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1. The prime factors of 45 are

and (a) What are the prime factors of 100? (b) Of 125 ? 2. Two, 3, and 5 are the prime factors of (c) Of what number are 3, 3, 2, and 7 the prime factors ? 3. Reduce 1% to its lowest terms. i = (d) Reduce go and 1 to their lowest terms. 4. Reduce 8 to an improper fraction. 8 = (e) Reduce 57 and 723 to improper fractions. 5. Reduce 38 to a mixed number. (f) Reduce 334 and 543 to mixed numbers.

6. Reduce s and to equivalent fractions having a common denominator. =

(g) Reduce and to equivalent fractions having a common denominator.

7. Multiply 60 by .7. This means, find 7 tenths of 60. One tenth of 60 = .; 7 tenths of 60 =

(h) Multiply $537 by .07. This means, find 7 hundredths of $537. (See page 43.)

8. At $1 per ton, 4240 lb. of coal cost
(i) Find the cost of 4240 lb. of coal at $7 per ton.

9. The volume of a rectangular solid 5 inches by 4 inches by 2 inches is - cubic inches.

() Find the volume of a rectangular solid 9 inches by 7 inches by 7 inches.

10. If 9 lb. of tea are worth $6, 12 lb. are worth

(k) If 9 acres of land are worth $346.50, how much are 12 acres worth at the same rate ? (12 is 14 times 9.)

MISCELLANEOUS PROBLEMS.

1. From June, 1881, to June, 1899, it is years. (a) How many years from Aug., 1492, to Aug., 1897 ?

2. A man sold a horse for one hundred twenty dollars; this was seventeen dollars and twenty cents more than the horse cost him; the horse cost him

(b) A man sold a farm for fourteen thousand seven hundred fifty dollars ; this was eight hundred seventy-five dollars more than he paid for the farm. How much did the farm cost him?

3. From 5460 lb. of coal there were sold 2 tons. pounds remained.

(c) From 18940 lb. of coal there were sold 8 tons. How many pounds were left?

4. From Mendota to Galesburg it is 80 miles; a train going 30 miles an hour, that leaves Mendota at 8:30, should arrive at Galesburg at

(d) From Chicago to Denver it is about 1000 miles. If a train leaves Chicago for Denver at 9 o'clock Monday morning and goes at the rate of 30 miles an hour, when will it arrive at Denver ?

5. If butter is 25° a pound and coffee is 30¢ a pound 3 lb. butter will pay for lb. coffee.

(e) Fifteen and one half pounds of butter at 24¢ a pound, will pay for how many pounds of coffee at 32¢ a pound

6. If of a yard of lace is worth 15%, 24 yards are worth

cents. (f) If of a yard of cloth costs $1.05, how much will 271 yards cost ?

SIMPLE NUMBERS. *

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1. Four, 6, 8, 10, 12, etc., are multiples of 2.
2. Six, 9, 12, 15, 18, etc., are multiples of 3.

3. Twelve is a multiple of 2. 12 is also a multiple of 3, and of 4, and of 6.

4. Twelve is a common multiple of 2, 3, and
5. Fifteen is a common multiple of

and 6. Common multiples of 4 and 6 are, 12, 24, etc. The least common multiple of 4 and 6 is 12.

7. Common multiples of 6 and 8 are, 24, 48, etc. The least common multiple of 6 and 8 is 8. Common multiples of 8 and 12 are,

etc. The least common multiple of 8 and 12 is 9. Common multiples of 6 and 9 are,

etc. The least common multiple of 6 and 9 is

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10. The prime factors of 18 are

and 11. The prime factors of 70 are

and (a) What are the prime factors of 140. (b) Of 160 ? (c) Of 135 ? (d) Of 175 ? (e) Of 250 ? (f) Of 225?

Multiply.
Divide.

Divide. (g) 635 by 53. (h) 944 lb. by 56 lb.† (i) 1536 lb. by 12.1 (j) 728 by 54. (k) 846 lb. by 56 lb. (1) 1445 lb. by 12. (m) 834 by 52. (n) 739 lb. by 56 lb. (0) 1374 lb. by 12. (p) 947 by 51. (9) 873 lb. by 56 lb. (r) 1653 lb. by 12. , (s) 836 by 55. (t) 965 lb. by 56 lb. (u) 1738 lb. by 12.

* Do much oral work in preparation for this page. By using these terms, make the pupil as familiar with multiple, common multiple, and least common multiple, as he is with house, schoolhouse, and stone schoolhouse.

+ Tell the meaning. Tell a number story.

COMMON FRACTIONS.

11 = 24

194

L. C. m. is the abbreviation for least common multiple. 1. Add g and 1.1.

(a) + g. L. C. m. of 8 and 12 is

(b) +.

(c) 5 + H + 4 = 24

(d) $ +4. 2. From subtract .

(e) 6 - 1 L. C. m. of 9 and 6 is

(f) Ž-5.

- å 18 - 18 = 18

(h) 3. Divide 11 by š.

(i) } = L. C. m. of 12 and 8 is

(j) = .
13

(k) = 5
(1) = $.

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18

(g) 12

24*

27

3

24

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(s) 64 dollars)375 dollars. This means, find how many times

$64 are contained in $375. (Change 61 and 375 to fourths.) Story.

(t) 6)756 dollars. This means, find 1 sixth of $756. Story.

DECIMAL FRACTIONS.

1. One tenth of $432 is

2 of $432 2. One hundredth of $432 is

.02 of $432 (a) Multiply $432 by .25. This means, find 2 tenths of $432, plus 5 hundredths of $432. Operation.

Explanation. $432

One hundredth of $432 is .25

5 hundredths of $432 are

One tenth of $432 is $21.60*

2 tenths of $432 are $86.4 †

$21.60 + $86.4 = $108.00

NUMBER STORY.

If 1 acre of land is worth $432,

1 hundredth of an acre is worth
5 hundredths of an acre are worth
1 tenth of an acre is worth
2 tenths of an acre are worth
25 hundredths of an acre are worth

(b) Multiply $325 by .23. (c) $482 X.32.
(d) Multiply $278 by .43. (e) $356 x.36.
(f) Multiply $536 by .07. (g) $351 x .7.
(h) Multiply $249 by 2.6. (i) $426 X.45.
(j) (k)

(1)

(m) (n) Add. Subtract. Multiply. Divide. Divide. 8.75 56} 675 $.05)$6. 5)$6. 7.324 12.9 .36

* The pupil should understand that he multiplies $4.32 (not $432) by 5, and should be taught to write the decimal point in the partial product immediately after writing the tenths' figure, 6, of the partial product.

+ The pupil should understand that he multiplies $13.2 (not $432) by 2, and should be taught to write the decimal point in the partial product immediately after writing the tenths' figure, 4, of the partial product.

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