10 years, bearing a semi-annual compound interest, at 2 per cent. ? MISCELLANEOUS EXAMPLES IN ANNUITIES. II. 6. A young man has bought a farm for $3500. He pays $2000 cash, and gives his note on interest at 6 per cent. for the remainder, the payment of which is secured by a mortgage of the whole farm. How much must he save yearly, in order to pay his interest money annually, and clear his farm in 10 years? NOTE. The amount of all his yearly savings, at compound interest, must be equal to the amount of the whole note for 10 years at compound interest. Therefore, divide the amount of the note, $1500, at compound interest for 10 years, by the amount of an annual payment of $1 for 10 years (Quest. 1), the quotient will be the answer. 7. How much must he save yearly, to clear his farm in 8 years? in 5 years? 8. A man wishes to put at compound interest such a sum of money as will afford him annually $100 for 10 years, at the end of which time the principal and interest shall be exhausted. What sum must be put at interest, the rate being 6 per cent.? NOTE. Divide the amount of an annuity of $100 for 10 years by the amount of $1 at compound interest for 10 years. 9. How much must he put at interest at 5 per cent., to yield an annuity of $200 for 20 years? 10. What is the present worth of an annuity of $150 to continue 10 years, allowing compound interest at 6 per cent.? NOTE. This question is the same as the preceding. Why? When an annuity is not to commence till some specified time has elapsed, or till the occurrence of some future event, it is called an annuity in reversion. 11. A person leaves an estate, the annual rent of which is $400, to his widow, during her life, and the reversion of the same to his son for 10 years after her death. What is the present value of each legacy, allowing compound interest at 5 per cent., supposing the widow to live 15 years after the death of her husband? NOTE. The present worth of the annuity to continue 25 years will be the present value of both their legacies. Its present worth for 15 years is the present value of the widow's, and this subtracted from the value of both will give the value of the son's. Express the rule for this in your own language. 12. What would be the present worth of each of the above. legacies, if the widow should survive her husband 10 years? 5 years? 151. PERMUTATION. Permutation is the method of finding in how many ways any number of things may be arranged. To do this, we have the following RULE. Multiply continually together all the terms of the natural series, from 1 up to the given number; the product will be the answer required. 1. For how many days can 5 persons be placed in a different position around a table at dinner? 1×2×3×4×5 120. For how many days can 10 persons? 12? 2. How many changes can be made of the letters in the word Charlestown? New York? Manchester? QUESTIONS. What is arithmetical progression? What is the common difference? What is an ascending series? a descending series? Give examples. What are the terms of the series? the extremes? the means? What 5 things are to be considered in an arithmetical series? Give the rule when the first term, common difference and number of terms are given, to find the last term ;-the two extremes and the number of terms being given, to find the common difference. What is the rule for finding the sum of all the terms? Why? Demonstrate the rule. What is an annuity? an instalment? the present worth of an annuity? the amount of an annuity? Show that an annuity at simple interest is an example of arithmetical progression. (See quest. 1, note.) What is the first term? the common difference? the number of terms? What is meant by an annuity in arrears? What is geometrical progression? What is the common ratio? What is an ascending series? a descending series? Give examples. What are the 5 things to be considered in a geometrical series? What is the rule when the first term, ratio, and number of terms are given, to find the last term? Give an example. The first term and ratio given, to find the sum of all the terms? Demonstrate the rule. Show that an annuity at compound interest is an example of a geometrical series. What is the first term? the ratio? the last term? the sum of all the terms? How may the present worth of an annuity be found? What is an annuity in reversion? How do you find the present value of an annuity in reversion? A C E SECTION XVIII. - SURFACES. B D F M T 1. A POINT is position only, without dimensions. 2. A LINE has one dimension only, length. 3. A STRAIGHT LINE is extension in one direction only; it is the shortest distance between two points. 4. A CURVE LINE constantly changes its direction. 5. PARALLEL LINES are equally distant in every point, and never meet, though ever so far extended. 6. OBLIQUE OR INCLINED LINES change their distance from each other, and would meet if sufficiently extended. 7. AN ANGLE is the opening between two lines which meet in a point. The point of meeting is called the VERTEX of the angle. Thus, the opening between the lines PQ and QR is an ANGLE, and the angular point at Q is the VERTEX of the angle. Angles are denoted by three letters, the middle letter denoting the angular point; as, P Q R, or R Q P, denotes the angle at Q. 8. RIGHT ANGLES are angles which are made by two lines meeting so as to form equal angles. STU and V T U are right angles. 9. PERPENDICULAR LINES are lines which meet at equal angles. The lines W X and Y Z are perpendicular to each other. 10. HORIZONTAL LINES are lines parallel to the plane of the horizon. 11. VERTICAL LINES are lines perpendicular to the plane of the horizon. 12. AN OBLIQUE ANGLE is either greater or less than a right angle. When greater, it is called an OBTUSE angle, as EFG; when less, it is called an ACUTE angle, as HIK. 13. A SURFACE is the outside of anything. It has two dimensions, length and breadth. 14. A PLANE SURFACE does not change its direction; that is, it is perfectly flat or level. 15. A TRIANGLE is a figure bounded by three sides. Its altitude or height is the perpendicular distance between one of the angles and the opposite side. The side to which the perpendicular is drawn is called the base. Thus, P Q R is a triangle; Q S is the altitude, and P R the base. 16. AN EQUILATERAL TRIANGLE is a figure that has its three sides equal. Y 17. AN ISOSCELES TRIANGLE is one that has two of its sides equal. B A C 18. A SCALENE TRIANGLE is one that has its three sides unequal. 19. A RIGHT-ANGLED TRIANGLE is one that has one right angle, as at E. The hypothenuse is the side opposite the right angle; as, the side D F. G 20. AN OBTUSE-ANGLED TRIANGLE has one obtuse angle. 21. An acute-angled triangle has all its angles acute. H 22. A QUADRILATERAL or QUADRANGLE is a figure bounded by 4 straight lines; as, G H IK. 23. A PARALLELOGRAM is a quadrilateral that has its opposite sides parallel. 24. A RECTANGLE is a right-angled parallelogram; as, GHIK. 25. A SQUARE is an equilateral rectangle; as, LM NO. The line L O is a diagonal. 26. A RHOMBOID is an oblique-angled parallelogram; as, P Q R S. 27. A RHOMBUS is an equilateral rhomboid; as, TUV W. 28. A TRAPEZOID is a quadrilateral which has only one pair of its opposite sides parallel; as, A BČ D. 29. A TRAPEZIUM is a quadrilateral, neither pair of whose opposite sides are parallel. 30. PLANE FIGURES bounded by more than four straight lines are called polygons. A polygon of 5 sides is called a pentagon; of 6 sides, a hexagon; of 7, a heptagon; of 8, an octagon. 31. A REGULAR POLYGON has all its sides and all its angles equal. If they are not all equal, the polygon is irregular. The figure I K L M N is a regular polygon. The equilateral triangle and square are regular figures. |