EXERCISES IN ANALYSIS.*

1. If 11 tons of coal cost $99, what will 73 tons cost?
SOLUTION.
If 1 ton cost $9,75 tons will cost 75 times $9, or $68.
2. If 20 barrels of flour cost $210, what will 27

If 11 tons cost $99, 1 ton will cost

of $99, or $9. Therefore, etc. barrels cost? Ans. $283.

3. If 27 barrels of flour cost $283, what will 20 barrels cost?

4. If of a yard of cloth cost $2.80, what will 5 yards cost?

SOLUTION. If of a yard of cloth cost $2.80, of a yard will cost of $2.80, or $.70, and 7, or 1 yard, will cost 7 times $.70, or $4.90. If 1 yard cost $4.90, 54 yards will cost 5 times $4.90, or $28.42. Therefore, etc.

5. If of a pound of tea cost $.60, what will 5534 pounds Jost? Ans. $442.60. 6. When of an acre of land cost $75, what will 7 acres cost? Ans. $1560.

7. If 7 acres of land cost $1560, what will of an acre cost?

8. If $2.80 will buy of a yard of cloth, how many yards will $28.42 buy?

SOLUTION. If $2.80 will buy of a yard of cloth, of $2.80, or $.70, will buy ‡ of a yard, and 7 times $.70, or $4.90, will buy 7, or 1 yard.

If $4.90 will buy 1 yard of cloth, as many yards can be bought for $28.42 as $4.90 is contained times in $28.42, or 5 yards. Therefore,

etc.

9. If $7 will buy 53 bushels of rye, how many bushels will $15 buy? Ans. 1138 bushels. 10. If $5.60 will pay for 3 of a ton of coal, what part of a ton will $5.40 purchase? Ans. of a ton.

REVIEW QUESTIONS. How are fractions reduced to equivalent fractions having a common denominator? (149) The Rule for Addition of Fractions? (152)

* Optional.

11. When 19 pounds of coffee cost $114, how many pounds can be obtained for $23? Ans. 4 pounds.

12. When $3311 will pay for 4§ barrels of flour, how much can be purchased with $27.50?

13. When 4 tons of hay will suffice certain time, for how many horses will 7 same time?

Ans. 33 barrels.

for

11 horses for a tons suffice for the Ans. 18 horses.

14. If A can do a piece of work in 7 days, and B the same work in 5 days, in what time can both do it by working together?

SOLUTION. If A can do a piece of work in 7 days, he can do of it in 1 day; and if B can do the same in 5 days, he can do of it in 1 day.

If A can do of it in 1 day, and B in 1 day, they can, by working together, do +, or 1, of it in one day.

If by working together they can do 1 of the work in 1 day, they can do of it in of a day, and 35, or the whole, in 35 times of a day. which is of a day, or 21 days. Therefore, etc.

15. A man can trench a garden in 13 days, and his son can do the same in 10 days; in what time can both working together do it? Ans. 5 days.

16. A can mow of a field in a day, and B ; in what time can both, by working together, mow it?

17. A cistern has 3 pipes; the first will fill it in 10 hours, the second in 15 hours, and the third in 16 hours. What time I will it take them all to fill it? Ans. 4 hours.

18. In an orchard, of the trees bear apples, pears, and the remainder, which is 38, cherries. trees are there in the orchard?

peaches,

How many

SOLUTION. Since of the trees bear apples, peaches, and pears, 1 + 1 + 4 or 8, bear apples, peaches, and pears, and 185, or the whole orchard, less, or 1, must bear cherries.

REVIEW QUESTIONS. The Rule for Subtraction of Fractions? (155) General Rule for Multiplication of Fractions? (162) Division of Fractions?

(169)

*This page is optional.

If the 38 trees bearing cherries are 1 38 trees, or 2 trees, and if 2 trees are must be 105 times 2 trees, or 210 trees.

of the orchard, is of 11, 105, or the whole orchard, Therefore, etc.

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19. A person having bought a house for a certain sum, found that after he had paid one-half and one-third of the cost, he then owed $400. How much had he paid?

20. A farmer had 3 of his sheep in one field, in a second field, and the remainder, 75, in a yard. How many had he in each of the two fields?

21. A, B, and C $2000 of the cost.

Ans. In the first, 135; in the second, 150.

purchase a mill; A pays , B %, and C
What are the sums paid by A and B ?
Ans. By A, $903; by B, $1161.

22. A and B together have $136, and of A's money is equal to of B's. How much has each?

SOLUTION. Since of A's money equals of B's,

of A's equals

of, or, of B's, and §, or the whole of A's, equals 3 times & of B's, or of B's.

If A's money equals & of B's, as B's must equal & of itself, A and B together have + §, or 17, of B's money.

If 17 of B's money is $136,

is of $136, or $8, and §, or A's money, is 9 times $8, or $72, while §, or B's money, is 8 times $8, or $64. Therefore, etc.

23. The sum of two numbers is 350, and 3 of the larger number is of the smaller; what are the numbers?

24. My carriage is worth 2 both together are worth $420.

Ans. 200 and 150.

times as much as my horse, and What is the value of each? 25. A gentleman bought a horse, chaise, and harness, for $300. The chaise cost as much as the horse, and the harness as much as the chaise and horse both. What was the cost of each ? Ans. Horse, $150; chaise, $75; harness, $75.

REVIEW QUESTIONS.

What is meant by the Relations of Numbers?

(172) What is the Rule for finding what part one number is of another?

(174)

This page is optional.

DECIMAL FRACTIONS.

175. If a unit be divided into ten equal parts, each of these parts will be 1 tenth.

If each tenth be divided into ten equal parts, each part will be of, or 1 hundredth.

If each hundredth be divided into ten equal parts, each part will be of roo, or 1 thousandth.

In like manner, we may, by dividing by ten, continue to obtain fractions, each of whose values is one tenth of the fraction preceding it; such fractions are called Decimal Fractions. Hence,

176. A Decimal Fraction is a fraction whose unit is divided into tenths, hundredths, thousandths, etc.

A decimal fraction, for brevity, is usually called a decimal.

NOTATION AND NUMERATION.

177. Decimal fractions are commonly written without the denominator, and distinguished from whole numbers by having the decimal point (.) at the left.

The figures at the right of the point are called decimal figures.

The first order to the right of the decimal point expresses tenths.

Thus,

The second order to the right of the decimal point expresses hundredths. Thus,

When a unit is divided into ten equal parts, what is each of the parts called? What is a Decimal Fraction? What is a decimal fraction usually called? How are decimal fractions usually written? What does the first order at the right of the point express ? Second?

ro may be written .01, and read 1 hundredth;

The third order to the right of the decimal point expresses

thousandths. Thus,

Tobo may be written .001, and read 1 thousandth;

The fourth order to the right expresses ten-thousandths; the fifth hundred-thousandths, and so on. Hence, the following

GENERAL PRINCIPLES.

1. The value expressed by decimal figures is determined by the place of each with reference to the decimal point.

2. The denominator of a decimal is understood to be 1, with as many ciphers annexed as there are orders in the decimal expression.

3. Ten of any lower order of decimals are always equal to one of the next higher.

178. A Mixed Number may be a whole number and a decimal expressed together, with the decimal point between them. Thus,

5.34, read five units and thirty-four hundredths, is a mixed number.

179. A whole number may be regarded as a decimal by placing the decimal point on the right of the order of units; and the expression may be read, according to the decimal

What does the third order at the right of the point express? Fourth? How is the value expressed by a decimal determined? What is the denom inator of a decimal? How many of one decimal order make one of the next higher? How may a mixed number be expressed decimally? How may a whole number be regarded as a decimal?