Part 32

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 19199 368601601 ----- x ----- = --------- and so on. 960 960 921600

Many adopt the following method:--

£20 x £20 = £400

£ s d 400 0 0 ¼d x ¼d = 1/16 less 1/16 ---------------- £399 19 11-15/16 Ans.

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.