majority everywhere, and many even of the few who do know better, will not resist the temptation to reap a small present advantage at the expense of a vastly greater future loss to themselves or descendants, and where new land is so cheap and accessible as it is in this country the temptation is strong to get as much as possible from the land in the shortest time and move to new regions. All the bluff region of the Mississippi valley is in danger of having a damage done that nothing can repair. The harm to navigation will be most severely felt in the future. The same is true of the probable derangement of climate and rain-fall all over the West; the evil will be cumulative. Teachers of Wisconsin, will you not take pains to inform yourselves on this most important matter, and do all you can to sow correct ideas about it in the communities where you teach? Marsh's Man and Nature, of which the Nation says that it is the best book on the subject, is published by Scribner at $3.00. Bryant's Forest Trees, an excellent practical treatise on trees, is published by N. T. Williams, N. Y., office of the Horticulturist. Price $1.50. THE PROBLEM IN INTEREST. RICHLAND CITY, Wis., Aug. 3, 1872. EDITORS OF JOURNAL-Dear Sirs-In the last issue of the JOURNAL I find an algebraic solution for what I consider a nice arithmetical problem in interest. It is the following: A man has (or owes) a note of five hundred dollars, which he wishes to pay in five equal annual installments, at five per cent. per annum. What are the installments? The conditions of the problem show that the first installment will contain less of the principal than either of the others, and we will proceed to divide the principal into parts whose unit shall be the amount of principal contained in the first installment. The first year he pays of the principal, 1. Or each year he will pay the amount he paid the preceding year and the interest on the same, as he will have that amount less to pay interest on. The five hundred dollars then contains as many times the amount of principal paid the first year as the sum of these payments is contained in it; which is 3.281-90.4874 dollars. 500 This equals the amount of principal paid the first year, and as he had the interest on the whole to pay beside, it would make the installment 90.4874+(500 x .05)=115.4874, which is the correct result instead of 115.51+ as rendered in the JOURNAL. Below I submit the full working of the problem and the proof of the same carried out close enough for all practical purposes. 500+25=525 = 115.4874 PROOF. the amount due at the end of the first year. 409.5126 the amount due at beginning of second year. .05 The last payment is a trifle less, owing to the form being a shade too great in the hundredths of cents in the installments. I find that the result given in the JOURNAL will make the last payment thirteen cents less than either of the others, which is contrary to the conditions of the problem. Whole amount of princip'l=5.52563125 times the first payment of principal. The 500 divided by this number gives the payment of principal the first year 5.52563125 | 500.000000000000 | 90.5874(y)=Am't prin. paid 1st yr. 497 3068125 2693187500 2210252500 4829350000 4420505000 4088450000 3867941875 2205081250 2210252500 500 x .05=25.00 amount of interest paid first year; 90.4874+25=115.4874, whole amount of first year's payment, and answer sought. ANOTHER METHOD. The problem may be performed by geometrical progression, after having a thorough understanding of its parts and their relations, asThe first term equals one, 1. To find the sum of the terms, raise the ratio to a power one less than the number of terms: (1.05)'=1.21550625; this multiplied by the first term (1) equals the last term. The last term multiplied by the ratio and divided by the difference between the ratio and the first equals the sum of the terms, as— 1.21550625 x 1.05÷(1.05-1.05)=5.52563125. From this stage it would be worked the same as the other solution. ANOTHER SOLUTION. EDITORS JOURNAL-In the August number of the JOURNAL is a solution of a problem in interest, which, although plain to those who understand Algebra, is very difficult to explain to pupils unacquainted with that branch of mathematics. Similar problems are given in dif ferent arithmetics, and it is desirable that in their solution the wellknown rules and principles of arithmetic, only should be used. Below is an arithmetical solution of the same problem: 1.00 1.1025 1.157625 1.21550625 25 5.52563125J 500.0000000090.487+ 497.3068125 115.487+ 2.693187500 2.210252500 4829350000 4420505000 4088450000 3867941875 220408125 Explanation-Take $1 as the proportional part of the principal to be paid at the end of the first year. It is obvious that when interest is reckoned at the end of the second year, it will be .05 less than at the end of the first year; the proportional part of the principal included in the second payment must be 1.05, for what is lost in interest must be made up in principal. When interest is reckoned at the end of the third year, it will be less than at the end of the first year-by the interest on the sum of the proportional parts of the principal already paid ($2.05)—the reduction in interest will be .1025, and this must be added to $1 to form the proportional. part of the principal paid at the end of the third year. The sum of the proportional parts thus far paid is $3.1525, and when interest is reckoned at the end of the fourth year it will of course be less than at the end of the first year, by the interest on this sum, which is .157625, add this to $1 to form the proportional part of the principal included in the fourth payment. The sum of the proportional parts now paid is $4.310135, and when interest is reckoned at the end of the fourth year it is obvious that it will be less than at the end of the first year, by the interest on this sum, which is .21550625, add to this $1 to form the proportional part of the principal included in the fifth payment. The sum of these proportionals is to each proportional as $500 is to the corresponding parts of the principal in the several payments. The part of the principal included in the first payment is $90.487+; add to this the interest reckoned at the end of the first year, which is $25, and the first payment is found to be $115.487 +. The others of course are the same.-WM. B. MINAGHAN, Chilton. Official Department. CORRESPONDENCE. The following letter is practical in its queries, but illustrates the divigulty" encountered by our German citizens in mactering our pronunciation, orthography and idioms: Samuel Fallows State Supt Dir Sir Clerk and August 12-1872 Director by the last Wee hav a divigulty in oure School district No. 2 of the town of wer im Treasurer and annal Meeting in September wee tac a wott that 9 Monts School shall by taud in the district-6 Monds in English and 3 Monds in German Langvis and 203 Dollars we rest in the District-the ditsher of the English School his School is desmist and the German School gomenses agin in first Monday in September-two of our woters forbitt me to pay any from the puplic Mony to the German ditsher-af i pay he will sue mee-the Clerk and Director heir the German Ditsher and the contract said that tree monds School shall bey ditsht by a a German ditsher in German Langvis as a qualifitt ditsher-our Supt tal me thut thas ditsher is qualifitt in English not in German Langvis-i refyvust to pay the ditsher for his forst two monds and the Clerk and Director will su uy of i nott pay the ditsher and will su me af i pay any of the puplic Mony to the Germann ditsher Now therfor i lik to hav your dicishon watt i shal thu-af i hav the louer to pay the puplic Mony to German Ditsher or not-i lic that you shall sent mee a School Law-ten i get them and all orter Laws-i lic you decishen in this case Yours Respecfuly Answer. We advise our German friend not to pay the German teacher, because a school in German is not contemplated by law. The next letter is of older date and less practical, and we are inclined to think is of Hibernian origin; in both, punctuation is left to the reader: To Samuel fallows St. Supert -- Sir in looking carefully over your State report & hearing the few remarks in relation to progressing education in all its various forms and colours it behooves me to address you simply on my views as to the Town old System,-there is a strong feeling because the general sentiment of the people Reproves it-that is the tax payers consider there is no justice in their County measures-the cry we out to have more Teachers to lessen the tax especially the Summer term-a word for district clerks-their Lamps dont Burn at all-I noticed in the some not getting certified & some has-those that has aint any more entitled to qualifications -thats absurd for Encouragement-Schools are for Encourageing both parent & child—a word on the Constitution-the Teacher [who] taught in my District the winter term could not teach it & was No 1 [meaning 1 on a scale of 10] or No none in all his branches-I presume the same thats the way the country is stifled this Spring-thinking of Justice to Extend-I noticed in my hiring a Teacher the same old tune-poorly qualified-no Constitution at all-as to the balance of Branches poor-this is the way applicants ought to be ajusted-to give most according to credit-Justice to all and friendship to none Respectfully yours OFFICIAL OPINIONS. Prepared by the Assistant Superintendent. Q. Is it the legal duty of the director to approve the treasurer's bond, the clerk having already done so, and the security being sufficient? |