{"id":8920,"date":"2023-11-22T21:52:15","date_gmt":"2023-11-22T12:52:15","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?p=8920"},"modified":"2023-11-25T20:01:44","modified_gmt":"2023-11-25T11:01:44","slug":"persian-folk-method-of-figuring-interest","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/persian-folk-method-of-figuring-interest\/","title":{"rendered":"\uc774\ub780\uc758 \uc804\ud1b5 \uc6d4\ubcc4 \uc0c1\ud658\uae08 \uacc4\uc0b0 \ubc29\ubc95"},"content":{"rendered":"<h3>\uc774\ub780\uc758 \uc804\ud1b5 \uc6d4\ubcc4 \uc0c1\ud658\uae08 \uacc4\uc0b0 \ubc29\ubc95<\/h3>\n<p>\uae00\uc4f4\uc774: \ud398\uc774\ub9cc \ubc00\ub780\ud30c(Peyman Milanfar)<br \/>\n\uc62e\uae34\uc774: \uc774\uc2ac\ube44(designeralice\uff20daum.net)<\/p>\n<p>\ucd5c\uadfc\uc5d0 \ub098\ub294 \ub300\ucd9c \uae08\uc561\uc758 \uc6d4\ubcc4 \uc0c1\ud658\uae08\uc744 \ucd94\uc815\ud558\ub294 \ub9e4\uc6b0 \ube60\ub974\uace0 \ud6a8\uacfc\uc801\uc778 \ubc29\ubc95\uc744 \ubc30\uc6e0\ub2e4. \uc774 \ubc29\ubc95\uc740 \uc544\ubc84\uc9c0\uaed8\uc11c \uac00\ub974\uccd0 \uc8fc\uc168\ub294\ub370, \uc544\ubc84\uc9c0\uaed8\uc11c\ub294 19\uc138\uae30 \uc774\ub780\uc5d0\uc11c \ubb34\uc5ed\uc744 \ud558\uc168\ub358 \ud560\uc544\ubc84\uc9c0\ub85c\ubd80\ud130 \ubc30\uc6b0\uc168\ub2e4\uace0 \ud55c\ub2e4. \uc774 \uacf5\uc2dd\uc774 \ucc98\uc74c \ub9cc\ub4e4\uc5b4\uc9c4 \uae30\uc6d0\uc740 \ubbf8\uc2a4\ud14c\ub9ac\uc774\uc9c0\ub9cc, \uc774 \ubc29\ubc95\uc740 \uc774\ub780\uc744 \ud3ec\ud568\ud55c \ub9ce\uc740 \uc9c0\uc5ed\uc5d0\uc11c \uc0ac\uc6a9\ub418\uace0 \uc788\ub2e4.<\/p>\n<p>\uc544\ubc84\uc9c0\uaed8\uc11c \uc54c\ub824\uc8fc\uc2e0 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<p>\\[(\\text{\uc6d4\ubcc4 \uc0c1\ud658\uae08}) = \\frac{1}{(\\text{\uc6d4\uc218})} [(\\text{\uc6d0\uae08}) + (\\text{\uc774\uc790})].\\]<\/p>\n<p>\uc5ec\uae30\uc11c \uc774\uc790\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ud55c\ub2e4.<\/p>\n<p>\\[(\\text{\uc774\uc790}) = \\frac{1}{2} (\\text{\uc6d0\uae08}) \\times (\\text{\uc5f0\uc218}) \\times (\\text{\uc5f0\uac04\uc774\uc790\uc728}).\\]<\/p>\n<p>\uc6d4\ubcc4 \uc0c1\ud658\uae08\uc758 \ucc38\uac12\uc744 \\(C\\)\ub77c\uace0 \ub450\uba74, \uc5ec\ub7ec \uae08\uc735 \uad50\uacfc\uc11c\uc5d0\uc11c \\(C\\)\ub97c \uad6c\ud558\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uacf5\uc2dd\uc744 \ucc3e\uc744 \uc218 \uc788\ub2e4.<\/p>\n<p>\\[C = \\frac{r(1+r)^N P}{ (1+r)^N &#8211; 1 }.\\tag{1}\\]<\/p>\n<p>\uc5ec\uae30\uc11c \\(r\\)\uc740 \uc6d4 \uc774\uc790\uc728(\uc5f0\uac04 \uc774\uc790\uc728\uc758 1\/12), \\(N\\)\uc740 \ucd1d \uc6d4 \uc218, \\(P\\)\ub294 \uc6d0\uae08\uc774\ub2e4. \uc774 \ud45c\uae30\ubc95\uc744 \uc0ac\uc6a9\ud558\uba74, \uc55e\uc11c \uc18c\uac1c\ud55c \uc0c1\ud658\uae08\uc758 \uc804\ud1b5\uc801\uc778 \uadfc\uc0bf\uac12 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<\/p>\n<p>\\[ C_f = \\frac{1}{N} \\left( P + \\frac{1}{2} PNr \\right) .\\tag{2}\\]<\/p>\n<p>\uaf64 \ub9ce\uc740 \uacbd\uc6b0 \\(C_f\\)\ub294 \\(C\\)\uc5d0 \ub300\ud574 \ub180\ub78d\ub3c4\ub85d \uc88b\uc740 \uadfc\uc0ac\uac12\uc774\ub2e4. \uc608\ub97c \ub4e4\uc5b4, 4\ub144\uc9dc\ub9ac \ub300\ucd9c\uae08 1\ubc31\ub9cc \uc6d0\uc5d0 \ub300\ud574 \uc5f0\uac04 \uc774\uc790\uc728\uc744 7%\ub85c, \uc6d4\ubcc4 \ubcf5\ub9ac\ub85c \uacc4\uc0b0\ud560 \ub54c, \uc815\ud655\ud55c \uc6d4\ubcc4 \uc0c1\ud658\uae08\uc740 23,946\uc6d0\uc774\uba70, \uc804\ud1b5\uc801\uc778 \uadfc\uc0bf\uac12\uc740 23,750\uc6d0\uc774\ub2e4.<\/p>\n<p>\uc774\uc640 \uac19\uc740 \uacf5\uc2dd (2)\uac00 (1)\uc758 \uadfc\uc0bf\uac12 \uacf5\uc2dd\uc73c\ub85c\uc11c \uc798 \uc791\ub3d9\ud558\ub294 \uc774\uc720\ub97c \uc0b4\ud3b4\ubcf4\uc790. \\(C\\)\ub97c \\(r\\)\uc758 \ud568\uc218\ub85c \uac04\uc8fc\ud558\uace0, \ub2e4\ub978 \ubaa8\ub4e0 \uac12\uc740 \uc0c1\uc218\ub77c\uace0 \ud558\uc790. (\uc2dd (1)\uc5d0\uc11c \\(r=0\\)\uc77c \ub54c\uc758 \ud568\uc22b\uac12\uc740 \\(r\\rightarrow 0\\)\uc77c \ub54c\uc758 \uadf9\ud55c\uac12\uc778 \\(P\/N\\)\ub85c \ub300\uccb4\ub418\uba70, \uc774 \uac12\uc740 \uc774\uc790\uc728\uc774 \\(0\\)\uc77c \ub54c\uc758 \uade0\ub4f1\uc0c1\ud658\uae08\uacfc \uc77c\uce58\ud55c\ub2e4.) \ud568\uc218 \\(C(r)\\)\uc758 1\ucc28 \ub9e5\ud074\ub77c\ub9b0 \ub2e4\ud56d\uc2dd(\uc911\uc2ec\uc774 \\(0\\)\uc778 \ud14c\uc77c\ub7ec \ub2e4\ud56d\uc2dd)\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<p>\\[ C(r) \\approx \\frac{1}{N} \\left( P + \\frac{1}{2} P(N+1)r \\right) .\\tag{3}\\]<\/p>\n<p>\uc5ec\uae30\uc11c \uc6b0\ubcc0\uc740 \\(C_f\\)\uc758 \uc815\uc758\uc640 \ub9e4\uc6b0 \uc720\uc0ac\ud558\ub2e4. \\(P\\)\uac00 \uace0\uc815\ub41c \uc0c1\uc218\uc774\uace0 \\(r\\)\uc774 \ucda9\ubd84\ud788 \\(0\\)\uc5d0 \uac00\uae4c\uc6b0\uba70 \\(N\\)\uc774 \ucda9\ubd84\ud788 \ud074 \uacbd\uc6b0, \uc2dd (2)\uc640 (3) \uc0ac\uc774\uc758 \ucc28\uc774\ub294 \\(0\\)\uc5d0 \uac00\uae4c\uc6cc \uc9c4\ub2e4.<\/p>\n<p><!--\n\n\n<p>\u203b \uace0\ub4f1\ud559\uc0dd\uc744 \uc704\ud55c \uc5ed\uc790 \uc8fc: \uc774\uc790\ub97c \ubcf5\ub9ac\ub85c \uacc4\uc0b0\ud560 \ub54c, \uc6d0\ub798 \uc5f0 \uc774\uc790\ub294 \uc6d4 \uc774\uc790\uc758 12\ubc30\uac00 \uc544\ub2c8\ub2e4. \uadf8\ub7ec\ub098 \uc608\uae08\uc774\ub098 \ub300\ucd9c\uae08\uc758 \uc774\uc790\ub97c \uacc4\uc0b0\ud560 \ub54c\ub294 \"\uc6d4\uc774\uc728\"\uc744 \"\uc5f0\uc774\uc728\"\uc758 1\/2\ub85c \"\uc815\uc758\"\ud55c\ub2e4.<\/p>\n\n\n--><\/p>\n<p>\uc6d0\ubb38: <a href=\"\/blog\/wp-content\/uploads\/2023\/11\/A-Persian-Folk-Method-of-Figuring-Interest.pdf\">A Persian Folk Method of Figuring Interest.pdf<\/a><\/p>\n<p>\ucd9c\ucc98: Peyman Milanfar (1996) A Persian Folk Method of Figuring Interest,<br \/>\nMathematics Magazine, 69:5, 376-376, DOI: 10.1080\/0025570X.1996.11996479<\/p>\n<p><!--\n\nA Persian Folk Method of Figuring Interest\n\nPeyman Milanfar\n\nI recently learned a very quick and effective way of estimating monthly payments on a loan. My father showed me the method, having learned it himself from my grandfather, who was a merchant in nineteenth century Iran. While its origins remain a mystery, the method is still in use among merchants all around Iran, and perhaps elsewhere.\n\nMy father used the formula:\n\n\\[(Monthly payment) = \\frac{1}{(Number of months)} [(Principal) + (Interest)]\\];\n\nhe calculated the interest as\n\n\\[(Interest) = \\frac{1}{2} (Principal) \\times (Number of years) \\times (Annual interest rate)\\].\n\nThe exact fonnula, assuming interest accrued monthly, can be found in any basic finance textbook:\n\n\\[C = \\frac{r(1+r)^N P}{ (1+r)^N - 1 },\\]\n\nwhere \\(C\\) is the (exact) monthly payment, \\(r\\) is the monthly interest rate (1\/12 the annual interest rate), \\(N\\) is the total number of months, and \\(P\\) is the principal. With this notation, the folk formula becomes\n\\[ C_f = \\frac{1}{N} \\left( P + \\frac{1}{2} PNr \\right) .\\]\n\nIn many cases, \\(C_f\\) is a surprisingly good approximation to \\(C\\). As an example, for a 4-year auto loan of $10,000 at an annual rate of 7% compounded monthly, the exact formula gives monthly payments of $239.46 while the folk estimate gives $237.50.\n\nTo see why the approximation works, we regard \\(C\\) as a function of \\(r\\), with all other quantities held fixed. (The singularity in (1) at \\(r = 0\\) can be cancelled out.) A straightforward calculation shows that the first order Maclaurin polynomial for \\(C(r)\\) has the form\n\n\\[ C(r) \\approx \\frac{1}{N} \\left( P + \\frac{1}{2} P(N+1)r \\right) ,\\]\n\nwhich closely resembles the definition of \\(C_f\\). For a fixed \\(P\\), when \\(r\\) is sufficiently small and \\(N\\) sufficiently large, the difference between (2) and (3) is small.\n\n--><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\uc774\ub780\uc758 \uc804\ud1b5 \uc6d4\ubcc4 \uc0c1\ud658\uae08 \uacc4\uc0b0 \ubc29\ubc95 \uae00\uc4f4\uc774: \ud398\uc774\ub9cc \ubc00\ub780\ud30c(Peyman Milanfar) \uc62e\uae34\uc774: \uc774\uc2ac\ube44(designeralice\uff20daum.net) \ucd5c\uadfc\uc5d0 \ub098\ub294 \ub300\ucd9c \uae08\uc561\uc758 \uc6d4\ubcc4 \uc0c1\ud658\uae08\uc744 \ucd94\uc815\ud558\ub294 \ub9e4\uc6b0 \ube60\ub974\uace0 \ud6a8\uacfc\uc801\uc778 \ubc29\ubc95\uc744 \ubc30\uc6e0\ub2e4. \uc774 \ubc29\ubc95\uc740 \uc544\ubc84\uc9c0\uaed8\uc11c \uac00\ub974\uccd0 \uc8fc\uc168\ub294\ub370, \uc544\ubc84\uc9c0\uaed8\uc11c\ub294 19\uc138\uae30 \uc774\ub780\uc5d0\uc11c \ubb34\uc5ed\uc744 \ud558\uc168\ub358 \ud560\uc544\ubc84\uc9c0\ub85c\ubd80\ud130 \ubc30\uc6b0\uc168\ub2e4\uace0 \ud55c\ub2e4. \uc774 \uacf5\uc2dd\uc774 \ucc98\uc74c \ub9cc\ub4e4\uc5b4\uc9c4 \uae30\uc6d0\uc740 \ubbf8\uc2a4\ud14c\ub9ac\uc774\uc9c0\ub9cc, \uc774 \ubc29\ubc95\uc740 \uc774\ub780\uc744 \ud3ec\ud568\ud55c \ub9ce\uc740 \uc9c0\uc5ed\uc5d0\uc11c \uc0ac\uc6a9\ub418\uace0 \uc788\ub2e4. \uc544\ubc84\uc9c0\uaed8\uc11c \uc54c\ub824\uc8fc\uc2e0 \uacf5\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4. \\((\\text{\uc6d4\ubcc4 \uc0c1\ud658\uae08}) = \\frac{1}{(\\text{\uc6d4\uc218})} [(\\text{\uc6d0\uae08}) + (\\text{\uc774\uc790})].\\) \uc5ec\uae30\uc11c&hellip;<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[581,360],"class_list":["post-8920","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-taylor-series","tag-360"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8920","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=8920"}],"version-history":[{"count":16,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8920\/revisions"}],"predecessor-version":[{"id":8938,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/8920\/revisions\/8938"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=8920"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=8920"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=8920"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}