ADDITION.

MENTAL EXERCISES.

§ XLVI. 1. George gave of an orange to his sister, to his brother, and then had himself. How many fourths had he at first?

2. William bought of a quart of chesnuts at one place, at another, and at another. How many sixths did he buy in all ?

3. A man has 2 of an acre of ground in his yard, in his garden, and in his corn lot. How many fourths

has he in all?

4. A man gave & of a bushel of rye to one person, to another, and to another. How many fifths did he give away?

5. How many eighths in 6. How many twelfths in +,},{?

++? In 7+8+8+f? tat? In

T2

3

In the above examples, the denominators are alike, and the process of Addition is very easy, since we only have to add the numerators. But when the denominators are different, we cannot add in this way. But, by § XLIV, we can make the denominators alike that is, we can reduce the Fractions to a common denominator, 7. Add and . } and }.

and }.

and .

and 1. † and 1.

and

and 7.

8. Add The following are to be written. It is always best to reduce Fractions to their lowest terms before adding, The sum may often be reduced to a whole or mixed number. When this cannot be done, it may often be reduced to lower terms.

and. A.

10. Add 2, 3 and 18.

81

A. 2.

11. Add %, 72 and 3. A. 2118.

26 60 120 158,

12. Add 1,, 18, 33 and 3. A. 241.

24 1,102,

When mixed numbers occur, it is plain that we must add the whole, and the Fractional parts separately. The two sums must then be united into one.

13. Add 13 and 53 13+5=18. &+2=18=11⁄25. 18 +1=19 A,

14. Add 17 and 164. A. 3311.

15. Add 15, 143, and . A. 30.

16. Add 7, 19, 1, 9, 252 and 1%.

A. 28319.

17. Add, 8, 259, 54788, 32938, 54 and 3. A. 1,19418. 18. Add 13, 17%, 30%, 99, 101, 204, 19, and 15. A. 216.

19. Add 1373, 26, and 243 4. A. 4072.

In this example, reducing to its lowest terms, it becomes which added to makes, or 1. It is therefore unnecessary to reduce and to the same denominator with 4. Such artifices may often be employed.

19

108

20. Add 33, 414, 324, 943, 16, and. A. 38. 21. Add 183, 13, 172, 11, 6,3 and 5. A. 59. 22. Add 131, 17, 238, 139, 14 and 5311. A. 102. 23. Add 4, 335, 10,5 727, 6,27, 84, 134, 247, 47, 57 and 85. A. 604. 51

19

30

MULTIPLICATION.

MENTAL EXERCISES.

XLVII. 1. If 1 dollar will buy of a bushel of wheat how many thirds will 2 dollars buy?

2. If 1 cent will buy of an orange, how much will 5 cents buy?

3. If 1 bushel of oats cost of a dollar, what will 2 bushels cost?

4. If 1 apple cost of a cent, what will 3 apples cost? 5. A boy gave of a pine-apple to one of his com panions, and 3 times as much to another, how much did he give the last?

6. 4 times 7. 3 times

8. 10 times 9. 5 times

are how many 6ths? 7 times ?
are how many 5ths? 6 times } ?
are how many 4ths? 5 times?
are how many 3ds? 6 times?

In the above examples, a Fraction is the multiplicand, and a whole number the multiplier; in other words, it is required to multiply a Fraction by a whole number. As Multiplication is only a repeated Addition of the multiplicand, (§ x1.) it is plain, that, in this case, it may be performed like Addition of Fractions, in the last section. Thus,

10. A man had 2 of a dollar in each pocket, and he had 4 pockets. How many fourths of a dollar had he? Repeat, 4 times by Addition, thus, +++1=Y

Ans.

Hence it appears, that Multiplication of a Fraction by a whole number, may be performed by a repeated Addition of the Numerator of the multiplicand. But this repeated Addition is Multiplication of the Numerator. (§ XI.)

Hence, to multiply a Fraction by a whole number,

1. MULTIPLY THE NUMERATOR OF THE FRACTION BY THE WHOLE NUMBER.

This was also shown in § XLI. In the same §, likewise, it was shown, that dividing the denominator multiplies the value.

Hence, another rule.

II. DIVIDE THE DENOMINATOR OF THE FRACTION BY THE WHOLE

NUMBER.

NOTE. This is altogether the best rule, and should always be used when the denominator admits of Division.

In performing the following examples for practice, let the pupil divide the denominator, in all cases, where it is practicable. The results should be reduced, if possible, to whole or mixed numbers as in Addition.

11. Multiply 5 by 4. A. 25=11⁄2·

7853

24

12. Multiply 328 by 7,953. A. 328=328. Hence, A Fraction is multiplied into a number equal to its denominatør, by removing the denominator.

14. How much is 40 times A. 23.

120

13. How much is 27 times 3. 31.

27 is a composite number, having the factors 9 and 3. We may multiply by these factors, successively, instead of multiplying by 27 at once. (§ xI.) Now 9 will divide the denominator 45, though 27 will not. Therefore, 31 X9, and X3=18 A. This mode is often Convenient.

15. A man bought 12 yards of cloth at 5 yard. How much did the whole come to.

dollars a

A. $62.

To multiply a mixed by a whole number, multiply the whole, and fractional parts separately, and unite the products.

16. Multiply 35 by 42.

A. 129. 17. Multiply 7% by 40. A. 281.

18. Multiply by 471.

13 237

A. 2004.

19. Multiply by 138. A. 133.

000

20. What will 1233 hogsheads of wine

$1.23 per gallon? A. $115.34.

come to at

21. What will 43 tons of iron come to at $4.00 pr. cwt.? A. $350.

22. What will 83 hhds. of molasses come to at $0.48 pr. gal. A. $360.06%.

23. What cost 7 cwt. of sugar at $0.08 pr. lb. ? A. $63.66, 24. What cost 2313 pipes of brandy at $1.43 pr. gal.? A. $4.231.653

§ XLVIII. In the last section you were taught to multiply a Fraction by a whole number. Multiplying a whole number by a Fraction, is exactly the same thing, for, (§ xI.) it matters not which factor is made the multiplicand nor which the multiplier. This case was noticed, § xxxiii.

To this case belong such questions as these: what is of 24; what is of 40, &c. that is all questions where a whole is given to find a certain part.

EXAMPLES FOR PRACTICE.

1. Three men, A. B. and C. drew a prize in a lottery, of $25, 000.00, A. owned the ticket; B.; and C. . What amount ought each to have? Ans. A. of $25,000-$12,500. B. of $25,000-$8,333.33%, and C. 8 of $25,000-$4,166.66}.

2. For the same ticket was paid $16.00. What did each pay? Ans. A. $8.00. B. $5.33. C. $2.663. 3. A man, worth $32,750.25, wished to invest of it in bank capital. How much would he have left? A. $14,328.23,7. 4. At of a dollar a yard, what cost 164 yards of linen?

A. $123.00.

5. At 75 cts. a yard, what cost 164 yards of linen? 75 cts. is of a dollar; therefore the question is the same as the last.

6. At 50 cts a yard, what cost 230 yards of cambric ? ·

50 cts. is a dollar, and of 230-$115.00 A. 7. At 4d. a pr. pt. what cost 135 pts. of molasses ? 4d. is of a shilling, and of 135=459.=£2; 5, A. cts. pr. lb. what cost 15,618 lbs. of coffee? A. $5,206. of a dollar, and of 15,618=$5,206, A. cts. pr. lb. what cost 5,766 lbs. of sugar? A. $991. of a dollar, and of 5,766 $991.00 A.

8. At 33 33 cts. is 9. At 16

163 cts. is

A. £914.

10. At 25 cts. pr. lb. what cost 9,496 lbs. of raisins? A. $2,374. 25 cts. is of a dollar, and of 9,495 $2,374 A. 11. At 5s. a yard what cost 3,656 yds. of cloth? 12. At 6d. pr. lb. what cost 2,864 lbs. of sugar? A. £71; 12In the above examples, the price is an even part of a higher denomination, as a dollar, a shilling, &c. An even part is called an ALIQUOT part. Hence, when the price of a single thing, (as a gallon, pound, &c.) is an aliquot part of a higher denomination of money,

MULTIPLY THE QUANTITY GIVEN, BY THE FRACTION EXPRESSING

THAT PART.

NOTE. This rule may be found in other books, under the head of PRACTICE; which name is given it on account of its practical utility.

NOTE. When the price is not an aliquot part, but some other simple fractional part, (as in ex. 5 above,) that Fraction may be used as a multiplier in the same manner. The pupil's ingenuity will enable him to discover many abbreviations, which we have not room to mention.

13. At 5 cts. pr. lb., what cost 360 lbs. of pork? A. $18.00. 14. At 8 cts. pr. lb., what cost 6 bls. of beef? A. $100.00. 15. At 25 cts. pr. qt. what cost 478 gals. of brandy? A. $478.00. 16. At 6 cts. pr. pt. what cost a barrel of rum? A. $15.75.

MENTAL EXERCISES.

§ XLIX. 1. A man having divided a lot of ground into 4 equal parts, divided each part into 2 equal portions. What part of the whole lot was one of these portions? What is

of?

The figure represents the lot, cut by the black lines into 4 equal parts, or fourths. These fourths are each cut into 2 equal parts, or halves, by the dotted lines. It will be seen that there are 8 of these halves. But when a thing is divided into 8 equal parts, each part is called an eighth. of, then is. This is multiplying a Fraction by a Fraction.

2. A boy had 3 of an orange, and told his brother he would give him of what he had, if he could tell what part of the whole orange that would be. How much did he give him? What is of 2?

3. A boy had of an apple, and gave away of what he had. How much did he give away? What is of? 4. A man having of a bushel of oats, fed his horse of them. What part of a whole bushel did he give the horse? What is of?

5. A man owning of a bed of ore, sold of his share. What part of the whole did he sell? What is of 2?

6. A man owning of the capital of a trading establishment, sold of his share. What part of the whole capital did he sell? What is of 4 ?