3. A person, who can walk forwards four times as fast as he can walk backwards, undertakes to walk a certain distance, (one-fourth of it backwards,) in a stated time. He finds that, if his speed per hour backwards were one-fifth of a mile less, he must walk forwards 2 miles an hour faster, to gain his object. What is his speed?

4. How are Ratios compared? Shew that the ratio

(a+b)2 (a+2b)2

:

a-b a-1b

will be one of greater or less inequality, according as 762 is greater or less than 3a2.

5. If a, b, c, d be in proportion, and a the greatest, and d the least, shew that a+d>b+c.

(a) Ifa: ba+b: a−b, then a=b(√2+1).

(3) If ad=bc, then (a-b): √(c-d):: Ja- √b: √c-d.

(7) Find two numbers which are to each other as 3: 4, and their sum: sum of their squares :: 7:50.

6. If xy when ≈ is given, and xxx when y is given, shew that when neither y nor z are given, ∞ ∞ yz.

(a) Having given the relation xy ∞ a2x2 + b2y2, and the circumstance that when xa, y=b, find the exact equation between x and y.

(3) If the time of describing any space the space, and if the time of describing 144 feet be 3′′, what will be the time of describing 400 feet?

2. How are ratios compounded together? Shew how the value of a ratio is affected by being compounded with one of less inequality.

3. (a) A piece of work was done by a certaín number of men in a certain time. Had there been 5 men more, or 4 fewer, the time would have been altered by 1 day. What was the number of men?

(B) If it is between 1 and 2 o'clock, and 6 minutes hence the minute-hand of my watch will be exactly opposite to the point where the hour-hand was 7 minutes since; find the time.

4. In a match of Cricket, if each side had obtained 7 runs fewer, the scores would have been as 4: 3; if each had obtained 3 more, the scores would have been :: 13: 10; find the majority for the winner.

5. State the geometrical definition of proportion, and shew that if four quantities be proportionals according to the algebraical, they are also proportionals according to the geometrical definition. Why cannot the algebraical definition be adopted in geometry? Shew from your definition whether the numbers 2, 3, 4, 5 are proportionals.

If mp2=nq2, then will m: q::n p2 and m+q2 : q3 :: n+p2 : p2. Supposing p and q to represent lines containing p and q feet respectively, what will m and n represent?

6. If y =

C

where C is a constant, shew that y varies inversely as x.

(a) In a triangle, the area = base × perpendicular altitude: if the base continue the same whilst the area varies, find how the area varies with respect to the altitude.

(3) Assuming that the expences vary jointly as the number of persons and the time; if 7 persons spend £425. 16s. 8d. in a tour of 5 months, how much would it cost 12 persons, after the same rate of expenditure, to continue 7 months on their travels?

D.

1. Shew from your definition of a Ratio, that in all calculations it is fully and sufficiently represented by a fraction.

Shew that if a be small when compared with a, the ratio of (a + x)3: a3 is nearly equal to the ratio a + 3x : a.

Give an approximate value of the ratio (8961)5: (8958).

2. A ship sails with a supply of biscuit for 60 days, at a daily allowance of 1 lb. a head: after being at sea 20 days she encounters a storm, in which 5 men are washed overboard, and damage sustained that will cause a delay of 24 days, and it is found that each man's allowance must be reduced to lb. Find the original number of the crew.

4. When is one quantity said to be a mean proportional between two others? Find a mean proportional between a and b.

(a) If a : b :: c : d, prove that

(1) a±b: a :: c±d : c,

1

1

1

c+

c

(2) a+ba-be+d: e-d

(3) If a2+2b2: 2a2+362 :: c2+2d2: 2c2+3d, then a, b, c, d, are proportionals.

5. If x x xy, then shew that x= Cxy, where C is a constant quantity. If A & B, and C x D, shew that AC & BD.

In a triangle, the area = base × perpendicular altitude: supposing the area to remain the same whilst the base and altitude vary, determine how the base varies with respect to the perpendicular altitude.

6. A's money is 3 times that of B's when they begin to play: A loses £5 the first hour, and his money continues to vary inversely as the time from the beginning, and after 3 hours B's money is to A's money as 31 is to 9. How much had each of them at first?

ARITHMETIC.-RULE OF THREE.

Arith. Art. 142. The RULE OF THREE is a method by which we are enabled, from three quantities which are given, to find a fourth quantity which shall be the same multiple, part or parts, of the third, as the second is of the first; or, it is a Rule by which, when three terms of any proportion are given, we can determine the fourth.

In Alg. Art. 139 we have seen that if a, b, c, x are proportional, so that a bc: x, then

We defined Proportion to be the relation of equality subsisting between two ratios, and we have seen that, for estimating the value of a ratio, its terms must not merely be of the same kind, but of the same denomination of that kind, and then they may be treated as abstract numbers, whatever be the denomination to which they are reduced.

This being so, it follows that the terms of the above proportion may be multiplied together, notwithstanding the remark in Note, Arith. Art. 26; and the value of x will be given in that denomination to which the third term is reduced.

In every proportion derived from a practical question, two of the terms are quantities of one kind, and the other two, either quantities of a different kind, or if of the same kind, then distinguished from the others by the nature of the question; and in every such question, if we put a (the unknown quantity) in the fourth term, there is such a relation between the first and third terms, that one varies directly, inversely, or jointly as the other; and the Rule of Three is termed the RULE OF THREE DIRECT, INVERSE, or DOUBLE, according as such variation is direct, inverse, or joint.

143. We may lay down the following Rule for working examples in the Rule of Three, Direct or Inverse.

RULE. "Leaving out of consideration superfluous quantities, find, out of the three quantities which are given, that which is of the same kind as the fourth or required quantity; or that which is distinguished from the other terms by the nature of the question: place this quantity as the third term of the proportion.

"Now consider whether, from the nature of the question, the fourth term will be greater or less than the third; if it be greater, then put the

larger of the other two quantities in the second term, and the smaller in the first term; but if less, put the smaller in the second term and the larger in the first term.

"Take care to reduce the first and, second terms to the same denomination, and also the third and fourth terms to the same denomination; remembering, however, that if the quantities involved be all of the same kind, it is unnecessary to reduce all the four terms to the same denomination, but only the first and second terms to one denomination, and the third and fourth terms to one denomination. When the terms have been properly reduced, multiply the second and third together, and divide by the first, treating all three as abstract numbers. The quotient will be the answer to the question, in the denomination to which the third term was reduced."

The following Examples are worked out in an explanatory form, so as not only to illustrate, but also more fully to explain the above Rule.

Rule of Three Direct.

Ex. 1. Find the value of 37 yards of silk, when 25 yards cost £4. 7s. 6d. Here we have given three terms of the proportion; viz. 25 yards, and 37 yards, which are of one kind; and £4.7s. 6d. which is of another kind; and we have a connexion between 25 yards (a quantity of the first kind) and £4. 7s. 6d. (a quantity of the second kind), viz. that 25 yards cost £4.78. 6d.: our object is now to find the value in money of 37 yards, the other given quantity of the first kind.

From the nature of the question it is evident, that the price of any given no. of yds. = cost of one yd. x no. of yds.

= constant x no. of yds. ;

therefore, price of any given number of yds. a number of yds.; therefore, denoting the given number of yds. by x, we have

no. of yds. in 1st case: no. of yds. in 2nd case

:: money in 1st case: money in 2nd case,

or 25: 37: £4. 7s. 6d. : x

:: £43 x (pounds);

Ex. 2. If a workman earn £17. 6s. in 1021 days, how long will he

be in earning 50 guineas?

Let a denote the number of days required.