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them is found too difficult for beginners, they can be deferred till review.

102. If 7 men can reap 42 acres in 6 days, how many men can reap 100 acres in 5 days ?

103. If 14 men can build 84 rods of wall in 3 days, how long will it take 20 men to build 300 rods?

104. If 1000 barrels of provisions will support a garrison of 75 men for 3 months, how long will 3000 barrels support a garrison of 300 ?

105. If a man travels 320 miles in 10 days, traveling 8 hours per day, how far will he go in 15 days, traveling 12 hours per day?

106. If 24 horses eat 126 bushels of oats in 36 days, how many

will 32 horses eat in 48 days ? 107. A lad returning from market being asked how many peaches he had in his basket, replied that }, }, and * of them made 52 : how many peaches had he ?

Analysis.-The sum of , }, and 1=13. (Art. 127.) The question then is this : 52 is 13 of what number? Now if 52 is 13, 1 is 52--13=4; and 12 is 4 x 12:48.

Ans. 48 peaches. Proof. of 48 is 24; } is 16; and 4 is 12. Now 24 and 16 are 40, and 12 are 52.

303. This and similar examples are often placed under a rule called Position, or Trial and Error.

OBs. The shortest and easiest method of solving them is by Analysis.

108. A farmer lost ¢ of his sheep by sickness ; } were destroyed by wolves; and he had 72 sheep left: how many had he at first ?

109. A person having spent and f of his money, finds he has $48 left: what had he at first?

110. After a battle a general found that d of his army had been taken prisoners, s were killed, had desert ed, and he had 900 left : how many had he at the commencement of the action ?

111. What number is that } and 4 of which is 80 ?

112. What number is that of which if į and } be added to itself, the sum will be 110 ?

113. A certain post stands } in the mud, & in the water, and 10 feet above the water : how long is the post ?

114. Suppose I pay $85 for } of an acre of land: what is that per acre ?

115. A man paid $2700 for of a vessel: what is the whole vessel worth?

116. A gentleman spent } of his life in Boston, 4 of it in New York, and the rest of it, which was 30 years, in Philadelphia : how old was he?

117. What number is that / of which exceeds of it by 10 ?

118. In a certain school of the scholars were studying arithmetic; & algebra, and the remainder, which was 30, were studying grammar :

how
many

scholars were there in the school?

119. A owns g, and B 12 of a ship; A's part is worth $650 more than B's: what is the value of the ship?

120. In a certain orchard } are apple-trees; d peach trees; plumb-trees, and the remaining 15 were cherrytrees : how many trees did the orchard contain ?

SECTION XII.

RATIO AND PROPORTION.

Art. 305. Ratio is that relation between two numbers or quantities, which is expressed by the quotient of the one divided by the other. Thus, the ratio of 6 10 2 is 6-2, or 3; for 3 is the quotient of 6 divided by 2.

MENTAL EXERCISES.

Ex. 1. What is the ratio of 14 to 7? Ans. 2. 2. What is the ratio of 10 to 2? Of 16 to 4 ?

QUEST.---305. What is ratio ?

3. What is the ratio of 18 to 9? Of 18 to 6?

4. What is the ratio of 24 to 3 ? Of 24 to 4? Of 24 to 6 ? Of 24 to 8? Of 24 to 12?

5. What is the ratio of 30 to 6? Of 25 to 5 ? Of 27 to 9? Of 40 to 8? Of 56 to 7 ? Of 84 to 12 ?

6. What is the ratio of 3 to 7?

7. What is the ratio of 5 to 8? Of 7 to 10 ? Of 9 to 13? Of 10 to 17 ? Of 21 to 43 ?

Ans.

3

306. The two given numbers thus compared, when spoken of together, are called a couplet; when spoken of separately, they are called the terms of the ratio.

The first term is the antecedent; and the last, the consequent.

307. Ratio is expressed in two ways:

First, in the form of a fraction, making the antecedent the numerator, and the consequent the denominator. Thus, the ratio of 8 to 4 is written ; the ratio of 12 to 3, ?, &c.

Second, by placing two points or a colon ( : ) between the numbers compared. Thus, the ratio of 8 to 4, is written 8:4; the ratio of 12 to 3, 12 : 3, &c.

Obs. 1. The expressions, and 8: 4 are equivalent to each other, and one may be exchanged for the other at pleasure.

2. The English mathematicians put the antecedent for the numerator and the consequent for the denominator, as above ; but the French put the consequent for the numerator and the antecedent for the denominator. The English method appears to be equally simple, and is confessedly the most in accordance with reason.

3. In order that concrete numbers may have a ratio to each other, they must necessarily express objects so far of the same nature, that one can be properly said to be equal to, or greater, or less than the other. (Ari. 280.) Thus a foot has a ratio to a yard; for one is three times as long as the other; but a foot has not properly a ratio to an hour, for one cannot be said to be longer or shorter than the other.

Quest.-306. What are the two given numbers called when spoken of together? What, when spoken of separately? 307. How many ways is ratio expressed? What is the first? Second? Obs. How do the English express ratio ? How do the French ? In order that concrete numbers may have a ratio to each other, what kind of objects must they express?

8. What is the ratio of 15 lbs. to 3 lbs.? Of 21 lbs. to 7 lbs ? Of 35 bu. to 7 bu. ? Of 27 yds. to 9 yds. ?

9. What ratio is £1 to 10s. ?

Note.-£1 is 20s. The question then is this: what is the ratio of 208. to 10s. ? Ans. 2.

10. What is the ratio of £2 to 5s.? Of £3 to 12s. ? 11. What is the ratio of 23 yds. to 4 yds.? Of 21 yds. to 34 yds.?

308. Direct ratio is that which arises from dividing the antecedent by the consequent, as in Art. 305.

309. Inverse or reciprocal ratio, is the ratio of the reciprocals of two numbers. (Art. 280. Def. 11.) Thus, the direct ratio of 9 to 3, is 9 : 3, or ş; the reciprocal ratio is : }, or ] = }=;; (Art. 139;) that is, the consequent 3 is divided by the antecedent 9. Hence,

A reciprocal ratio is expressed by inverting the fraction which expresses the direct ratio; or when the notation is by points, by inverting the order of the terms. Thus, 8 is to 4, inversely, as 4 to 8.

310. Compound ratio is the ratio of the products of the corresponding terms of two or more simple ratios. Thus, The simple ratio of

9: 3 is 3;
And
" of

8: 4 is 2;
The ratio compounded of these is 72: 12 = 6.

OBs. Compound ratio is of the same nature as any other ratio. The term is used to denote the origin of the ratio in particular cases.

311. From the definition of ratio and the mode of expressing it in the form of a fraction, it is obvious that the ratio of two numbers is the same as the value of a fraction whose numerator and denominator are respectively equal to the antecedent and consequent of the giv

Quest.-308. What is direct or simple ratio ? 309. What is inverse or recipr ratio ? How is a red ocal ratio expressed by a fraction ? How by points ? 310. What is compound ratio ? Obs. Does it differ in its nature from other ratios ? 311. What in effect is the ratio of two numbers ?

en couplet; for each is the quotient of the numerator divided by the denominator. (Arts. 305, 110.)

Note.Hence, from the principles of fractions already established, we may easily deduce the following truths respecting ratios.

312. To multiply the antecedent of a couplet by any number, multiplies the ratio by that number; and to divide the antecedent, divides the ratio : for, multiplying the numerator multiplies the value of the fraction by that number, and dividing the numerator divides the value. (Arts. 111, 112.)

Thus, the ratio of 16 : 4 is 4;
The ratio of 16 x2:4 is 8, which equals 4x2;
And
16--2 : 4 is 2,

4:2. 313. To multiply the consequent of a couplet by any number, divides the ratio by that number; and to divide the consequent multiplies the ratio : for, multiplying the denominator divides the value of the fraction by that number, and dividing the denominator multiplies the value. (Arts. 113, 114.)

Thus the ratio of 16 : 4
The

16 : 4X2 is 2, which equals 4-2 ; And

16 : 4-2 is 8, which equals 4x2. 314. To multiply or divide both the antecedent and consequent of a couplet by the same number, does not alter the ratio: for, multiplying or dividing both the numerator and denominator by the same number, does not alter the value of the fraction. (Art. 116.)

Thus, the ratio of 12:4 is 3;
The

12 X2 : 4X2 is 3;
And

12=-2 : 4:2 is 3. 315. If the two numbers compared are equal, the ratio is a unit or 1: for, if the numerator and denominaQuest.–312. What is the effect of multiplying the antecedent of a

plet by any number? Of div ding the antecedent? How does this appear? 313. What is the effect of multiplying the consequent by any number? Of dividing the consequent? 314. What is the effect of multiplying or dividing both the antecedent and consequent by the same number?

is 4 ;

66

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