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ANNUITIES AT COMPOUND INTEREST.
473. An Annuity at compound interest constitutes a geometrical progression whose first term is the annuity itself; the common multiplier is 1 plus the rate per cent for one interval expressed decimally ; the number of terms is the number of intervals for which the annuity is taken; and the last term is the first term multiplied by 1 plus the rate per cent for one interval raised to a power 1 less than the number of terms.
474. The Present Value of an Annuity is such a sum as would produce, at compound interest, at a given rate, the same amount as the sum of all the payments of the annuity at compound interest.
Hence, to find the present value; - First find the amount of the annuity at the given rate and for the given time, by 461; then find the present value of this amount by dividing it by the amount of $1 at compound
The present value of a reversionary annuity is that principal which will amount, at the time the reversion expires, to what will then be the present value of the annuity.
The present value of a perpetuity is a sum whose interest equals the annuity.
475. Hence, it will be seen that all questions in Annuities at compound interest can be solved by the rules of Geometrical Progression, by substituting for the terms given the corresponding terms used in Progression.
1. What is the amount of an annuity of $500 which is 7 years in arrears, at 6% compound interest?
$500 × (1.06' - 1)
1.06 - 1
SOLUTION. The payment now due, $500, is the first term of a geometrical ratio, 1.06, the amount of $1 for 1 year, is the ratio, and 7 the number of terms. Solving by 461, we find the sum of the series, which is the amount of the annuity, to be $4196.91}.
= $251.815÷.06 = $ 4196.913, Ans.
2. An annual pension of $500 is in arrears 10 years. What is the amount now due, allowing 6% compound interest? Ans. $6590.40.
NOTE. Consult the compound interest table. The amount of $1 at 6% for 10 years will be equal to the 10th power of 1.06.
3. What is the present worth of the annuity in Ex. 1?
$ 1.50363 = $ 2791.18+.
SOLUTION. The amount of the annuity is $4196.913. The amount of $1 for 7 years at 6% is 1.50363. Hence the present worth is 4196.913 ÷ 1.50363 = $2791.18+, Ans.
4. Find the annuity whose amount for 5 years, at 6% compound interest, is 2818.55. Ans. $500.
5. What is the present value of a reversionary lease of $100 commencing 14 years hence and to continue 20 years, compound interest at 5%. Ans. $629.426.
6. Allowing 6% compound interest on an annuity of $200, which is in arrears 20 years, what is its present amount? Ans. $7357.11. 7. What is the present worth of an annuity of $500 for seven years, at 6% compound interest?
8. An annuity of $200 for 12 years is in reversion 6 years. What is its present worth, compound interest at 6%? Ans. $1182.05 +.
476. Mensuration is the process of finding the number of units in extension.
477. Extension denotes that property of bodies, by virtue of which they occupy definite portions of space. Its dimensions are length, breadth, and thickness.
478. A point is that which has position only. 479. Direction is relative position of points.
480. A line has length, but neither breadth, nor thickness. A surface has length and breadth, but no thickness. A solid has length, breadth, and thickness.
LINES AND ANGLES.
481. A Straight Line is a line that does not change its direction. It is the shortest distance between two points.
482. A broken or crooked line is one made up of two or more straight lines. It changes its direction at one or more points.
483. A Curved Line changes its direction at every point.
* Some of the problems under Mensuration have been explained before as applications of Denominate Numbers, Involution, Evolution, etc. They are here repeated.
484. Parallel Lines have the same direction; and being in the same plane and equally distant from each other, they can never meet.
485. A Horizontal Line is a line parallel to the horizon or water level.
486. A Perpendicular Line is a straight line drawn to meet another straight line, so as to incline no more to the one side than to the other.
A perpendicular to a horizontal line is called a vertical line.
487. Oblique Lines approach each other, and will meet if sufficiently extended.
488. An Angle is the opening between two lines that meet each other in a common point, called the vertex.
489. A Right Angle is an angle formed by two lines perpendicular to each other.
A right angle is always equal to 90°.
490. An Obtuse Angle is greater than a right angle.
It may be equal to any number of degrees more than 90° and less than 180°. At 180° the two lines forming the angle merge into one, and become a straight line.
491. An Acute Angle is less than
a right angle.
An acute angle may be any number of degrees less than 90°.
492. A Plane Figure is a portion of a plane surface bounded by straight or curved lines.
493. A Polygon is a plane figure bounded by straight lines.
494. The Perimeter of a polygon is the sum of its sides.
495. The Area of a plane figure is the surface included within the lines which bound it.
1. A regular polygon has all its sides and all its angles equal. 2. A polygon of three sides is called a triangle; of four sides, a quadrilateral; of five sides, a pentagon, etc.
Pentagon. Hexagon. Heptagon. Octagon.
496. A Triangle is a plane figure bounded by three sides, and having three angles.
497. A Right-angled Triangle is a triangle having one right angle.
498. The Hypotenuse of a rightangled triangle is the side opposite the right angle.
499. The Base of a triangle, or of any plane figure, is the side on which it may be supposed to stand.
500. The Perpendicular of a right-angled triangle is the side which forms a right angle with the base.