4. The distance from London to Cambridge is 57 m.; and from Yarmouth to Norwich 20m. The second class fares between the same places are 11s. and 2s. respectively what would have to be added to the present fare per mile (second class) between Cambridge and London, so as to make it exactly double the second class fare per mile between Yarmouth and Norwich?

5. Multiply £721. Os. 51d. by 96; and divide 1283 cwt. 4lbs. by 75. Reduce of £1 to the fraction of 13 of £3. 5s. Prove the rule for the multiplication of two

fractions, taking as an example &× 4.

6. When are four quantities said to be in proportion? Shew by means of your definition that £191. 12s. 6d. : £31. 10s. :: 365 days: 60 days; and deduce the method of working the following question: "If 3 workmen earn between them £191. 12s. 6d. in a year, in what time would they earn £31. 10s.?"

12

7. Reduce 2s. 6d. to the decimal of 5 of £1; and of 5 of £1000. respectively. Find the value of 875 of 15s. 6d.

8. Divide 12.55 by 01004; 1255 by 10·04; and 001255 by 1004.

Reduce 1018,, 42000 to decimals, and then add them together.
Reduce of 1.375 and 285714 to vulgar fractions in their lowest terms.

9. Shew that the fraction is not altered in value by multiplying 5 into numerator and denominator. How is it that we do not alter the value of a decimal fraction by bringing down any number of cyphers to the right hand of the last figure?

10. After paying an income-tax of 10 per cent. a person has £1250 a year, what was his entire income?

11. Find the difference between the simple and compound interest of £3300 at 3 per cent. for 2 years.

12. In what time will £537. 16s. 8d. amount to £591. 12s. 4d. at 23 per cent. simple interest?

13. What must be the rate of interest in order that the discount on £387. 7s. 7 d. payable at the end of 3 years may be £41. 10s. 1d.?

14. At what price must the 3 per cents. be, in order that a person may obtain an equal rate of interest by investing in them, as he would by investing in the 3 per cents. at 72?

15. A person taking two tickets (a 1st and a 2nd class) from Norwich to Stowmarket receives 7s. 6d. change out of a sovereign; how much had he to pay for each ticket separately, supposing that the 1st and 2nd class fares from Norwich to Diss are 3s. 6d. and 2s. 9d. respectively? Of course the fares throughout are supposed proportional to the distance.

16. In extracting the square root of ⚫003 you have by mistake "pointed" thus 00300 &c.; and proceeded with the operation and marked off the decimals accordingly. Without extracting the root of 003 over again, there is a certain

quantity which if multiplied into your erroneous result, will give a correct value of ✓003; find the first three decimal places of this multiplier.

MARCH 4, 1856.

1. Prove that 29 multiplied by 15-15 multiplied by 29.

Likewise that of 1 of 3.

=

2. (a) If a person's estate be worth £1384. 16s. a year, and the land be assessed at 2s. 93d. per £, what is his clear annual income?

(6) What is the cost of 39 cwt. 3 qrs. 26lbs. at £4. 17s. 10d. per cwt.?

N.B. (a) and (6) both by "Practice." To what class of examples does the "Rule of Practice" apply, and why is it so called?

What is the meaning of an "aliquot part"?

3. Easter Sunday is always the Sunday directly following the first full moon which falls after March 20th: there are 29 days between any two consecutive full moons: February 1852 (being a "leap" year) had 29 days, and there was a full moon on April 18th, 1848 (a Tuesday).

From these data, find when Easter fell in 1855.

4. Find the area of a room 8ft. 4 in. long, by 12 ft. 2 in. broad, by duodecimals or cross multiplication. If in this example the room were not supposed to be a rectangular parallelogram, how would the answer have to be interpreted?

5. Add together § of 16s. 6d. +1 of 12s. 101d. +† of £2. 4s. 83d.

If A be of 23 of B, and C be 13 of B, what fraction is A of C?

What is meant by "reducing one quantity to the fraction of another"?

7. A person rows a distance of 13 miles down a stream in 20 minutes; but without the aid of the stream, it would have taken him half an hour; what is the rate of the stream per hour? and how long would it take him to return against it? 8. Divide '01 by 1000; 202 by '01; and 13099-52 by 0011008; and prove your results by vulgar fractions.

Reduce (of 118-5 of 11·02)÷0.1 to a decimal.

9. Reduce 18s. 4 d. to the decimal of £1, and likewise to that of £1000. Find the value of 785 of £10.

10. When are four quantities said to be in proportion? and apply your definition to ascertain whether the four quantities 3 lb. 2 oz.; ls. 1ąd.; ls. 71⁄2d.; 4 lb. 2 oz. can be so arranged as to form a proportion. Can pounds and ounces be multiplied into shillings and pence?

12. If 20 men can perform a piece of work in 12 days, how many men will perform a piece of work half as large again in a fifth part of the time, if they work the same number of hours a day; supposing that 2 of the second set can do as much work in an hour as 3 of the first set?

13. A person investing in the 4 per cents. receives 5 per cent. for his money; what is the price of stock?

14. When the 3 per cents. are at 80, how much stock must be sold out to pay a bill of £690. 3s. 9d. due 9 months hence at 3 per cent. simple interest?

15. (a) Given that the square of 10129 is 102596641; find the square of 101293, without going through the operation of squaring.

(B) Given that the square root of 105625 is 325, find that of 10582009. (7) Extract the cube root of 5 to 3 places of decimals.

MARCH 6, 1856.

CANDIDATES FOR HONORS.

1. Distinguish between the addition of Algebraical and that of Arithmetical quantities.

Add 16a2-7ub-8b+3c, 4b2-8c+ab, 12ab — 8a2 +5c.

2. Find the value of 20ab7bc+16ac - 5d2 when a, b, c, d are equal to 1, 3, 4, 5 respectively.

3. Explain the multiplication of -a by-a.

4. Multiply a2+2ab+b2 — c2 by a2-2ab+b2+c2; and shew that the result may be expressed under the form (a2 — b2)2 — c2 (c2 - 4ab).

5. State the preliminary steps to be adopted in the division of one algebraical quantity by another.

Divide (a3 + b3)2+2u3b (a3 — a3b — b3) by a2+b2.

(3)

=27.

8. A hasth share in a concern and sells th of th of this share for £500; what is the value of th of th of the concern?

9. A is 40 years older than B and in 4 years A will be 3 times the age of B. What are the respective ages of A and B ?

10. A can do a piece of work in 6 days, but with the help of B he can do it in 2 days. How long would B alone take to do the work?

11. Prove that a ratio of greater inequality is diminished and of less inequality increased by adding the same quantity to both its terms.

12. The quantities a, b, c, d are in proportion. Express their relations to each other by an equation.

Find a mean proportional between 7 and 25.

13. Solve the following equations:

14. A farmer buys a number of oxen for 200 guineas, and, after losing 4 of them, sells the remainder for £7 a head more than they cost him, and gains by the transaction 20 guineas. What number of oxen did he purchase?

15. Two vessels A and B contain different mixtures of wine and water, the one in the proportion of 2:5, and the other in that of 5:9. What quantity must be taken from each to form a mixture which shall contain 5 quarts of wine and 12 of water?

MISCELLANEOUS QUESTIONS AND EXAMPLES

from Cambridge Examination Papers.

I.

1. WHAT decimal of a guinea is 60 francs, £1 being equivalent to 25 francs? 2. Find the length of a street in which the wheel of a barrow revolves exactly 150 times, the diameter of the wheel being 11⁄2 ft., and the ratio of the circumference to the diameter 3.14159: 1.

3. France is 128 millions of English acres, and the Pyrenees spread over it would cover it to the depth of 115 feet; find the bulk of the Pyrenees in cubic feet.

4. (a) If 28=a+b+c; shew that

=

2hc+(b2 + c2 - a2) S(s-a)
2bc-(b2+c2 - u2) (S-b) (s−c)*

(B) Prove that the expression a2 b+b2 c + c2 a − ab2 – bc2 - ca2 is divisible without remainder by the difference of any two of the quantities a, b, c.

5. The number of shares in a railway is 45000, and the expense of working it is equal to ths of the gross receipts, which annually amount to £581250. What should a person give for a share, that he may make exactly 5 per cent. of his money? 6. Explain the method of pointing in extracting the cube root of whole numbers, and also of decimals.

A cube contains 2.370 cubic yds. How many linear feet are there in (1) an edge, (2) a diagonal? and what is the area of one of its faces?

7. Give the algebraical and geometrical definitions of Proportion; and shew that if quantities be proportional according to the former, they are proportional according to the latter definition.

(a) Find a fourth proportional to 1.34, 001, and

⚫201.

8. What is the height of a closet 8 ft. by 6 ft., which will exactly contain 12 boxes 4 ft. long, 3 ft. wide, 2 ft. deep?

9. The National Debt in Consols being 375 millions sterling, what would be the annual saving, if the interest thereon were reduced from 3 to 23 per cent.? After this reduction, on what sum would the interest be £3?

10. The price of posting in Germany being 1 florins per German mile, which =4 English miles; find the cost in English money of posting 381 English miles in Germany.