only a Girl: What would have been her Dowry had fhe only had a Son? 20. A Cistern holds 103 Gallons, and being brim-full, has two Cocks to run off the Water: by the first of which, a three Gallon Pail will be filled in 60 Seconds, by the other in 75; in what Time will this Cistern be emptied through both thefe Apertures together, fuppofing the Efflux of the Water all the fame?

21. A Politician having about him a certain Number of Crowns, faid, If++ of what he had, were added together, they would make just Wilkes's Number (45) ; how many Crowns had he about him?

22. A Gentleman has an Orchard of Fruit Trees, one half of the Trees bearing Apples, one fourth Pears, one fixth Plumbs, and one twelfth of them bearing Cherries: How many Fruit Trees in all grow in the said Orchard?

23. A School Master being asked how

many Scholars he had, anfwered, If I had as many, and as many, and as many, I fhould have 99. How many had he?

24. In the Year I wrote this, if to my Age you add ,,, (thereof), with more,

1

The Number 74 will then be had.

Ingenious Youths, my Age explore.

25. A. in a Scuffle, feized on of a Parcel of Sugar-Plums, B. catched of it out of his Hands, and C. laid hold on more; D. ran off with all A. had left, except, which E. afterwards fecured flily for himself; then A. and C. jointly fet upon B. who, in the Conflict, fhed

he had, which were equally picked up by D. and E. who lay perdue. B. then kicked down C.'s Hat, and to work they all went anew for what it contained; of which A. got, B., D., and C. and E. equal Shares of what was left of that Stock; D. then struck of what A. and B. last acquired out of their Hands ; they with Difficulty recovered of it in equal Shares again, but the other three carried off a piece of the fame. Upon this they called a Truce, and agreed, that theof the Whole left by A. at firft fhould be equally divided among them: How much of the Prize, after this Diftribution, remained with each of the Competitors? THE

P

THE

TUTOR'S GUIDE.

PART III.

A

44.

DECIMAL FRACTIONS.

Decimal Fraction, is a Fraction whose Denominator is always Unity or 1, with one or more Cyphers: Thus, an Unit may be imagined to be equally divided into 10 Parts, and each of these into 10 more; fo that by a continual Decimal Sub-divifion, the Unit may be fuppofed to be divided into 10, 100, 1000, and fo on without End, all being equal Parts, called tenth, hundredth, thousandth Part of an Unit or 1.

In Decimal Fractions, the Figures of the Numerator are only expreffed, the Denominator being omitted, because it is always known to confift of an Unit with so many Cyphers as there are Places in the Numerator.

A Decimal Fraction is diftinguished from an Integer with a Point or Comma prefixed, thus, ,5 wich ftands for, or ,75 for 75%, or ;,2752 for 275.2 ; and 12,005 for

1009

12, Too, or 12, &c.

5

10000

Cyphers at the Right Hand of a Decimal Fraction alter not its Value for ,5 or ,50 or ,5000 is each of them the fame Value, and are equal to or; but Cyphers at the Left Hand, in a Decimal Fraction, decrease the Value in a tenfold Proportion, for ,05 is, alfo, 0005 is Tooo, &c. all of which will plainly appear by the following

TABLE.

By the above Table it also plainly appears, that as whole Numbers increase towards the Left Hand by a ten-fold Proportion, fo Decimal Parts decrease towards the Right Hand by the fame Proportion.

A finite Decimal is that which ends at a certain Number of Places; but an infinite is that which no where ends.

A circulating or recurring Decimal, is that wherein one or more Figures are continually repeated.

Thus 64,766666, &c. or 64,76, is called a fingle circulate or recurring Decimal,

And 147,642642, &c. or 147,64%, is called a compound recurring Decimal.

Note.-In all Operations, if the Result confifts of several Nines, reject them, and make the, next fuperior Place a Unity more. Thus, for 17,1999 write 17,2; and for 12,99 write 13, &c.

45. ADDITION of DECIMAL S.

1. Addition and Subtraction in Decimals, are performed after the fame Manner as Sect. 2, 3, of whole Numbers, Care being taken that like Parts be placed under one another, and from their Sum or Difference cut off fo many, Decimal Parts as there are the moft in any of the given Numbers.

EXAMPLE S.

1. What is the Sum of ,0476, 21,476,,0067, 64, 17,6, and,20764?

P 2

2. Add

2. Add ,427, 64,075, 27,6421, 10,8, ,0074, and 104,

,046842, together.

3. What is the Sum of ,274, ,076, ,64762, ‚0706, 147, ,007, and 968,42 ?

2d. To add Decimals, wherein there are fingle Repetends.

RULE.

Make every Line end at the fame Place, filling up the Vacancies by the repeating Digits, and annexing a Cypher or Cyphers to the finite Terms; then add as before; only increate the Sum of the Right Hand Row, with as many Units as it contains Nines, and the Figure in the Sum under that Place will be a Repetend.

4. What is the Sum of 47,674, 4,02642, 32,5, 6,14, and 27,0646?

5. Add 11,4, 6,14274, 91,78, 37,67%, and 146,476742. 6. What is the Sum of 14,276421, 7,4, 21,645, 9,27, and 31,1474?

3d. To add Decimals, having compound Repetends.

RULE.

Make the Repetends fimilar and conterminous; then add as before, only increase the Right Hand Figure by as many Units as are carried from the Column of Figures, wherein all the Repetends begin together; laftly, dafh off for a Repetend as many places as were fo in the Numbers, added together.

7. What is the Sum of 14,2472, 768,246, 7,064, and 26,006%? 8. Add ,246, 3,67, 27,0427694, and 9,943, together.

46. SUBTRACTION of DECIMALS.

EXAMPLE S.

1. What is the Difference between 176, and 10,764 ?

2. From

2. From 647, take ,00746.

3.

What is the Difference between 74,6407, and 69,5 ? 2d. To fubtract Decimals that have a fingle Repetend.

RULE.

Make both end together as in Addition; and if the Repetend of the Number to be fubtracted, be greater than the Repetend of the Number it is to be, taken from, then the Right Hand Figure of the Remainder must be less by Unity than it would be; or inftead of borrowing Ten, as in whole Numbers or Infinites, borrow in this Place 9, the rest_as usual, and the Right Hand Place or Figure will be a Repetend.

4. What is the Difference between 41,74, and 21,94648? 5. From 24,1466, take 19,9.

6. What is the Difference between 16,126, and 4,1942764?

47. MULTIPLICATION of DECIMALS.

f. Multiplication in Decimals is alfo performed as in whole Numbers, no Regard being had to the Decimals as fuch, till the Product is obtained, then obferve the following

RULE S.

1. Strike off fo many Figures from the Right Hand of the Product as there are Decimal Places in the Multiplier and Multiplicand.

2. But if there are not fo many Figures in the Product, fupply the Deficiency by prefixing Cyphers to the Left Hand to make them equal.

3. If the Number is to be multiplied by 10, 1000, &c. remove the feparating Point in the Multiplicand fo many Places towards the Right Hand, as there are Cyphers in the Multiplier.