Multiplication

Domestic Multiplication , or Multiplication of miseries. This rule is performed by taking unto oneself a wife for better  or worse ; then, multiplying as usual, and, at the end of fifteen or twenty years, having the young "olive branches" round about our tables.

Multiplication of Laws.—This is a favourite amusement with our modern legislators. It naturally leads to the multiplication of lawyers, whose proper calling is to set people together by the ears, for the multiplication of dissensions. The original type of this order was the plague of locusts.

Multiplication of Money.—This is the most universal case in the whole rule. The multipliers  are the operatives , who are placed at the bottom, instead of the top of the arithmetical scale. They may be ranged, in general, as in the following:—

  • A cotton spinner, 3½d. a-day.
  • Spitalfields weaver, 4½d.
  • Brummagem, 5½d.

These digits are to be worked  from fourteen to sixteen hours a-day at the lowest possible fraction of pay. The product is to be set down in the 3½ per cents. or invested in the first unjust war in which this nation may be engaged; or the whole aggregate of sums may be multiplied by monopoly.

9 × 1 = 9.

Multiplication teaches a short way of adding one number together any number of times. Its sign is a cat o'-nine-tails; its symbol a whipping-post. Since the wonderful powers of the number nine have been publicly discussed, we have had no more shooting at her Majesty, (Heaven preserve her!) which shows the transcendant powers of arithmetical argument. The Egyptian plague of frogs and flies exemplifies this rule. In Modern Rome we have multiplication of fleas. In Modern Babylon we have multiplication of bugs, particularly humbugs. In the West Indies we have multiplication of musquitoes and piccaninies, and in the East, multiplication of oneself, as in the case of Abbas Mirza and his 1000 sons for a body guard.

Multiplication of Fractions

(i) When the multiplier is a whole number. This, as in the case of whole numbers, means that we have to find the sum of a given number of repetitions of the fraction.

Example 1 :

7 9  × 4 means 7 9  + 7 9  + 7 9  + 7 9 i.e.28 9  or 7 × 4 9

Hence, to multiply a fraction by a whole number, simply multiply the numerator by that number.

Since the multiplier thus becomes a factor of the numerator, we cancel any common factors contained in the multiplier and the denominator; and this may be done before we perform the actual multiplication:

Example 2 : Multiply 19 46  by 69.

19 46  × 69 = 19 × 69 46  = 19 × 3 2  (cancelling 23), = 57 2  = 28 1 2  Ans.

It follows that if the multiplier be itself a factor of the denominator, we may, to multiply a fraction by a whole number, divide the denominator by that number.

(ii) When the multiplier is a fraction.

Example : In performing the operation 7 × 9, it is plain that we do to 7 what we do to a unit to obtain 9. Similarly, 3 5  × 4 11  may be looked upon as doing to 3 5  what we do to the unit to obtain 4 11.

Now, to obtain 4 11  from the unit, we must divide the unit into 11 equal parts and take 4 of them.

Therefore, to find the value of 3 5  × 4 11  we must divide 3 5  into 11 equal parts and take 4 of them.

But 3 5  = 33 55  = 3 55  × 11, so that, the eleventh part of 3 5  is 3 55 ; and, if we take 4 of these parts, we get 3 55  × 4 or 12 55.

Thus, 3 5  × 4 11  = 12 55 . Now 12 = 3 × 4, and 55 = 5 × 11.

Hence we have the following rule: To multiply two fractions together, multiply the numerators for a new numerator and the denominators for a new denominator.

As in Example 2 the work is shortened if we cancel common factors from the numerators and denominators.

Example : Multiply 22 91  by 13 77.

The  product = 2 22   ×  13 91 7  ×  77 7  = 2 49  Ans.

Here, the 22 of the numerator and the 77 of the denominator contain a common factor, 11. Therefore, we cross out the 22 and write 2 above it, and cross out the 77 and write 7 under it. Similarly, we cancel the factor 13 from 13 and 91. There is now 2 left for numerator and 7 × 7 for denominator.

To multiply more than two fractions together, we proceed in the same way.

In multiplication of fractions, mixed numbers must first be expressed as improper fractions.

Example : Simplify 5 1 7  × 11 27  × 1 11 24.

Given  expression = 3 36 7   ×  11 27 9  × 5 35 24 2  = 55 18  = 3 1 18