 3. How many variations may be made in the position of the 9 digits ?
4. Find how many changes may be rung on a chime of 8 bells; by dividing the answer to the last question by a certain number.
Ans. 40320. 5. A gentleman once offered to sell a township of land in Illinois, 6 miles square, for 1 cent for every different order in which the letters of the English alphabet could be arranged. What would be the price per acre at that rate?
PROPERTIES OF NUMBERS. The following exercises on the properties of numbers will not only assist the student in the art of computation, but, what is of vastly more importance, aid in the development of his faculties, by affording admirable materials for thought.
1. The sum, or the difference, of two even, or of two odd numbers will be an even number. Why? But the sum, or the difference of an even and an odd number will be an odd number. Why?
2. The sum of any number of even numbers, or of an even number of odd numbers, will be even. Why? But the sum of an odd number of odd numbers will be odd. Why?
3. If one, or both, of two factors be even numbers, the product will be even ; but if both be odd, the product will be odd. Why?
4. If any two numbers be, severally, divisible by a third (without a remainder), their sum, and their difference, will also be divisible by the same. Why?
5. If several different numbers he each divisible by any other number, then their sum will be divisible by the same number, and their product by the same number, or by any power thereof whose index does not exceed the number of factors. Why?
6. If several different numbers be each divisible by 3 or by 9, then their sum, and also the sum of their digits (see p. 112), will also be divisible by the same numbers respectively. Why?
7. If any number be multiplied by a number divisible by 9 or by 3 ; then the product, and also the sum of its digits, will also be divisible by the same numbers respectively. Why?
8. If any number be divisible by 3, by 9, or by 11, then its reverse (the same figures written backwards) will also be divisible by the saine numbers respectively. Why? See No. 6.
9. Any number diminished by the sum of its digits will become divisible by 9 and also by 3. Why? Any number divided by 9 will leave the same remainder as the sum of its digits when divided by 9. Why? The same remark also applies to 3. Why?
10. The sum of any number, consisting of an even number of places, and its reverse, will be divisible by 11, and their difference by 9. But, if the number of places be old, then the difference of the number and its reverse will be divisible both by 11 and by 9. Why?
11. If the sum of the digits in the odd places of any number, beginning at the place of units, be equal to the sum of the digits in the even places, then both the whole number and its reverse will be divisible by
11 ; their sum will also be divisible by 11, and their difference both by 11 and by 9. Why? When, in the above case, the number and its reverse (divisible by 11) consist of an odd number of places, then the sum of the quotients will be divisible by 11, and their difference by 9. But when the number, as above, consists of an even number of places, then the difference of the quotients will be divisible both by 11 and by 9. Why?
12. Any prime number (greater than 3) divided by 6, must have a remainder of 1 or of 5. Why? See p. 88.
13. Every prime number greater than 3 is divisible by 6, if 1 be either added to it or subtracted from it, according to circumstances. Thus 371, and 41+1, form numbers divisible by 6, and so of all other primes. Why?
14. Every number is either a prime number, or composed of prime numbers. Why?
15. An odd number cannot be divided by an even number without a remainder. Why?
16. A square number, or a cube number, arising from an even root, is even. Why? See No. 2.
17. The square and the cube of an odd number are odd. Why? See No. 3.
18. If an odd number measure an even number, it will also measure the half of it. Why?
19. If a square number be either multiplied or divided by a square, the product or quotient is a square. Why?
20. If a square number be either multiplied or divided by a number that is not a square, the product or quotient is not a square. Why?
21. The product arising from two different prime numbers cannot be a square. Why?
22. The product of no two different numbers, prime to each other, can make a square, unless each of these numbers be a square. Why?
23. The difference between an integral cube and its root is always divisible by 6. Why? See the synthesis of a cube, p. 165.
24. Every prime number above 2 is either 1 greater or 1 less than some multiple of 4. Why? And every prime number above 3 is either 1 greater or 1 less than some multiple of six. Why? See p. 88.
THE ELEMENTARY OPERATIONS OF ARITHMETIC
PERFORMED WHOLLY BY INSPECTION.
THE principal object of the following exercises is to develop the power of attention, to accustom the pupil to restrain his wandering thoughts, to fix them steadily upon one point, to the complete exclusion of all extraneous matters. Before such a power of concentration, all the seeming difficulties and obscurities of science disappear ; and that this power may be gained by a faithful practice of exercises like these, will hardly be disputed.