Amusements in Mathematics
(Part 4)
31.--DOMESTIC ECONOMY.
Young Mrs. Perkins, of Putney, writes to me as follows: "I should be
very glad if you could give me the answer to a little sum that has been
worrying me a good deal lately. Here it is: We have only been married a
short time, and now, at the end of two years from the time when we setup housekeeping, my husband tells me that he finds we have spent a third
of his yearly income in rent, rates, and taxes, one-half in domestic
expenses, and one-ninth in other ways. He has a balance of £190
remaining in the bank. I know this last, because he accidentally left
out his pass-book the other day, and I peeped into it. Don't you think
that a husband ought to give his wife his entire confidence in his money
matters? Well, I do; and--will you believe it?--he has never told me
what his income really is, and I want, very naturally, to find out. Can
you tell me what it is from the figures I have given you?"
Yes; the answer can certainly be given from the figures contained in
Mrs. Perkins's letter. And my readers, if not warned, will be
practically unanimous in declaring the income to be--something absurdly
in excess of the correct answer!
32.--THE EXCURSION TICKET PUZZLE.
When the big flaming placards were exhibited at the little provincial
railway station, announcing that the Great ---- Company would run cheap
excursion trains to London for the Christmas holidays, the inhabitants
of Mudley-cum-Turmits were in quite a flutter of excitement. Half an
hour before the train came in the little booking office was crowded with
country passengers, all bent on visiting their friends in the great
Metropolis. The booking clerk was unaccustomed to dealing with crowds of
such a dimension, and he told me afterwards, while wiping his manly
brow, that what caused him so much trouble was the fact that these
rustics paid their fares in such a lot of small money.
He said that he had enough farthings to supply a West End draper with
change for a week, and a sufficient number of threepenny pieces for the
congregations of three parish churches. "That excursion fare," said he,
"is nineteen shillings and ninepence, and I should like to know in just
how many different ways it is possible for such an amount to be paid inthe current coin of this realm."
Here, then, is a puzzle: In how many different ways may nineteen
shillings and ninepence be paid in our current coin? Remember that the
fourpenny-piece is not now current.
33.--PUZZLE IN REVERSALS.
Most people know that if you take any sum of money in pounds, shillings,
and pence, in which the number of pounds (less than £12) exceeds that of
the pence, reverse it (calling the pounds pence and the pence pounds),
find the difference, then reverse and add this difference, the result is
always £12, 18s. 11d. But if we omit the condition, "less than £12," and
allow nought to represent shillings or pence--(1) What is the lowest
amount to which the rule will not apply? (2) What is the highest amount
to which it will apply? Of course, when reversing such a sum as £14,
15s. 3d. it may be written £3, 16s. 2d., which is the same as £3, 15s.
14d.
34.--THE GROCER AND DRAPER.
A country "grocer and draper" had two rival assistants, who prided
themselves on their rapidity in serving customers. The young man on the
grocery side could weigh up two one-pound parcels of sugar per minute,
while the drapery assistant could cut three one-yard lengths of cloth in
the same time. Their employer, one slack day, set them a race, giving
the grocer a barrel of sugar and telling him to weigh up forty-eight
one-pound parcels of sugar While the draper divided a roll of
forty-eight yards of cloth into yard pieces. The two men were
interrupted together by customers for nine minutes, but the draper was
disturbed seventeen times as long as the grocer. What was the result of
the race?
35.--JUDKINS'S CATTLE.
Hiram B. Judkins, a cattle-dealer of Texas, had five droves of animals,
consisting of oxen, pigs, and sheep, with the same number of animals in
each drove. One morning he sold all that he had to eight dealers. Each
dealer bought the same number of animals, paying seventeen dollars for
each ox, four dollars for each pig, and two dollars for each sheep; and
Hiram received in all three hundred and one dollars. What is the
greatest number of animals he could have had? And how many would there
be of each kind?
36.--BUYING APPLES.
As the purchase of apples in small quantities has always presented
considerable difficulties, I think it well to offer a few remarks on
this subject. We all know the story of the smart boy who, on being told
by the old woman that she was selling her apples at four for threepence,
said: "Let me see! Four for threepence; that's three for twopence, two
for a penny, one for nothing--I'll take _one_!"
There are similar cases of perplexity. For example, a boy once picked up
a penny apple from a stall, but when he learnt that the woman's pears
were the same price he exchanged it, and was about to walk off. "Stop!"
said the woman. "You haven't paid me for the pear!" "No," said the boy,
"of course not. I gave you the apple for it." "But you didn't pay for
the apple!" "Bless the woman! You don't expect me to pay for the apple
and the pear too!" And before the poor creature could get out of the
tangle the boy had disappeared.
Then, again, we have the case of the man who gave a boy sixpence and
promised to repeat the gift as soon as the youngster had made it into
ninepence. Five minutes later the boy returned. "I have made it into
ninepence," he said, at the same time handing his benefactor threepence."How do you make that out?" he was asked. "I bought threepennyworth of
apples." "But that does not make it into ninepence!" "I should rather
think it did," was the boy's reply. "The apple woman has threepence,
hasn't she? Very well, I have threepennyworth of apples, and I have just
given you the other threepence. What's that but ninepence?"
I cite these cases just to show that the small boy really stands in need
of a little instruction in the art of buying apples. So I will give a
simple poser dealing with this branch of commerce.
An old woman had apples of three sizes for sale--one a penny, two a
penny, and three a penny. Of course two of the second size and three of
the third size were respectively equal to one apple of the largest size.
Now, a gentleman who had an equal number of boys and girls gave his
children sevenpence to be spent amongst them all on these apples. The
puzzle is to give each child an equal distribution of apples. How was
the sevenpence spent, and how many children were
