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sult is d times too small, and consequently must be multiplied by d; thus,

a d
x d =
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b c
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x + y
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x + y

Y
1
1

y
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x +

2 + y
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1

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(x + y)
x(x y)
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Y y(x - y) y(x + y) y(y + y)2 x(x + y) x(x + y)? = 2*x + y) + x*(x-y)-y*(x + y)yoxy), brought to a common denominator.

xy(x + y)2
2 x - - 2 xy?

2 x(x2 - y2)
xy(x + y) xy(x + y)
2(x - y)
y(x + y)

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"2 x

X

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a

SECTION III.
Find the value of x in each of these equations."

- 2 (1)

= 2 x -7" 5

3

b (2)"

a + 2 x a + x 2 x

2 (1)

= 2 x -7 5

3 6 x + 5 x 10 = 30 x 105 ; transposing and . changing all the signs, 30 x - 6 x — 5 x = 105 10

19 x = 95

..x = 5

X

+

a

=0

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= ab

a

26

b (2)

a + 2 x a + 2
ala + x - ba + 2x) = 0

ab 2 b 2 =0
x(a 26)

az

ab a2

.. x = See Appendix, Algebra.

2. (1) “What number is that, the double of which exceeds its half by 6 ?”

“ The difference of two numbers, and a quarter part of their sum are each equal to 2; find the numbers." (1) Let x = the number Then 2 x = its double

X
And its half

2

(2) “

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= 6

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.. 2 x

2

x = 12
3 x = 12

.. X = 4
(2) Let x = the greater
Then a

- 2 = the less Now the difference of these numbers is equal to the fourth part of their sum.

x + (x - 2)
..x - (x - 2)

4
4 x - 4x + 8 = x + 2 — - 2

2x = = 10

ix = 5, the greater And x

2 = 3, the less. 3. (1) “If A does a piece of work in 10 days, which A and B can do in 7 days, working together ; how long would B take to do it alone ?''

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(2) “ Find the amount of £P. at compound interest for n years, the interest being payable yearly." (1) Let x = the number of days in which В can

complete the work by himself, Then 1 = the part B would do in 1 day. Now A does it in 10 days

1

= the part A does in 1 day.

10 The part done in 1 day by A and B working to

1 1 gether would be, +

10 But they can together complete it in 7 days; and

1 therefore they can do

7

per day.
1 1 1
Hence,
10

7
7 x + 70 = 10 x

7 x = 70

.. = 23} days. (2) Letr = the interest of £1 for a year; the amount of £1 for a year will then be 1 + r.

To find the amount of succeeding years, we have the following proportions : as the principal of the first year is to the principal of any succeeding year, so is the amount of the first year to the amount of that succeeding year, Thus1:17 9::1 +:(1+r)? = the amount of 2nd

year 1:(1 + r): :: 1+r:(1!+ r)= ,

3rd

+

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10 x

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*

*

*

nth year

1:(1+r)::(1+r):(1+r)"= (1 + r)n being the amount of £1 for n years, the amount of £P will be P.(1 + r)".

See Appendix, Algebra.

GEOMETRY.

SECTION I. 1. Define a plane superficies, and a circle. 2. Draw a straight line perpendicular to a given

straight line from a given point in it. 3. The angles at the base of an isosceles triangle are

equal to each other, and if the equal sides be produced the angles on the other side of the base shall be equal.

SECTION II. 1. Prove that the sum of the three angles of a triangle

equals two right angles. Given the value of two angles of a triangle, how is the value of the third

ascertained ? 2. Prove that parallelograms on the same base and be

tween the same parallels are equal to one another. 3. In any right angled-triangle, the square which is

described upon the side subtending the right angle is equal to the squares described upon the sides containing the right angle.

SECTION III. 1. If a straight line be divided into two equal parts,

and also into two unequal parts, the rectangle contained by the unequal parts together with the square of the line between the points of section is

equal to the square of half the line. 2. In any triangle the square of the side subtending

any acute angle is less than the squares of the sides containing that angle by twice the rectangle

contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle and the acute angle ;-prove only the first case of this proposition.

SECTION IV. 1. Construct a triangle whose area shall be equal to

that of a given trapezium. 2. Show how to make a square double a given square. 3. Show that the diagonals of a parallelogram bisect

each other.

N.B.—As it may fairly be assumed that every school

master is possessed of a copy of Euclid's Elements ; and as the following questions may be answered in the very words of Euclid ; it will scarcely be necessary to give other solutions than direct reference to the particular proposition, the knowledge of which is presupposed in the questions. Schoolmasters will find the recently published " Geometry and Mensuration," by Mr. Tate, a most valuable auxiliary in acquiring a rapid knowledge of the essentials of the science. It may be briefly characterised as Euclid

divested of undue technicalities and redundancy. A regard to economy in publication requires the use of

as few diagrams as possible; hence the reason of aiming to dispense with them, as in this paper.

SECTION I. 1. Define a plane (1) superficies, and (2) a circle."

(1) A plane superficies is a surface, any two points of which being joined the straight line between them will be wholly within the superficies.

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