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SUPPLEMENT to Heduction.


1. WHAT is Reduction,

2. Of how many kinds is Reduction ? what are they called ? wherein do these

kinds differ one from the other? Which of the fundamental rules are employed

in their operations ? 3. How is Reduction Descending performed ? 4. How is Reduction Ascending performed ? 5. When it is required to know how many sorts of coin, weight or measure of

different values, of each an equal number, are contained in any other number of each an equal number of another kind, what is the method of procedure ?

EXERCISES. 1. IN 36 guineas how many crowns? Ans. 153 crowns, and 9d. over.

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2. How many steps of 2 feet and 5 inches each, will it require a man to
take, going from Leominster to Boston, it being 43 miles.
Ans. 93948 steps ;(the last step will carry him into the town 12 inches.



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4. Ir a vinter be desirous to draw off a pipe of Canary into bottles containing pints, quarts, and 2 quarts, of each an equal number, how many must he have?

Ans. 144 of each. 1 1-2 Gat

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5. THERE are three fields; one contains 7 acres, another 10 acres and the other 12 acres and 1 rood ; how many shares of 76 perches each, are contained in the whole ? Ans. 61 shares and 44 perches over.

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6. There are 106 lbs. of silver, the property of 3 men ; of which Å receives 1716. 10oz. 19 puuts. 19grs. of what remains B shares loz. 7 grs. so often as Cahares 13/rwts. What are the shares of B. and C?

Answer, B's share 53 lb. &oz. 5pwls. 5grs. C's share 3416. 4oz. 15/19viş.

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§ 2. Fractions.

When the thing or things signified by figures are whole ones, then the figo úres which signify them are called Integers, or whole numbers. But when only some parts of a thing aże signified by figures, as two thirds of any thing, five sixths, seven tenths, &c. then the figures which signify these parts of a thing, being the expression of some quantity.less than one, are called TRACTIONS.

FRACTIONS are of two kinds, Vulgar and Decimal ; they are distinguished by the manner of representing them; they also differ in their modes of operation.

VULGAR FRACTIONS. To understand Vulgar Fractions, the learner must suppose an integer (or the number 1) divided into a number of equal parts; then any number of these parts being taken, vould make a fraction, which would be represented by two numbers placed one directly over the other, with a short line between them, thus two thirds, three fifths, seven eighths, &c.

Each of these figures have a different name and a different signification. The figure below the line is called the Denominator and shews into how many parts an integer', or one individual of any thing is divided.....the figure above the line is called the Numcrator and shews how many of those parts are siga nified by the fraction.

For illustration, suppose a silver plate to be divided into nine equal parts. Now, one or more of these parts make a fraction, which will be represented by the figure 9 for a denominator placed underneath a short line, shewing the plate to be divided into nine equal parts ; and supposing two of those parts to be taken for thc fraction, then the figure 2 must be placed directly above the 9 and over the line () for a Numerator, shewing that two of those parts are signified by the fraction, or to ninths of the plate. Now let 5 parts of this plate, which is divided into 9 parts, be given to John, his fraction woulú be five ninths ; let 3 other parts be given to Harry, his fraction would be three minths ; there would then be one part of the plate remaining still (5 and 3 are 8) and this fraction would be expressed thus i one ninth.

In this way are all vulgar fractions written ;. the Denominator, or number below the line shewing into how many parts any thing is divided, and the nuinerator, or number above the line, shewing how many of those parts are taken, or signified by the fraction.

To ascertain whether the Learner understands what has now been taught him of fractions, let us again suppose a dollar to be cut into 13 equal parts ;--let 2 of those parts be given to A ; 4 to B ; and 7 to C,

A's fraction-

131 REQUIRED of the Learner that he should write B's. fraction


C's fraction

It is from Division only that fractions arise in Arithmelical operations: the remainder after division is a portion of the Dividend undivided ; and is always the Numerator to a fraction of which the Divisor is the Denominaior. The Quotient is so many integers. Tile Arithmetic of Vulgar Fractions is tedious and even intricate to begin

Besides, they are not of necessary use. We shall not, tijerefore, enter into any further consideration of them here. This difficulty arises chicfly from the variety of denominators ; for when numbers are divided into differn


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