COMPOUND INTEREST In all states, both law and business usage are opposed to the collection of compound interest. There are many problems, however, especially in connection with insurance, annuities, and reserve funds, where this computation is necessary. The principal bears interest, the interest on the principal at the end of each year bears interest, and all other interest at the end of each year bears interest. See p. 216. 118. What is the amount at compound interest of $1 for 3 yr. at 6%. During the first year $1 is on interest; it amounts to $ 1.06. During the second year $1.06 is on interest; since $1 amounts to $1.06, $1.06 will amount to 1.06 times $ 1.06, or $ (1.06)2. During the third year $ (1.06)2 is on interest; since $1 amounts to $ 1.06, $(1.06)2 will amount to (1.06)2 times $ 1.06, or $ (1.06)3, or $1.191016. 119. What is the amount at compound interest of $1 for 4 yr. at 6%? Explain as in example 118. AMOUNT OF $1 AT COMPOUND INTEREST YEARS 5% 6% 10% 3 = (1.05)3 = 1.157625 (1.06) 1.191016 (1.10)8 1.331000 = 4 5 120. Explain as in example 118 and fill out the blanks in the above diagram. 121. Prove that the amount of $1 for n years at r% is $(1+r)". 122. What is the amount at compound interest of $1 for 2000 yr. at 6%? Express the answer as suggested in Ex. 121. How many decimal places are there in the exact answer? 123. Using the table just prepared, find the compound interest of $129.36 for 3 yr. 3 mo. 3 da. @ 6%. $(1.06)3, or $1.191016, amt. $ 1, 3 yr. 154.07, amt. $129.36 27.10 int. whole time $1 in 3 yr. at 6% will amount to $ (1.06)3, or $1.191016; $129.36 will amount to 129.36 times $1.191016, or $154.07. At the end of 3 yr., $154.07 is on interest for 3 mo. and 3 da. $1 in this time will amount to $1.0155; $154.07 will amount to 154.07 times $1.0155, or $156.46. 124. In solving example 123, would it not be better to multiply $1.0155 by 1.191016 to find the amount of $1 for the whole time, and to multiply this product by 129.36? Ans. No; because the product of the first two is a very long decimal. 125. Without the table, find the amount of $2630 for 3 yr. 4 mo. 4 da. at 4%, compound interest. Using the table on p. 236, find the: 126. Compound int. of $365.25 for 4 yr. 5 mo. 5 da. at 5%. 127. Compound int. of $762.28 for 3 yr. 3 mo. 3 da. at 10%. 128. Compound int. of $625.50 for 5 yr. 6 mo. 4 da. at 9%. Without the table, find the: 129. Compound int. of $72.38 for 3 yr. 7 mo. 2 da. at 7%. 130. Compound int. of $65.96 for 3 yr. 2 mo. 18 da. at 8%. 131. Compound int. of $ 85.50 for 1 yr. 9 mo. 16 da. at 6%. Using the table, find the : 132. Compound amt. of $455 for 10 yr. 6 mo. 10 da. at 24%. 133. Compound amt. of $366 for 15 yr. 9 mo. 20 da. at 44%. 134. Compound amt. of $930 for 18 yr. 4 mo. 4 da. at 34%. AMOUNT OF $1 AT COMPOUND INTEREST FROM 1 TO 20 YEARS 1234 5 7 8 9 1.0200000 1.0250000 1.0300000 1.0350000 1.0400000 1.0450000 1.0404000 1.0506250 1.0609000 1.0712250 1.0816000 1.0926250 1.0612080 1.0768906 1.0927270 1.1087178 1.1248640 1.1411661 1.0824321 1.1038128 1.1255088 1.1475230 1.1698585 1.1925186 1.1040808 1.1314082 1.1592740 1.1876863 1.2166529 1.2461819 1.1261624 1.1596934 1.1940523 1.2292553 1.2653190 1.3022601 1.1486856 1.1886857 1.2298738 1.2722792 1.3159317 1.3608618 1.1716593 1.2184029 1.2667700 1.3168090 1.3685690 1.4221006 1.1950925 1.2488629 1.3047731 1.3628973 1.4233118 1.4860951 1.2189944 1.2800845 1.3439163 1.4105987 1.4802442 1.5529694 1.2433743 1.3120866 1.3842338 1.4599697 1.5394540 1.6228530 1.2682417 1.3448888 1.4257608 1.5110686 1.6010322 1.6958814 1.2936066 1.3785110 1.4685337 1.5639560 1.6650735 1.7721961 1.3194787 1.4129738 1.5125897 1.6186945 1.7316764 1.8519449 1.3458683 1.4482981 1.5579674 1.6753488 1.8009435 1.9352824 16 1.3727857 1.4845056 1.6047064 1.7339860 1.8729812 2.0223701 17 1.4002414 1.5216182 1.6528476 1.7946755 1.9479005 2.1133768 18 1.4282462 1.5596587 1.7024330 1.8574892 2.0258165 2.2084787 19 1.4568111 1.5986501 1.7535060 1.9225013 2.1068491 2.3078603 20 1.4859474 1.6386164 1.8061112 1.9897888 2.1911231 2.4117140 10 11 12 13 14 15 YEARS 5% 123456♪ 7 7% 8% 9% 10% 6% 1.0500000 1.0600000 1.0700000 1.0800000 1.0900000 1.1000000 1.1025000 1.1236000 1.1449000 1.1664000 1.1881000 1.2100000 1.1576250 1.1910160 1.2250430 1.2597120 1.2950290 1.3310000 1.2155063 1.2624770 1.3107960 1.3604890 1.4115816 1.4641000 1.2762816 1.3382256 1.4025517 1.4693281 1.5386240 1.6105100 1.3400956 1.4185191 1.50073041.5868743 | 1.6771001 1.7715610 1.4071004 1.5036303 1.6057815 1.7138243 1.8280391 1.9487171 1.4774554 1.5938481 1.7181862 1.8509302 1.9925626 2.1435888 1.5513282 1.6894790 1.8384592 1.9990046 2.1718933 2.3579477 10 1.6288946 1.7908477 1.9671514 2.1589250 2.3673637 2.5937425 1.7103394 1.8982986 2.1048520 2.3316390 2.5804264 2.8531167 1.7958563 2.0121965 2.2521916 2.5181701 2.8126648 3.1384284 1.8856491 2.1329283 2.4098450 2.7196237 3.0658046 3.4522712 14 1.9799316 2.2609040 2.5785342 2.9371936 3.3417270 3.7974983 15 2.0789282 2.3965582 2.7590315 3.1721691 3.6424825 4.1772482 16 2.1828746 2.5403517 2.9521638 3.4259426 3.9703059 4.5949730 17 2.2920183 2.6927728 3.1588152 3.7000181 4.3276334 | 5.0544703 2.4066192 2.8543392 3.3799323 3.9960195 4.7171204 5.5599173 2.5269502 3.0255995 3.6165275 4.3157011 5.1416613 6.1159090 2.6532977 3.2071355 3.8696845 4.6609571 5.6044108 | 6.7275000 11 12 13 18 19 20 From the table, the amount for any time may be found. 135. What is the amount of $100 for 90 years at 6% compound interest? $3.2071355, amt. $1, 20 yr. 3.2071355 $10.2857179, amt. $1, 40 yr. 10.2857179 $105.7959927, amt. $1, 80 yr. 1.7908477 189.4645099, amt. $1, 90 yr. $18946.45, amt. $100 $3.20 is on interest at the end of 20 yr. During the next 20 yr., $1 will amount to $3.20; $3.20+ will amount to 3.20+ times $3.20, or $10.28. $10.28 is on interest at the end of 40 yr. During the next 40 yr., $1 will amount to $10.28; etc. The pupil should finish the explanation. If interest is payable at intervals of less than a year, the example should be reduced to an equivalent example, interest payable annually. 136. What is the amount at compound interest, of $100 for 3 yr. 2 mo. 4 da. at 8%, interest payable quarterly? 3 yr. 2 mo. 4 da. 4 12 yr. 8 mo. 16 da. Paying quarterly is paying each time for a period one fourth of a year. The interest for a period one fourth of a year @ 8% is the same as the interest for a whole year @ 2%; . the interest @ 8%, payable quarterly, is the same as the interest for 4 times 3 yr. 2 mo. 4 da., or for 12 yr. 8 mo. 16 da. @ 2%, payable annually. The pupil should finish the explanation. 137. Show that the amount for 5 years at 12%, payable semiannually, is the same as the amount for 2 times 5 years, at of 12% payable annually. 138. Find the amount at compound interest, of $100 for 25 yr. 5 mo. 7 da. at 4%. 139. Find the amount at compound interest, of $100 for 8 yr. 3 mo. 2 da. at 12%, interest payable 3 times a year. INVOLUTION AND EVOLUTION INVOLUTION-TERMS AND RELATIONS 8 = 2 × 2 × 2, or 8 = 23. If the product is wanting, this becomes what = 23? It means "what is the product when 2 is used 3 times as a factor?" See p. 146. Involution is the process of finding the product when the same number is used several times as a factor. The result is the power. 1. Read: 52; 5; 5; 5*. What is the product when 2 is used 3 times as a factor? Ans. 8; 2 x 2 x 2 = 8. 8 is the third power of 2. Ans. 52, 5 square, or 5 to the second power; 58, 5 cube, or 5 to the third power; 54, 5 to the fourth power; 5, 5 to the xth power. 2. Write the squares of the integers from 13 to 25, and memorize the results. 3. Write the cubes of the integers from 1 to 10, and memorize the results. 4. Declare the results rapidly: 162; 252; 132; 172; 212; 222; 242; 142; 152; 182; 192; 202; 232; 93; 73; 23; 43; 63; 33; 53; 83. To raise a factor to any power, write the base, and over it, the product of the exponent by the number denoting the required power. Illustration: (43)2 = 46; (43)2 = 43 × 43 = (4 × 4 × 4) × (4 × 4 × 4) = 46. 5. Find the value of: (2); (24); (32); (3')2. 6. The square of 12 is 144; what is its 6th power? 7. The cube of 12 is 1728; what is its 6th power? Ans. 1443. Ans. 17282. |