5. A man bought a house containing 24 windows, at 3 mills for the first window, 6 for the second, 12 for the third, &c. What did he pay for the house? ANNUITIES. Ans. $50331.645. 529. An Annuity is a sum of money payable annually, or at regular periods of time. 530. The Amount of an annuity is the sum of the payments and their interest for the specified time. To find the amount of an Annuity at Compound Interest. EXAMPLE. To what will an annuity of $10 amount in 3 yr., at 6%? the given rate as the ratio; and the number of years, the number of terms, and perform the example by (527, 528). To facilitate computations, the following table answers a very good purpose. It gives the amount of any unit for any number of periods from 1 to 40, at 5, 6, and 7%. Years. TABLE Of amounts of $1 or £1 annuity per annum, at compound interest. 5 per cent. 6 per cent. 7 per cent. Years. 5 per cent. 6 per cent. 7 per cent. To use this table we have the following 531. RULE. Find the amount of 1 unit for the given time and rate, and multiply this amount by the number of units. EXAMPLE. To what sum will an annuity of $225 amount in 12 yr., at 7%? SOLUTION. $17.888451 × 225 = $4024.901. EXPLANATION. - We find in the table, under the heading 7%, and opposite 12 in the left-hand column, the number 17.888451, which is the number of dollars to which an annuity of $1 will amount in 12 yr., at 7%. Multiplying this by the given number of dollars, 225, we have $4024.901, the value of an annuity of $225 running 12 yr., at 7%. PROBLEMS. 1. A gentleman deposited for his son $150 for 10 consecutive years in a savings bank that paid 5% compound interest. To what sum did the annuity amount immediately after the 10th payment? Ans. $1886.684. 2. A gentleman paid $400 a year rent, for 15 yr. To what sum did his 15 yr. rent amount, at 7%? Ans. $10051.61. 3. A merchant at the age of 30 insured his life for $10000, paying a premium of $23.30 on $1000. He died just after making the 30th annual payment. How much more money would his family have received had this premium been improved at 6% compound interest ? Ans. $8420.56. 4. A father desired to deposit annually in a savings bank such a sum as would amount in 10 payments, at 5%, to $1886.684. What must be the annual deposit ? SOLUTION. Since in 10 yr., at 5%, $1 annuity would amount to $12.5779+, it will require as many dollars annually to amount to $1886.684, as $12.5779 + is contained times in $1886.684, or 150 times. Therefore, he must deposit annually $150, Ans. 5. A dealer in real estate sells $600 lots for $100 cash, and the balance in 10 equal annual payments, which pay both principal and interest, at 7%. How much is each payment? SOLUTION.-In 10 years, at 7%, compound interest, $500 will amount to $983.5755 (378). An annuity of $1 for 10 yr., at 7%, produces $13.81645 (see TABLE). It will, therefore, take an annuity, or yearly payment of as many dollars as $13.81645 is contained times in $983.5755 to yield $983.5955 in 10 yr. $983.5755 + $13.81645 = 71.189. The annual payment is, therefore, $71.189, Ans. 6. What would have been the annual payments at 6%? Ans. $67.934. OUTLINE OF MENSURATION. OF TERMS. AREAS. { 533. A Surface. 534. An Area. 535. A Triangle. 536. The Base. 537. The Altitude. 538. The Diagonal. 539. To find area of any Parallelogram. 532. MENSURATION. OF SURFACES. LINES. 546. To find Circumference. 549, A Solid. 550. Convex Surface. 551. Entire Surface. OF SOLIDS. SURFACE. 552. To find surface of Prism. 553. To find surface of Pyramid. 554. To find surface of Frustumт. 555. A Cylinder. 557. A Cone. 559. A Frustum. 561. A Sphere. VOLUME. 556. To find Surface. 558. To find Surface. 560. To find Surface. 562. To find Surface. 563. To find volume of Prism or Cylin der.. 564. To find volume of Pyramid or Cone. CHAPTER IX. MENSURATION 532. Mensuration treats of the measurement. of magnitudes. In reference to the kinds of magnitude, mensuration is of four kinds, namely, Lines, Angles, Surfaces, and Solids. A Straight Line is one that does not change its direction. Lines and Angles, and Surfaces and Solids in part, have been somewhat fully discussed in Arts. 202-224, and also 247-258. It is proposed in this Chapter simply to complete the subject of mensuration, so far as may be proper in a work of this kind, by adding some Definitions, Explanations, and Rules, which the pupil is now prepared to comprehend, with reference to surfaces and solids. SQUARE MEASURE. 533. A Surface is that which has length and breadth without thickness (204). 534. An Area is a definite amount of surface. 535. A Triangle is a plane surface bounded by three straight lines. (Fig. 1.) Triangles are equilateral, isosceles, and scalene, when the three sides are equal, when two sides are equal, and when no two sides are equal, respectively. They are also right-angled, obtuse-angled, acute-angled, and equiangular, when they have one right-angle, when they have one obtuseangle, when all the angles are acute, and when all the angles are equal, respectively. |