Number 7
The number 7 has always been considered the most sacred of all our figures. Its prominence in the Scriptures is very remarkable, from Genesis—where we read that the seventh day was consecrated as a day of rest and repose—to Revelations—where we find the seven churches of Asia; seven golden candlesticks; the book with seven seals; the seven angels with seven trumpets; seven kings; seven thunders; seven plagues, &c., &c., its frequent occurrence is most striking.
The Ancients paid great respect to the seven mouths of the Nile. The seven rivers of Vedic India; seven wonders of the world; seven precious stones; seven notes of music; seven colours of the rainbow, &c., &c. The “Lampads seven that watch the Throne of Heaven” led the Chaldeans to esteem the unit 7 as the holiest of all numbers, thereupon they established the week of seven days, and built their temples in seven stages. The temples and palaces of Burma and China are seven-roofed.
In modern times this number has kept up its reputation. Shakespeare paid special regard to it; the “seven ages” and every multiple of it is supposed to be a critical or important period in one’s life.
A modern philosopher as follows apportions—
Man’s Full Extreme.
| 7 years in childhood, sport and play, | (7) |
| 7 years in school from day to day, | (14) |
| 7 years at trade or college life, | (21) |
| 7 years to find a place and wife, | (28) |
| 7 years to pleasure’s follies given, | (35) |
| 7 years to business hardly driven, | (42) |
| 7 years for some wild-goose chase, | (49) |
| 7 years for wealth, a bootless race, | (56) |
| 7 years of hoarding for your heir, | (63) |
| 7 years in weakness spent and care, | (70) |
| And then you die and go—you know not | where. |
Very many superstitious and curious ideas have been and still are connected with all our figures. For those interested in this subject see page 146—“How To Become Quick At Figures” (Student’s Edition).
“What’s the difference,” asked a teacher in arithmetic, “between one yard and two yards?” “A fence,” said Tommy Yates. Then Tommy sat on the ruler 14 times.
179. What relation is a woman to me who is my mother’s only child’s wife’s daughter?
THE ADVANTAGES OF SKILFUL BOOK-KEEPING.
If a merchant wishes to get pretty deeply in debt, and then get rid of his liabilities by bankruptcy—if, in fact, he proposes to himself to go systematically into the swindling business, and engage in wholesale pecuniary transactions without a shilling of his own, the first thing he should take care to learn would be the whole art of book-keeping.
From what may occasionally be seen of the reports of the proceedings in bankruptcy, it is found that well kept books are regarded as quite a test of honesty, and though assets may have disappeared or never have existed, though large liabilities may have been incurred without any prospect of payment, the bankrupt will be complimented on the straight look of his dealings, if he has shown himself a good book-keeper.
To common apprehension it would seem that well kept books would only help to show a reckless trader the ruinous result of his proceedings, and that while the man without books might flatter himself that all would come out right at last, the man with exact accounts would only get into hot water with his eyes open. If a man may trade on the capital of others without any of his own, and get excused on the ground that he has kept his books correctly, it is difficult to see why a thief who steals purses, &c., may not plead in mitigation of punishment that he has carefully booked the whole of his transactions.
It would be interesting to know the effect of producing a ledger on a trial for felony, as well as curious to observe whether a burglar would be leniently dealt with on the ground that his house-breaking accounts gave proof of his experience in the science of “double-entry.”
Therefore it would be well for those interested to procure copies of “Re Accounts” and “Advanced Thought on Accounts.”
THE FIRM HE REPRESENTED.
A commercial traveller handed a merchant upon whom he had called a portrait of his sweetheart in mistake for his business card, saying that he represented that establishment. The merchant examined it carefully, remarked that it was a fine establishment, and returned it to the astonished and blushing traveller with the hope that he would soon be admitted into partnership.
180. A man and a boy being paid for certain days’ work, the man received 27s., and the boy, who had been absent 3 days out of the time, received 12s. Had the man, instead of the boy, been absent the 3 days they would both have claimed an equal sum. Find out the wages of each per day.
181. The extremes of an arithmetical series are 21 and 497, and the number of terms is 41. What is the common difference?
182. A wine which contains 7½ per cent. of spirit is frozen, and the ice which contains no spirit being removed the proportion of spirit in the wine is increased by 8¾ per cent. How much water in the shape of ice was removed from 504 gallons of the mixture?
THE SHARP SELECTOR.
183. A selector rented a farm, and agreed to give his landlord two-fifths of the produce, but prior to the time of dividing the corn the selector used 45 bushels. When the general division was made it was proposed to give to the landlord 18 bushels from the heap in lieu of the share of the 45 bushels which the tenant had used, and then to begin and divide the remainder as though none had been used. Would this method have been correct?
A GOOD “AD.”
A member of a certain firm appeared in a law court with a complaint that his partner would sell goods at less than cost price, and he desired to have him restrained. The defendant utterly denied the charge, and the case was adjourned for a fortnight. As the plaintiff went out of court he exclaimed in a tragic tone: “Then the sacrifice must still go on!” and “I’ll be ruined!” The story was noised abroad, and the result was that the shop was besieged by customers every day. There the case ended, for at the end of the fortnight the plaintiff failed to appear in court, having accomplished his purpose—advertisement.
184. I give 3 sovereigns for 2 dozen wine at different rates per dozen, and by selling the cheaper kind at a profit of 15 per cent. and the dearer at a loss of 8 per cent. I obtain a uniform price for both. What did each dozen cost me?
185. I have in my garden a shrub that grows 12 inches every day, but during the night it withers off to half the height that it was at the end of the previous day. How much short of 2 feet will it be at the end of a year?
TIT-FOR-TAT.
186. A farmer puts a 3 lb. stone in a keg of butter worth 11d. a pound. The merchant cheats him out of 1 lb. on the weight, and then does him out of 1s. 11d. on calico, tobacco, and a shovel. Who is ahead, and how much?
187. Trains leave London and Edinburgh (400 miles apart) at the same time and meet after 5 hours; the train which leaves London travels 8 miles an hour faster than that which leaves Edinburgh. At what rate did the former travel, and at what speed must the latter travel after they have met, in order that they both may reach their destinations at the same time?
“GOOD ENOUGH!”
“Will you give me a glass of beer, please?” asked a rather seedy-looking fellow with an old but well-brushed coat and almost too shiny a hat. It was produced by the barmaid, frothing over the edge of the tumbler.
“Thank you,” said the recipient, as he placed it to his lips. Having finished it in a swallow, he smacked his lips and said, “That is very good beer—very! Whose is it?”
“Why, that Perkins’s——”
“Ah! Perkins’s, is it! Well, give us another glass.”
It was done; and holding it up to the light and looking through it, the connoisseur said:—
“’Pon my word, it is grand beer—clear as Madeira! What a fine color! I must have some more of that; give me another glass.”
The glass was filled again, but before putting it to his lips the imbiber said:—
“Whose beer did you say this was?”
“Perkins’s,” emphatically replied the barmaid.
The contents of the glass was exhausted, as also the vocabulary of praise, and it only remained for the appreciative gentleman to say, as he wiped his mouth and went towards the door:—
“Perkins’s beer, is it! I know Perkins very well; I shall see him soon, and will settle with him for three long glasses of his incomparable brew. Good morning.”
A Conspiracy.
188. Three gentlemen are going over a ferry with their three servants, who conspire to rob them if they can get one gentleman to two of them, or two to three, on either side of the ferry. They have a boat that will only carry two at once, and either a gentleman or a servant must bring back the boat each time a cargo of them goes over. How can the gentlemen get over with all their servants so as to avoid an attack?
189. Find two numbers whose product is equal to the difference of their squares, and the sum of their squares equal to the difference of their cubes?
190. Divide 1400 into such parts as shall have the same ratio as the cubes of the first four natural numbers.
This was the tempting notice lately exhibited in the window of a dealer in cheap shirts: “They won’t last long at this price!”
POSTING THE LEDGER.
The well known author of several works on account-keeping, Mr. Yaldwyn, tells a rather good thing which actually occurred in New Zealand some time back. Mr. Yaldwyn was at the time engaged examining the books in one of the offices in a country town, and enquired from one of the clerks standing near if the ledger were posted. The person appealed to answered that “he didn’t know,” whereupon Mr. Y. said that he required it done, and with as little delay as possible. A few minutes later the same individual came rushing in and informed him that the ledger was “posted.” Such a piece of “lightning book-keeping” so surprised Mr. Y. that he further questioned the man, who replied “You said you wanted the ledger posted, and, begorra, I posted it.” It then dawned upon Mr. Yaldwyn that the clerk, who was an Irishman, had actually posted the book in the post office!
THEY MANAGED IT.
191. Billy and Tommy, two aboriginals, killed a kangaroo in the bush, and began quarrelling over the weight of the animal. They had no proper means of weighing it, but, knowing their own weights, Billy 130 lbs. and Tommy 190 lbs., they placed a log of wood across a stump so that it balanced with one on each end. They then exchanged places, and, the lighter man taking the kangaroo on his knees, the log again balanced. What was the weight of the kangaroo?
192. A son asked his father how old he was, and received the following answer: “Your age is now one quarter of mine, but five years ago it was only one-fifth.” How old is the father?
193. Place three sixes together so as to make seven.
THE PASSING TRAINS PUZZLE.
194. If through passenger trains running to and from New York and San Francisco daily start at the same hour from each place (difference of longitude not being considered) and take the same time—seven days—for the trip, how many such trains coming in an opposite direction will a train leaving New York meet before it arrives at San Francisco?
THE SCHOOL-TEACHER “CAUGHT.”
Two of our Public Schools were engaged playing a football match one afternoon. The head master of one of them had generously given the boys a half-holiday; but the gentleman who held the same capacity in the other school, not being an ardent admirer of Australia’s national game, refused to do so. When school assembled in the afternoon, a boy volunteered to ask the master for the desired holiday. When the question was put, he firmly answered, “No, no!” whereupon the bright youth called out: “Hurrah! we have our holiday; two negatives make an affirmative.” The teacher was so pleased at the boy’s sharpness that he dismissed the school right away.
195. A man arrives at the railway station nearest to his home 1½ hours before the time at which he had ordered his carriage to meet him. He sets out at once to walk at the rate of four miles an hour, and, meeting his carriage when it had travelled eight miles, reaches home exactly one hour earlier than he had originally expected. How far was his house from the station, and at what rate was his carriage driven?
“OFF THE TRACK.”
196. A man starts to walk from a town, A, to a town B, a distance by road of 16 miles, at the rate of 4 miles an hour. There is a point C on the road, at which the road to B leads away to the right, and another road at right-angles to this latter goes to the left, “to no place in particular.” The unwary traveller gets on to this left hand road, and is walking for 2¼ hours since he left A, before he finds out his mistake, and he resolves not to go back to the junction, which is five miles away, but makes straight across the bush to B, and strikes it exactly. How long did it take to go from A to B?
GAMBLING.
197. Three friends, A, B, and C, sit down to play cards. As a result of the first game, A lost to each of B and C as much money as they started to play with; the result of the second game B lost similarly to each of A and C; and in the third, C lost similarly to each of A and B;—and they then had 24s. each. What had they each at first?
This Sticks Them Up.
198. A, who is a dealer in horses, sells one to B for £55. B very soon discovers that he does not require the animal, and sells him back to A for £50. Now, A is not long in finding another customer for the horse: he sells it to C for £60. How much money does A make out of this transaction?
This question has been the cause of endless discussion and argument.
It might be as well to state that when A first sold the horse to B he neither made nor lost any money by the deal.
SCRIPTURAL FINANCE.
199. What is the earliest banking transaction mentioned in the Bible? The answer generally given to this is, “The check which Pharaoh received on the banks of the Red Sea, crossed by Moses & Co.” There is still an earlier instance: see if you can find it out.
200. How much tea at 6s. per lb. must be mixed with 12 lbs. at 3s. 8d. per lb. so that the mixture may be worth 4s. 4d. per lb.?
201. Place 17 little sticks—matches, for instance—making six equal squares, as in the margin, then remove five sticks and leave three perfect squares of the same size.
FOR THE JEWELLER.
202. How much gold of 21 and 23 carats must be mixed with 30 oz of 20 carats, so that the mixture may be 22 carats?
LONDON GRAMMAR.
Three cockneys, being out one evening in a dense fog, came up to a building that they thus described. The first said, “There’s a nouse.” “No,” said the second, “It’s a nut.” The third exclaimed “You’re both wrong; it’s a nin!”
203. A draper sold 12 yards of cloth at 20s. per yard, and lost 10 per cent. What was the prime cost?
204. A jockey, on a horse galloping at the rate of 18 miles an hour on the Flemington racecourse, passes in 30 minutes over the diameter and curve of a semi-circle. What area does he enclose by the ride?
205. How many trees 20 feet apart cover an acre?
“Multiplication is vexation,
Division is as bad.
The rule of three, it puzzles me,
And fractions drive me mad.”
MULTIPLY £19 19s. 11¾d. BY £19 19s. 11¾d.
This very old question is continually cropping up, and will continue to do so as long as men are able to reckon. The answer generally given is £399 19s. 2d. and a fraction, and the method of working it out as follows:—
£19 19s. 11¾d. = 19199 farthings.
| 19199 | x | 19199 | = | 368601601 | and so on. |
| 960 | 960 | 921600 |
Many adopt the following method:—
£20 x £20 = £400
| ¼d x ¼d = 1⁄16 | less | £ s d | Ans. |
| £400 0 0 | |||
| 1⁄16 | |||
| £399 19 11 15⁄16 |
It would be possible to adopt other methods, each of which would give a different result.
Properly speaking, this sum cannot be done.
Multiplication is merely a contracted form of addition: it means taking a number or quantity a certain number of times. Every multiplication can be proved by addition. All numbers are abstract or concrete—3 is abstract, £3 is concrete.
Two abstract numbers can be multiplied together—as, 4 times 3 = 12.
Proof: 3
3
3
3
12
One abstract number and one concrete number can be multiplied together—as 2s. multiplied by 3 = 6s.
Proof: 2s.
2s.
2s.
6s.
Two concrete numbers cannot be multiplied together.
In the example just given, 2s. multiplied by 3, we see it simply means to write down 2s. three times, and by addition we discover the answer to be 6s. Suppose the reader lent a friend 2s. on Monday, 2s. on Tuesday, and 2s. on Wednesday, he has lent 2s. three times, making 6s. lent in all.
Now, we will attempt to multiply 2s. by 3s., but it is impossible to comprehend how many times is 3s. times. The answer to 2s. x 3s. usually given is 6s. On the same lines, we multiply 9d. by 10d., and our answer is—90d., that is 7s. 6d.—a greater product than 2s. multiplied by 3s.
Although it is stated that two concrete numbers cannot be multiplied together, it should be borne in mind that we can multiply yards, feet, and inches, by yards, feet, and inches (length by breadth), which will result in square or cubic measure: 12 inches make 1 foot, and 3 feet make one yard, 144 square inches make 1 square foot, &c. 12 pence make 1 shilling, but how many square pence make 1 square shilling?
The argument generally brought forward in favour of the performance of this problem is, that when the Rule of Three is applied to financial questions (such as interests, &c.) money is multiplied by money.
Example.—If the interest on £10 is 15s., what is the interest on £20?
As £10: £20:: 15s.: x
| 10 | 15 | Ans. 30s. |
| )300 | ||
| 30 |
The multiplication in the above is in appearance only, for all we get in the Rule of Three is the ratio between the sums of money and this ratio is an abstract number, and not concrete. On examination we find the ratio between £10 and £20; that the latter is double, or two times as much as the former, and not £2 times more than it.
We extend a general invitation to all our readers who hold a different opinion to multiply three pints of Dewar’s Whisky by 6 quarts of soda-water, but in case they might plead inability to perform this little feat, on conscientious grounds, we will extend the invitation to three cups of tea by six spoonfuls of sugar. And if any of them have a few pounds (say £10) in the Savings Bank we would advise “Don’t add any more deposits, but wait till you have £2, then proceed to the bank and multiply the £10 by the £2, and prove to the teller that you have £20 to your account. Be careful to take no less a sum than £2, or the result might be a little surprising, for if you take only £1, the teller might argue after he has received your sovereign that “ten ones are ten,” and then your £10 would remain the same.”
206. What is the difference between six dozen dozen and half a dozen dozen?
A TELL-TALE TABLE.
There is a good deal of amusement in the following table. It will enable you to tell how old the young ladies are. Ask a young lady to tell you in which column or columns her age is found, add together the figures at the top of the columns in which she says her age is, and you have the secret. Suppose a young lady is 19. You will find that number in the first, second and fifth columns; add the first figures of these columns—1, 2 and 16—and you get the age.
| 1 | 2 | 4 | 8 | 16 | 32 |
| 3 | 3 | 5 | 9 | 17 | 33 |
| 5 | 6 | 6 | 10 | 18 | 34 |
| 7 | 7 | 7 | 11 | 19 | 35 |
| 9 | 10 | 12 | 12 | 20 | 36 |
| 11 | 11 | 13 | 13 | 21 | 37 |
| 13 | 14 | 14 | 14 | 22 | 38 |
| 15 | 15 | 15 | 15 | 23 | 39 |
| 17 | 18 | 20 | 24 | 24 | 40 |
| 19 | 19 | 21 | 25 | 25 | 41 |
| 21 | 22 | 22 | 26 | 26 | 42 |
| 23 | 23 | 23 | 27 | 27 | 43 |
| 25 | 26 | 28 | 28 | 28 | 44 |
| 27 | 27 | 29 | 29 | 29 | 45 |
| 29 | 30 | 30 | 30 | 30 | 46 |
| 31 | 31 | 31 | 31 | 31 | 47 |
| 33 | 34 | 36 | 40 | 48 | 48 |
| 35 | 35 | 37 | 41 | 49 | 49 |
| 37 | 38 | 38 | 42 | 50 | 50 |
| 39 | 39 | 39 | 43 | 51 | 51 |
| 41 | 42 | 44 | 44 | 52 | 52 |
| 43 | 43 | 45 | 45 | 53 | 53 |
| 45 | 46 | 46 | 46 | 54 | 54 |
| 47 | 47 | 47 | 47 | 55 | 55 |
| 49 | 50 | 52 | 56 | 56 | 56 |
| 51 | 51 | 53 | 57 | 57 | 57 |
| 53 | 54 | 54 | 58 | 58 | 58 |
| 55 | 55 | 55 | 59 | 59 | 59 |
| 57 | 58 | 60 | 60 | 60 | 60 |
| 59 | 59 | 61 | 61 | 61 | 61 |
| 61 | 62 | 62 | 62 | 62 | 62 |
| 63 | 63 | 63 | 63 | 63 | 63 |
COIN PUZZLE.
207. Place four florins alternately with four pennies, and in four moves, moving two adjacent coins each time, bring the florins together and the pence together. When finished there must be no spaces between the coins.
208. If 2 be added to the numerator of a certain fraction, it is made equal to one-fifth, whilst if 2 be taken from the denominator it becomes equal to one-sixth. Find the fraction.
EUCLID.—The Famous Forty-Seventh.
“In any right-angled triangle, the square which is described upon the side opposite to the right-angle is equal to the squares described upon the sides which contain the right-angle.”
Here is a simple way of proving this proposition. Although perhaps not exactly scholastic, it is none the less interesting.
Draw an exact square, whose sides measure 7 in.; then divide it into 49 square inches. Having done this, cut the figure in following the big lines as shown by Fig 1. It will be observed that C is a complete square, and that A and B will form a square: but as D is 1 in. short of being a square, it is necessary to cut a square inch and add it on.
Then construct a right-angled triangle as shown by Figure 2.
We then see that the sum of the two small squares is equivalent to the large square.
D contains 9 small squares.
A & B do. 16 do.
25
And as we see that C has 25 small squares, it is thus proved that the sum of the squares upon the sides which contain the right angle are equal to the squares upon the side opposite the right angle.
Q.E.D.
THE GREAT FISH PROBLEM.
209. There is a fish the head of which is 9 in. long, the tail is as long as the head and half the back, and the back is as long as the head and tail together. What is the length of the fish?
210. How may 100 be expressed with four nines?
211. Two shepherds, A and B, meeting on the road, began talking of the number of sheep each had, when A said to B, “Give me one of your sheep, and I will have as many as you.” “Oh, no!” replied B; “give me one of yours, and I will have as many again as you.” How many sheep had each?
A BRICK PUZZLE.
One for Builders, Contractors, &c.
212. Suppose the measurements of a brick to be:—Length, 9 in.; breadth, 4½ in.; depth, 3 in. How many “stretchers, headers and closures” can be cut out of one, and what would be the face area of same?
For the benefit of the uninitiated we might say that
“stretcher” = length of brick x depth
“header” = breadth “
“closure” = half-breadth”
213. A woman has a basket of 150 eggs; for every 1½ goose egg she has 2½ duck eggs and 3½ hen eggs. How many of each had she?
The Great Chess Problem.
214. Move the Knight over all the 64 squares of the chess board so as to successively cover each square and, of course, not enter any square twice. This problem has always proved to be an interesting one. Mathematicians throughout all ages have devoted a good deal of time to it. To chess players it should be especially attractive.
215. If 3 times a certain number be taken from 7 times the same number the remainder will be 8. What is the number?
216. Divide £27 among 3 persons, A, B and C, so that B may have twice as much as A, and C 3 times as much as B.
ANSWER THIS.
217. Suppose it were possible for a man in Sydney to start on Sunday noon, January 1st, and travel westward with the sun, so that it might be in his meridian all the time, he would arrive at Sydney next day at noon, Monday, Jan. 2nd. Now, it was Sunday noon when he started, it was noon with him all the way round, and is Monday noon when he returns. The question is, at what point did it change from Sunday to Monday?
218. Start with 1 and keep on doubling for eight times, thus giving nine numbers, and arrange them in a square that when multiplied together, horizontally, vertically, or diagonally, the product of each row will be the cube of the number which must go in the centre of the square.
The happiest year in a man’s life is 40; for then he can XL.
Bound to Win!
219. A certain gentleman, who was employed in one of our city offices, purchased The Doctrine of Chance, which he studied in his spare time, with the result that he sent in his resignation to the head of the firm in order to try his luck on the racecourse.
At the first meeting he attended, there were only three horses in a race. His brother bookmakers were crying out the odds—
“Two to 1 bar one.”
The odds on this latter horse which was “barred” he discovered to be 6 to 4 on. He determined to give far more liberal odds, and called out—
“Even money, 2 to 1, and 3 to 1.”
How could he give such odds, and yet win £1, no matter which horse wins the race?
AN INCH OF RAIN.
How many people really consider what is contained in the expression? Calculated, it amounts to this:—An acre is equal to 6,272,640 square inches; an inch deep of water on this area will be as many cubic inches of water, which, at 277·274 inches to the gallon, is 22622·5 gallons. The quantity weighs 226,225 lbs. Thus, an “inch of rain” is over 100 tons of water to the acre.
Extract from a small boy’s first essay:—“Man has two hans. One is the rite han an one is the left han. The rite han is fur ritin, and the left han is fur leftin. Both hans at once is fur stummik ake.”
220. Find the side of a square whose area is equal to twice the sum of its sides?
“THE EVIDENCE YOU NOW GIVE, &c., &c.”
221. Smith, Brown, and Jones were witnesses in a law case. The first-named gentleman swore that a certain thing occurred; Brown, on being called, confirmed Smith’s statement, but Jones denied it. They are known to tell the truth as follows:—
Smith, once in 3 times
Brown, " " 5 "
Jones, " " 10 "
What is the probability that the statement is true?
When a man attains the age of 90 years, he may be termed XC-dingly old.