£ 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. Example.--If the interest on £10 is 15s., what is the interest on £20? As £10 : £20 :: 15s. : _x_ 15 ____ 10)300 ---- 30 Ans. 30s. 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.