NOTES AND QUERIES.

Otago Witness, Issue 1296, 30 September 1876, Page 13

NOTES AND QUERIES.

SIGMA, Dunedin, writes : — The mathematical questions published in your jpaper have, to my knowledge, been productive of considerable good by causing a number of youths to take an intelligent interest in a subject which, after leaving school, ifcis too often the custom wholly to neglect. The benefit they derive will, however, be marred if incorrect solutions are passed over without notice.' I have observed more than one wrong answer in the Witness within the last few weeks, but until to-day I have not seen any solution in which both the propounder of the question and several correspondents were alike at fault. Question jfo. 511, however, has been quite misunderatood by both. The question was : — A person hired a carriage to go a distance of 40 miles for £3 6s. At the eighth milestone he was joined by A., at the fifteenth by B. and C, and at the thirtieth by D. What proportion of the hire must "The solution appearing in to-day's Witness is as follows :— 40 32 25 25 10 132 :66 t : 40 :£1 First person's share 132 :66 : : 32 : 16/ A's share 132 :66 : : 25 : 12/6 B and C's share 132 :66 : : 10 : 5/ D's share." This is a correct answer to an entirely different question, but not to No. 511. If the vehicle had been hired at sixpence per mile per passenger, the above are the sums the travellers would respectively have had to payj and each person's liability would have been unchanged, whether he had travelled alone or in company with one or any other number of persons. But in question 511 the expense to each traveller depends not only on the distance travelled, but also on the number of persons with •whom he travelled ; and the fallacy of the solution given, consists in the latter condition not having been takeninto account. There were four stages of the journey ; each travelledunder different conditions, and each person's liability varied accordingly. The hirer travelled the first eight miles, or onefifth, alone, and must bear the whole proportionate cost. The second stage of seven miles, or seven-fortieths, he travelled with A., with whom lie must share the expense. The third of fifteen miles, or three-eighths, with A. B. and 0., each of whom must contribute one-fourth of the cost ; and for the last stage of ten miles, or one-fourth, the hirer and A. B. C. and T). were together, and must divide the cost equally between them. The hirer's proportion of the expense is therefore : —

There is some excuse for the error into which your correspondents have fallen, as some books on Arithmetic give the same I wrong formula that they have adopted. i EUREKA, Chi'ifltcUurch, writes*— ln answer*

to " E. R. G.," I observe that my solution of question 481 is incorrect. I regret that I am unable at the present time to remember my authority for the said question. I observed the distinction when roading, and am nearly certain that the authority is the "British Encyclopedia." I took a note at the time, the correctness of which I can vouch for; but will endeavour to afford "I.E. G." the information he asks. With the new era of the Riddler, which will commence on the 7th prox. ; may I ask if propounders of problems will be allowed to count answers amongst those accredited as correct ? Also, if there will be two different classes for the problems ? Whether arithmetical questions are to be reckoned a distinct branch from puzzles, &c. ? And again, in the event of an arbitrary question which admits two answers, are all thoso who comply with the stipulations, and whose answers are strictly correct to be accredited ? The interest involved in the foregoing questions may bo evident to your readers from the following : — Referring to question 511 ; with all due deference to Mr Smith, and those who answered correctly, suppose I had sent in the following answer, which at the risk of contributing another "amusing little episode" I incline to as correct. I submit this case without egotism ' or scepticism, but merely to prove how a different interpretation of the question may act prejudicially to those who intend competing for your awards. I would consider it a favour if you would find room for the whole of this letter in your valuable paper, though I must apologise for its unusual length. The person (A) hired the carriage for 40 miles for 665, therefore 66s for 40 miles =1*65, original rate per mile. He went 8 miles at ls*6s per mile = 135% when B joined him. Debit A with his amount (see under.) Subtract 13 2 from 665, leaving 52 "8 amount, they are jointly responsible for on account of remaining 32 miles. Therefore 525-8 for 32 miles == T65 x 7 (being number of miles the two travelled alone) = 11*55, amount they should pay for that distance. Debit their share to each (see under.) 52*8 —11 -55 = 41*25, the amount due for the remairh<j 25 rriles, when C and D joined. Therefore 41*25 for 25 miles x 15 = 24*75, amount due among four of them. Charge each their share (see under.) 41*25 — 24*75 = 16*50, amount due when E joins them for the last 10 miles. Therefore IGa'lO for 10 miles = 165, amount due among sof them. Debit to each share (see under)— and the answer should be (clear of decimals)— A 28 S U-20ths, C 9 5 17-20ths, E3 3 12-20tha B 15 3 3.2oths, D 9 5 17-20ths,

Total 66a. P. M., St. Bathans, writes : — Many thanks to Mr A. Beverley for the answer to my question. I hope Ihe will be kind enough to show how he found it j that is, to give a solution. ANSWERS. SUBSCRIBER, Rugged Ridge,— Mr Justice Chapman was on the Bench when Captain Jarvoy was condemned. R. N., Albertown. — In giving credit for late answers to riddles, we make an exception in favour of those who reside in the Wakatip district and Canterbury. PUZZLED SCHOOLBOY.— J. Stalker, Rangiora, and R. Norman, Albertown, send correct answers.

B. C. D. E. 13-2000 57750 6-1875 3-3000 57750 6-1875 3-30C0 6-1875 3-3000 6-1875 3-3000 3-8 28*4625 15-2625 9-4875 9-4875 3.3

he whole of one-fifth = 13 2 8-20 [alf of seven-fortieths = 5 9 9-20 Ine-fourth of three-eightks= 6 2 5-20 Ino-fifth. of one-fourth = 3 3 12-20 £18 5 11-20 l's share is the three last items = 15 3 3-20 t's „ two „ = 9 5 17-20 "a „ „ >, = 9 5 17-20 >'s „ last „ = 3 3 12-20 £1 17 6 9-20