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divisor, write a cipher in the quotient, and annex another figure.

LESSON CXXIX. 1. How many times is 13 contained in 4,056 ? 2. How many times is 15 contained in 5,475? 3. How many times is 16 contained in 6,804? 4. How many times is 18 contained in 9,221? 5. How many times is 21 contained in 8,757 ? 6. What is 1 thirty-fifth of 75,445? Of 150,890 ? 7. What number of times is 125 contained in 46,875 ? 8. What number of times is 276 contained in 89,700 ? 9. What number of times is 365 contained in 147,825 ? 10. If a man's income be 5,110 dollars a year, what is

that a day, allowing the year to contain 365 days ? 11. There are 144 square inches in 1 square foot; how many square feet there in 10,800 square

inches? 12. In one acre there are 160 square rods; how many

acres are there in 28,000 square rods ? 13. A man purchased a farm containing 275 acres for 6,875 dollars; how many dollars did he pay

for each acre ? 14. The circumference of the earth is about 25,000 miles;

if a railroad car travel at the rate of 480 miles a day, in how many days would it travel round the earth?

are

LESSON CXXX. When the divisor does not exceed 12, the operation may be shortened by performing the operation mentally, and writing only the quotient.

Illustration. - If 109,025 acres of land be equally divided

among 12 men, what number of acres will each man have? 12) 109,025

The two highest'orders of figo Ans. 9,085 acres. ures being less than the divisor, we take the three highest, 109 (thousand), and find that 12 is contained in 109 (thousand) 9 (thousand) times; we write the 9 under the order of thousands in the dividend, and, mentally, multiply the divisor by it; the product is 108 (thousand), which we subtract from 109 (thousand); the remainder is 1 (thousand); this remainder and next figure of the dividend, 10 (hundred), being less than the divisor, we write a cipher in the place of hundreds in the quotient: we then find that the divisor is contained in the remainder and two next figures of the dividend, 102 (tens), 8 times; we write 8 in the quotient, in the place of tens, and multiply the divisor by it; the product is 96 (tens), which we subtract from 102 (tens); the remainder is 6 (tens): we then find that the divisor is contained in this remainder and last figure of the dividend, 65, 5 times; we write 5 in the quotient, in the place of units ; we then multiply the divisor by it, and the product is 60, which we subtract from 65; the remainder is 5: this remainder will contain the divisor 5 twelfths

1 time, which we place at the right of the quotient figures before found, and the complete quotient is 9,085, the number of acres which each man will have. 1. Divide each of the following numbers by 2 and 4.

446; 682 ; 566; 874; 916; 1,276; 2,548. 2. Divide each of the following numbers by 4 and 6.

448; 816; 564; 728; 824; 936; 1,272. 3. Divide each of the following numbers by 6 and 8.

324; 648; 786; 672 ; 846; 936; 1,236. 4. Divide each of the following numbers by 8 and 9

848; 728; 928; 1,688; 1,806; 1,248; 1,848. 5. Divide each of the following numbers by 9 and 10.

819; 729; 1,242; 1,539; 1,863; 2,745. 6. Divide each of the following numbers by 10 and 11.

540; 624; 850; 1,260; 1,580; 2,460. 7. Divide each of the following numbers by 11 and 12.

372; 612; 720; 840; 1,800 ; 1,728. 8. If 1,296 dollars bé equally divided among 9

men, what number of dollars will each man receive ? 9. If 12 bales of cotton, each of the same size, weigh FRACTIONS.

INTRODUCTORY LESSON.

Definitions and Illustrations. WHEN any whole thing or number is divided into any number of equal parts, the parts are called fractions of the thing or number.

If an orange be divided into two equal parts, the parts are called halves. If it be divided into three equal parts, the parts are called thirds. If it be divided into four equal parts, the parts are called fourths. If it be divided into five equal parts, the parts are called fifths. If it be divided into six equal parts, the parts are called sixths. If it be divided into seven equal parts, the parts are called sevenths. If it be divided into eight equal parts, the parts are called eighths. If it be divided into nine equal parts, the parts are called ninths. If it be divided into ten equal parts, the parts are called tenths. If it be divided into eleven equal parts, the parts are called elevenths. If it be divided into twelve equal parts, the parts are called twelfths. The number of equal parts into which any whole thing or number is divided always indicates the name of the parts.

Fractions are usually expressed by two numbers written one above the other, with a short line between them;

吞。 Dinthe, is tenths, ti eleventhe, 13 e weirens.

The number below the line expresses the number of equal parts into which the whole thing or number is supposed to be divided ; and it is called the denominator, because it indicates the name of the parts, as thirds, fourths, fifths, &c.

The number above the line is called the numerator, because it shows the number of equal parts expressed by the fraction.

thus : 28

two

3 three

four

5 five

six

7seven

1 one

2 half,

nine

eight

LESSON I. 1. In a unit, or 1, there are 2 halves. How many halves are there in 2? _In 21 ? In 4?

In 41? In 6? In 6!? In 8? In 81? In 10? In 101? The 2 halves of any whole thing or unit are equal to the whole thing or unit; hence, in any number of ħalves, there must be 1 half as many whole ones or units as there are halves. 2. How many whole ones are there in 4 halves ? In 5

halves? In 8 halves? In 9 halves? In 12 halves ? In 13 halves? In 16 halves? In 17 halves? In

20 halves? In 21 halves? In 24 halves ? 3. In a unit, or 1, there are 3 thirds. How many thirds

are there in 2 ? In 21 ? In 4? In 43? In 6? Since the 3 thirds of any whole thing or unit are equal to the whole thing or unit, there must be 1 third as many whole ones or units as there are thirds. 4. How many whole ones are there in 6 thirds? In 7

thirds? In 12 thirds ? In 14 thirds ? In 18

thirds? In 19 thirds? In 24 thirds ? 5. In a unit, or 1, there are 4 fourths. How many

fourths are there in 3? In 54? In 7? Since 4 fourths are equal to a whole one, there must be 1 fourth as many whole ones as there are fourths. 6. How many whole ones in 12 fourths? In 21 fourths ?

In 28 fourths? In 38 fourths ? 7. In a unit, or 1, there are 5 fifths. How many

fifths are there in 2? In 31 ? In 4? In 5 ? In 6? Since 5 fifths are equal to a whole one, there must be 1 fifth as many whole ones as there are fifths. 8. How many whole ones are there in 10 fifths? In 16

fifths? In 20 fifths? In 27 fifths? In 30 fifths ? 9. In a unit, or 1, there are 6 sixths. How many sixths

are there in 3? In 41? In 5? In 63? In 7 ? Since 6 sixths are equal to a whole one, there must be 1 sixth as many whole ones as there are sixths. 10. How many whole ones are there in 18 sixths ? In 25

In 94 ?

4

MULTIPLICATION AND DIVISION OF FRACTIONS.

85

How many

LESSON II. 1. In a unit, or 1, there are 7 sevenths.

sevenths are there in 4? In 54 ? In 6? In 7%?

In 8? In 9 ? In 10 ? In 12? Since 7 sevenths are equal to a whole one, there must be 1 seventh as many whole ones as there are sevenths. 2. How many whole ones are there in 28 sevenths ?

In 36 sevenths ? In 42 sevenths ? In 51 sevenths ? In 56 sevenths ? In 66 sevenths ?

In 84 sevenths ? 3. In a unit, or 1, there are 8 eighths ? How many

eighths are there in 5? In 63? In 7? In 83?

In 9? In 10g? In 11 ? In 12? Since 8 eighths are equal to a whole one, there must be 1 eighth as many whole ones as there are eighths. 4. How many whole ones are there in 40 eighths? In

51 eighths? In 56 eighths? In 69 eighths? In 72 eighths? In 86 eighths? In 88 eighths ? In

96 eighths ? 5. In a unit, or 1, there are 9 ninths. How many ninths

are there in 6? In 73? In 8? In 9f? In 10 ?

In 105 ? In 11 ? In 12 ? Since 9 ninths are equal to a whole one, there must be 1 ninth as many whole ones as there are ninths. 6. How many whole ones are there in 54 ninths? In

64 ninths? In 72 ninths? In 84 ninths? In 90 ninths ? In 95 ninths? In 99 ninths? In 108

ninths? 7. In a unit, or 1, there are 10 tenths. How many

tenths are there in 5? In 61? In 8? In 90?

In 10? In 114? In 12? Since 10 tenths are equal to a whole one, there must be 1 tenth as many whole ones as there are tenths. 8. How many whole ones are there in 50 tenths ? In

63 tenths? In 80 tenths? In 96 tenths ? In 100 tenths? In 119 tenths? In 120 tenths ?

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