Answer, £6190. 7. 4¾.

There are four towns, in the order A, B, C, and D. If a man can go from A to B in 5ʰ 20ᵐ 33ˢ, from B to C in 6ʰ 49ᵐ 2ˢ and from A to D in 19ʰ 0ᵐ 17ˢ, how long will he be in going from B to D, and from C to D?

Answer, 13ʰ 39ᵐ 44ˢ, and 6ʰ 50ᵐ 42ˢ.

227. In order to perform the process of Multiplication, it must be recollected that, as in (52), if a quantity be divided into several parts, and each of these parts be multiplied by a number, and the products be added, the result is the same as would arise from multiplying the whole quantity by that number.

It is required to multiply £7. 13. 6¼ by 13. The first quantity is made up of 7 pounds, 13 shillings, 6 pence, and 1 farthing. And

1 farth. × 13 is 13 farth. or£0 . 0 . 3¼(219)
6 pence × 13 is 78 pence, or 0 . 6 . 6
13 shill. × 13 is 169 shill. or 8 . 9 . 0
7 pounds × 13 is 91 pounds, or91 . 0 . 0
The sum of all these is £99 . 15 . 9¼

which is therefore £7. 13. 6¼ × 13.

This process is usually written as follows:

228. Division is performed upon the same principle as in (74), viz. that if a quantity be divided into any number of parts, and each part be divided by any number, the different quotients added together will make up the quotient of the whole quantity divided by that number. Suppose it required to divide £99. 15. 9¼ by 13. Since 99 divided by 13 gives the quotient 7, and the remainder 8, the quantity is made up of £13 × 7, or £91, and £8. 15. 9¼. The quotient of the first, 13 being the divisor, is £7: it remains to find that of the second. Since £8 is 160s., £8. 15. 9¼ is 175s. 9¼d., and 175 divided by 13 gives the quotient 13, and the remainder 6; that is, 175s. 9¼d. is made up of 169s. and 6s. 9¼d., the quotient of the first of which is 13s., and it remains to find that of the second. Since 6s. is 72d., 6s. 9¼d. is 81¼d., and 81 divided by 13 gives the quotient 6 and remainder 3; that is, 81¼d. is 78d. and 3¼d., of the first of which the quotient is 6d. Again, since 3d. is ¹²/₄, or 12 farthings, 3¼d. is 13 farthings, the quotient of which is 1 farthing, or ¼, without remainder. We have then divided £99. 15. 9¼ into four parts, each of which is divisible by 13, viz. £91, 169s., 78d., and 13 farthings; so that the thirteenth part of this quantity is £7. 13. 6¼. The whole process may be written down as follows; and the same sort of process may be applied to the exercises which follow: