a translation of the enunciation, by supposing the interest additive to the capital, in which case, the sum bought to be subtracted.

This equation, treated as the preceding, shall give x=20b,

If the interest at 5 per cent be subtracted from 100, in which case 100 reduces itself to 95, we have the capital at the end of the year, by the proportion

95x

100

consequently, +b=x;

multiplying by 100, and transposing, we shall have 100b5x, .x=20b.

The negative isolated result, that is, the negative value of x, would announce rectification or a correction in the terms of the enunciation, and the problem proposed could be re-established in two ways.

Prob 5. What number is that, the double of which exceeds its half by 6?

...multiplying by 2, 4x-x12, or 3x=12, .. by division, x=4.

Prob. 6. From two towns which are 187 miles distant, two travellers set out at the same time, with an intention of meeting. One of them goes 8 miles, and the other 9 miles a day. In how many days will they meet?

Let x

the number of days required; then 8x the number of miles one travelled, the number the other travelled;

and 9x

and since they meet, they must have travelled together the whole distance,

consequently, 8x+9x=187,

or 17x=187, .. by division, x=11.

Prob. 7. What number is that, from which 6 being subtracted, and the remainder multiplied by 11, the product will be 121?

Let x=the number required;
Then by the problem (x-6) XH=121,

by transposition, 11x=121+66,
or 11x=187,

.. by division, x=17.

Prob. 8. A Gentleman meeting 4 poor persons, distributed five shillings amongst them: to second he gave twice, the third thrice, and to the fourth four times as much as to the first. What did he give to each ?

Let the pence he gave to the first,

..2x the pence given to the second, and 3x=

4x=

to the third,

... by the problem, x+2x+3x+4x=5X12=60,

to the fourth.

or 10x=60, =6,

by division, x= and therefore he gave 6, 12, 18, 24 pence respectively to them.

Prob. 9. A Bookseller sold 10 books at a certain price; and afterwards 15 more at the same rate. Now at the latter time he received 35 shillings more than at the former. What did he receive for each book?

Let the price of a book. then 10x

the price of the first set, and 15x the price of the second set; but by the problem, 15x=10x+35; .. by transposition, 5x=35;

and by division, x-7.

Prob. 10. A Gentleman dying bequeathed a legacy of 1400 dallars to three servants. A was to have twice as much as B; and B three times as much as C. What were their respective shares ?

Let x=C's share,

... 3x=B's share, and 6x=A's share;

then by the problem, x+3x+6x=1400,

or 10x1400,

.. by division, x=140 C's share.

... A received 840 dollars; B, 420 dollars; and C, 140 dollars.

Prob. 11. There are two numbers whose difference is 15, and their sum 59. What are the numbers?

As their difference is 15, it is evident that the greater number must exceed the lesser by 15. Let, therefore, x= the lesser number; then will x+15= the greater;

... by the problem, x+x+15-59, or 2x+15=59,

by transposition, 2x=59-15=44, ... by division, x=22 the lesser number, and x+15=22+15=37 the greater.

Prob. 12. What two numbers are those whose difference is 9; and if three times the greater be added to five times the lesser, the sum shall be 35?

Let x

the lesser number;

then x+9= the greater number.

And 3 times the greater=3(x+9)=3x+27, 5 times the lesser 5x.

=

... by the problem, (3x+27)+5x=35 ; by transposition, 3x+5x=35-27, or 8x=8;

.. by division, x=1 the lesser number, and x+9=1+9-10 the greater number.

Prob. 13. What number is that, to which 10 being added, ths of the sum will be 66 ?

Let x the number required;

then x+10= the number, with 10 added to it.

3x+30

5

Now ths of (x+10)=3(x+10)=

But, by the problem, gths of (x+10)=66;

5

Prob. 14. What number is that, which being multiplied by 6, the product increased by 18, and that sum divided by 9, the quotient shall be 20?

Let

the number required,

then 6x the number multiplied by 6;
6x+18= the product increased by 18;

6x+18

and

that sum divided by 9.

9

6x+18
9

=20;

Prob. 15. A post is feet out of the water.

.. by the problem,

by multiplication, 6x+18=20×9; by transposition, 6x=180-18, or 6x=162;

.. by division, x=27.

th in the earth, ths in the water, and 13 What is the length of the post?

and 13 the part of it out of the water.

But by the problem, part in earth + part in water +part out of water whole part;

3x

3x

·() + (3)

5

+13=x.

and 35+×35+13×35=35∞,

5

or 7x+15x+455=35x;

by transposition, 455-35x-7x-15x-13x,

or 13x=455;

.. by division, x=35 length of the post.

Prob. 16. After paying away 4th and 4th of my money, I had 850 dollars left. What money had I at first?

Let x the money in my purse at first;

4 7

Multiplying by 28 the product of 4 and 7, which is the least common multiple,

or 28x-7x-4x=23800,

.. 17x=23800; and by division, x=1400 dollars.

Prob. 17. What number is that, whose one half and one third plus 12, shall be equal to itself?

Let x= the number required;

fhen, by the problem, x=++12;

3