48. Two boys are each able to earn a penny an hour; they both go to bed at the same hour; one gets to work at 5 o'clock every morning, the other sleeps till 8: how much will the active boy be richer than the sluggard in 10 years?

Ans. £39 2s. 6d. (313 days).

49. Suppose a man to get for a stock-farm all the land he can see on an extensive plain, and he can see 28 English square miles, how many Irish acres in his farm?

Ans. 11062a. 3r. 174p. Irish.

50. I mix 50cwt. flour, third quality, at 12s. per cwt., with 40cwt., second quality, at 13s. 4d.; at what rate, per stone, must I sell the mixture to gain £15 6s. 8d.?

Ans. 2s.

MENTAL ARITHMETIC.

CHAPTER II.

III. To find the price of a SCORE, the price of one article being given.

RULE.-Reckon a pound for every shilling in the price of one.

For, a ton, at 6s. 6d. per cwt.

= £6 10s.

IV. To find the cost of a GROSS, from having the price of one

given.

RULE.-The number of pence in the price of a dozen will be the price of a gross in shillings.

Because 12doz., that is, 12 × 12=144, 1 gross.

V. To find the cost of 1 from the cost of any number.

RULE.-Divide the price by the number, for the cost of 1.
For, if 8 apples cost 24d., 24÷8=3d., the price of each.

VI. To find the price of 1 from the cost of a dozen.

RULE.-Reckon a penny for every shilling in the price of a dozen. Rules 5, 6, 7, and 8, are the reverse of 1, 2, 3, 4, respectively.

VII. Given the cost of a score to find the price of 1.

RULE.-Reckon a shilling for every pound, and 6d. for every

VIII. Given the price of a gross, to find the cost of 1.

RULE.-Reckon a penny for every shilling and divide these

pence by 12.

VULGAR FRACTIONS.

CHAPTER III.

71. A FRACTION is an expression for a part or parts of anything considered as a whole.

A fraction may be either greater or less than a unit; if it be onee-half of ten it is equal to five units.

A fraction is expressed by means of two numbers placed one over the other with a line between them.

Thus,,,, read, one-half, two-thirds, eight-fifths. 72. The number below the line is called the DENOMINATOR, because it shows the number of parts into which the whole or unit is divided. It gives the name or denomination to the fraction, just as cut. gives the name hundredweight to any number after which it is written. The number above the line is called the

NUMERATOR.

The numerator shows or enumerates the number of parts expressed by the fraction. If we divide anything into four equal parts, we express three of these parts by the fraction 3. The numerator and denominator are called the TERMS of the fraction.

73. A fraction corresponds to an example in division before the process is performed, the numerator corresponding to the dividend and the denominator to the divisor. Therefore the TRUE or REAL VALUE of any fraction is the quotient obtained by dividing its numerator by its denominator. Hence, the greater the numerator in comparison with the denominator, the greater the value of the fraction. For, the more parts we take of anything, the greater must be the particular fraction used to express these parts.

74. Increasing the numerator, the denominator remaining the same, increases the value of the fraction; it increases the number of parts; or, which amounts to the same thing, it increases the dividend, the divisor remaining the same (73). Increasing the denominator decreases the value of the fraction; as it makes a greater number of parts, each part must be smaller. It increases the divisor, the dividend remaining the same.

75. The terms of a fraction may be both multiplied or both divided by the same number, without altering the true or real value of the fraction. The quotient will remain the same, therefore the value of the fraction will not be changed. For, ==1‡, &c.

76. Two-thirds of one is equal one-third of two, as explained in the illustration.

In the same manner it may be shown that of 1: and of 1 of 3, &c.

=

77. A PROPER FRACTION has its numerator less than its denominator:-,,, are proper fractions.

An IMPROPER FRACTION has its numerator equal to or greater than its denominator :—4, 4, 3, are improper fractions.

A SIMPLE FRACTION refers to one or more units; as we have seen, may be considered of 1, or of 2.

A COMPOUND FRACTION is a fraction of a fraction, and is known by having "of" interposed; as, of 1, of, &c. It bears the same relation to a fraction that a fraction bears to unity.

Observe that the to the left is divided into two parts, each of which represents the fraction of. Again, since

A UNIT.

of a shilling

=

=

3d., and one-half of 3d. is 1d., and 11d. — § of a shilling, therefore of }.

=

A MIXED NUMBER consists of a whole number and a fraction; as, 31, 23, 61.

78. A COMPLEX FRACTION has a fraction or a mixed number in its numerator, or in its denominator, or in 343 of 3

both; as,

The first means that one-fourth of is to be taken, and not of 1. The second expresses of 4, and not 3 of 1.

At first sight these appear to be complicated expressions, but they will be easily understood from this explanation.

79. To reduce an improper fraction to a whole or mixed number.

RULE.-Divide the numerator by the denominator, the quotient is the whole number; under the remainder (if any) write the denominator, in the form of a fraction.

Ex. 1. Reduce to a whole number?

Here, 15÷4=3+3r.=33 Ans.

REASON. The denominator shows the number of parts to be taken to make up a unit; therefore, we must put as many parts together to make 1, as there are units in the denominator; this is effected by the rule (see hint 23). Since the operation of division does not alter the nature of the quantity divided, the remainder is a part of the given fraction, and is expressed as such by writing the denominator under it.