Exercises.

2. An annuity of $200 has been in arrears 8 years; what is the amount due, at 6 % simple interest? Ans. $1936. 3. To what will a rent of $450 per annum, payable quarterly, amount, if forborne for 11 years, at 6 % simple interest? Ans. $6546.37.

4. If a salary of $450 a year be in arrears 10 years, to how much will it amount at 7 % simple interest? Ans. $5917.50.

CASE II.

456. To find the amount of an annuity at compound interest.

1. Required the amount of an annuity of $100, forborne five years, at 6% compound interest.

At the end of the 5th year there will be due: the 5th year's payment, or $100; the 4th year's payment, $100, plus 1 year's interest, or $106; the 3d year's payment, plus 2 years' compound interest, or $112.36; the 2d year's payment, plus 3 years' compound interest, or $119.1016, and the 1st year's payment plus 4 years' compound interest, or $126.2476+.

Hence, the sums due are $100 + $106+$112.36 + $119.1016+ $126.2476, or $563.709+.

But the sums due at the end of the 5th year form a geometrical series, of which the annuity, $100, is the first term, the amount of $1 for 1 year, or $1.06, is the rate, and the number of years the number of terms. Hence,

Find the amount of the first payment at compound interest for the last term of a geometrical series, and then the sum of the series for the amount of the annuity.

Exercises.

2. What is the amount of an annuity of $200 a year, forborne 5 years, at 7 % compound interest? Ans. $1150.146+. 3. If a person expends for 30 years $40 per annum for cigars, how much will they cost him at 7 % compound interest? Ans. $3778.39+.

How do we find the amount of an annuity at compound interest?

4. If you should deposit $50 every 6 months in a savings bank, to how much would it amount in 25 years, at 3 % semiannual compound interest? Ans. $5639.84+.

CASE III.

457. To find the present value of an annuity at compound interest.

1. What is the present value of an annuity of $100, to continue for 5 years, at 6 % compound interest?

The amount of the given annuity for 5 years, by the preceding case, is $563.709, and the present value of the annuity must be the present value of the amount (Art. 302). The amount of $1 at compound interest for the given time and rate, from the table, Art. 319, is $1.338225.

Hence, the present value is $563.709 ÷ $1.338225, or $421.236+. Hence, to find the present worth of an annuity at compound interest,

Find the amount of the annuity, and divide it by the amount of $1 at compound interest for the given time and rate..

Exercises.

2. What is the present value of a pension of $1000 for 4 years, at 7%? Ans. $3387.207+. 3. What is the present value of an annual rent of $154 for 19 years, at 5% ? Ans. $1861.13.

4. Bought an estate for $30000, payable in equal yearly installments of $5000; how much ready money, at 6 %, should discharge the debt at the time of purchase? Ans. $24586.62.

REVIEW EXERCISES.

1. What is the third power of 11% ?

Ans. 14816.

2. Find the second power of the third power of 5.

Ans. 15625.

3. Sold a field for $484, receiving as many dollars per acre as there were acres; how many acres were there, and what was the price per acre? Ans. 22 acres, and $22 per acre.

How is found the present worth of an annuity at compound interest?

4. There is a certain room, of a cubical form, which contains 1953.125 cubic feet; what is the length of each of its equal sides?

Ans. 12.5.

5. I have 841 trees, which I wish to set out in a square grove; how many of the trees must be planted in each row?

6. What is the difference between of a solid foot and a solid foot, or a cube whose sides are each of a foot square ? Ans. 3 solid feet.

7. If a lead pipe 1 inch in diameter will fill a cistern in 4 hours, in what time will 2 pipes, each of an inch in diameter, fill the same? Ans. 8 hours.

8. A grocer has two kinds of tea, one at 75 cents a pound, and the other $1.10; how must he mix them in order to afford the mixture at $1 a pound?

9. A man had 10 children whose several ages differed alike, the youngest being 6 years old, and the oldest 51; what was the difference between the ages of the ninth and tenth ?

10. What will be the cost of painting a conical spire at of a dollar a square yard, if the slant hight of the spire be 50 feet, and the circumference at the base 26.7 feet? Ans. $14.83+.

11. If a horse be tethered equidistant from the four corners of a square lot containing exactly 10 acres, what must be the length of the rope to allow him to graze over every part of the lot? Ans. 28.27+rd.

12. Construct a geometrical series, of which 12 is the first term, and 3072 the 5th term. Ans. 12, 48, 192, 768, 3072. 13. I can purchase a farm for $700 cash down, or for $924 to be paid in the course of 7 years, part of the price at the end of each year. Allowing compound interest at 6 %, which erms will be the most advantageous to me?

Ans. Cash down, by $36.87+.

REVIEW QUESTIONS. What is a Series? (441) Terms of a Series? (441) Extremes? (441) The Means? (441) Arithmetical Series? (443) Geomet rical Series? (448) Rate or Ratio of a Geometrical Series? (448)

Exercises in Analysis.

1. Required the greatest common divisor of §,, and .

SOLUTION.,, and, changed to equivalent fractions having the least common denominator, become 24, 15, and 18.

The greatest common divisor of 24, 15, and 18 twentieths, is 3 twenti eths, or

Therefore, etc.

2. What is the greatest common divisor of §, 3%, and 63? Ans. 15

3. What is the greatest number that is contained an exact whole number of times in 3, 3, 4, and 21?

4. What is the least common multiple of 24, 41⁄21⁄2, and 33? SOLUTION. 24, 41, and 33, changed to equivalent fractions having the least common denominator, become 18, 36, and 27.

The least common multiple of 18, 36, and 27 eighths, is 108 eighths, or 108 13.

=

Therefore, etc.

5. What is the least common multiple of 2, 4, and § ?

Ans. 30.

6. What is the sum of money with which can be purchased a number of hens at $.75 each, a number of ducks at $.37 each, and a number of turkeys at $2.06 each? Ans. $8.25.

7. Find the square root of 225 by factoring.

SOLUTION. 225 factored is equal to 5 × 5 × 3 × 3.

Since the square root of a number is the factor which must be taken twice to form the number (Art. 391), 5 and 3, or one of every two equal prime factors of the number, must be taken to compose its square root.

Hence, 5 X 3, or 15, is the square root of 225.

8. Find the sixth root of 46656 by factoring. 9. Find the cube root of 28 by factoring.

Ans. 6.

Ans. .

10. A person being asked the hour of the day, said that the time past noon was equal to # of the time till midnight. What was the time?

REVIEW QUESTIONS. What is a Unit? (1) A Quantity? (2) A Num ber? (3) Figures? (18) Notation? (16) Numeration? (17) Addition! (37) Subtraction? (42) Multiplication? (47) Division? (57)

SOLUTION.

The time to midnight is of itself; then § +, or %, of the time to midnight, is equal to the time from noon to midnight, which is 12 hours.

If of the time to midnight is equal to 12 hours, is equal to ↓ of 12 hours, or 1 hour 20 minutes, and is equal to 4 times 1 hour 20 minutes, or 5 hours 20 minutes.

Hence, the time was 20 minutes past 5 o'clock, P. M.

11. If the time of day is such that of the time past noon is of the time past midnight, what is the hour?

Ans. 4 o'clock, P. M.

12. What is the hour, if of the time past 10 o'clock, A. M. is the time till 10 o'clock, P. M. ? Ans. 6 o'clock, P. M.

13. A, B, and C start at the same time from a given point, to travel in the same direction round an island 73 miles in circumference, A at the rate of 6, B of 10, and C of 16 miles per day; in what time will they be next together?

SOLUTION. Since B travels 4 miles a day faster than A, he will gain an entire round of the island, or 73 miles, in 4 of 73 days, or 18 days. Since C travels 10 miles a day faster than A, he will gain an entire round in of 73 days, or 7-3 days.

Hence, B cannot be with A except at the end of 181 days, or of some multiple of 184 days; and C cannot be with A except at the end of 7,3 days, or of some multiple of 73 days.

10

Therefore, C and B can both be with A for the first time, only after the lapse of a number of days expressed by the least common multiple of 184 and 7-8; and the least common multiple of 184 and 7-3 is 361⁄2. Therefore, etc.

10

14. There is an island 73 miles in circumference, and 3 men all start together to travel round it in the same direction; A goes 5 miles a day, B 8, and C 10; when will they all come together again? Ans. In 73 days.

15. A and B, at the opposite extremities of a wood, 135 rods in a circuit, begin to go round it in the same direction, at

REVIEW QUESTIONS. What is a Rule? (11) A Formula? (69) An Operation? (8) Which are the Fundamental Operations? (66) What is a Sign? (38) Symbols of Operation? (67) A Solution? (10) Analysis?

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