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4. Reduce 16 to a fraction whose denominator is 9.
| In 1 unit there are 9-ninths; 9
therefore, there are 9 times as
many ninths as there are units in 144 Ans. 144
ġ any number. 5. Reduce 75 to a fraction whose denominator is 13. 6. Reduce 3 to a fraction whose denominator is 342. 7. How many fifteenths are there in 74 ? 8. How many eighths of a dollar in $647 ? 9. Reduce 364 to an improper fraction.
In this example, we add the 364
3-sevenths to the sevenths pro
duced by the multiplication of 36 256 Ans. 250 by 7, and thus obtain 250. 10. Reduce 25to an improper fraction. 11. Reduce 615 734 to an improper fraction. 12. How many sixteenths of a dollar in $ 54116?
CASE III. To reduce an improper fraction to a whole number, or a mixed number.
RULE. Divide the numerator by the denominator, and the quotient will be the whole, or mixed number. 13. Reduce 362 to a whole, or mixed number. 8)362
1 Since are equal to 1 unit,
there are as many units in 362 as 45 =45ị there are times 8 in 362. 14. Reduce 4993 to a whole, or mixed number. 15. How many units are there in 453,15 ? 16. How many dollars in 262 of a dollar ?
CASE IV. To reduce a compound fraction to a simple, or single fraction.
RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator: then reduce the new fraction to its lowest terms.
When any numerator is equal to any denominator, the operation may be abbreviated by rejecting both.
If part of the compound fraction be an integer, or a mixed number, it must first be reduced to an improper fraction.
17. Reduce of į of of 6 to a simple fraction.
. Here the common terrn, 3, is ģX X;Xi=32 = Ở *X*XXf=33=8 .
omitted in the multiplication. 18. Reduce 5 of į to a simple fraction. 19. Reduce I of of li to a simple fraction. 20. Reduce la of 13 of 2 ý to a simple fraction. 21. Reduce I of of of 5 to a simple fraction.
CASE V. To reduce a fraction from one denomination to another.
RULE. Multiply the proposed denominator by the numerator of the given fraction, and divide the product by the denominator of the given fraction; the quotient will be the numerator of the proposed denominator.
22. Reduce & to a fraction whose denominator shall be 14: or, in other words change 5-sixths to fourteenths.
1 Ž is equal to į of 14, and is :
5 times as much : we therefore 6)70
find 5 times 14-fourteenths and Ans. 112 take 1 of this product for the 114
14 required fourteenths. 23. How many fifths are there in ? 24. ^ is equal to how many twenty-fourths ? 25. Reduce to a fraction whose denominator is 4. 26. How many twelfths of 1 shilling in 4 of 1s.?
CASE VI. To reduce the lower denominations of a compound number to the fraction of a higher denomination.
RULE. Reduce the given quantity to the lowest denomination mentioned, and this number will be the numerator: then reduce a unit of the higher denomination to the same denomination with the numerator, and this number will be the denominator.
27. Reduce 7 oz. 18 dwt. 13gr. to the fraction of a pound.
We find, that 702. 18 dwt. 13gr. when reduced to grains, gives 3805 for the numerator; and 1 pound when reduced to grains, gives 5760 for the denominator. Therefore, 1986=965) is the fraction required.
28. Reduce 4s. 9 d. 3qr. to the fraction of £1.
29. Reduce 3 inches to the fraction of a yard. 30. What fraction of a hogshead is 9 gal. 2 pt.? 31. Reduce 5 cwt. 8 lb. 4 oz. to the fraction of a ton.
CASE VII. To reduce the fraction of a higher denomination to its value in whole numbers of lower denomination.
RULE. Multiply the numerator by that number of the next lower denomination which is required to make a unit of the higher, and divide the product by the denominator; the quotient will be a whole number of the lower denomination, and the remainder will be the numerator of a fraction. Proceed with this fraction as before, and so on.
It will be readily perceived, that the fraction of a higher denomination is reduced to the fraction of a lower, by multiplying the numerator by the number of units of the lower, required to make a unit of the higher. Thus, of a bushel is 4 times as many fifths of a peck; that is, of a peck. Again, of a peck is 8 times 12-fifths, that is, ac of a quart; and again, of a quart is 2 times 96-fifths, that is, 132 of a pint. If the denominator be multiplied, instead of the numerator, the effect is the reverse, and the fraction is reduced to a higher denomination. Thus, of a pint, (the 5 being multiplied by 2,) becomes io of a quart; io of a quart, (the 10 being multiplied by 8,) becomes of a peck; and so of a peck, (the so being multiplied by 4,) becomes to of a bushel.
32. Reduce 1 of a gallon to its value in quarts, &c. 11
We find by multiplication, that id of a gallon is 11 of a quart;
and, by division, that 11 of a 12)44
quart is 3qt. and is of a quart. 3 8
We then find, that of a qt. is
la of a pint; and, that ia of a pt. 12)16
is 1 pt. and 1 of a pt. And thus,
by finding the units of one de1 4
nomination at a time, we finally
obtain the whole answer, which, 12)16 denoted as a compound number,
11=1} \is 3 qt. 1 pt. 1įgi. 33. Reduce of £l to its value in shillings &c.
34. Reduce is of a yard, to its value in feet, &c.
CASE VIII. To reduce fractions to a common denominator; that is, to change two or more fractions which have different denominators, to equivalent fractions, that shall have the same denominator.
RULE Ist. Multiply each numerator into all the denominators except its own, for a new numerator. Then multiply all the denominators together for a new denominator, and place it under each new numerator.
RULE 2nd. Find the least common multiple of the given denominators for the common denominator; then divide the common denominator by each given denominator and multiply the quotient by its given numerator; the scveral products will be the several new numerators. (See PROBLEM X, page 23.)
The 1st. of the above rules is convenient when the terms of the fractions are small numbers, but the 2nd. is otherwise to be preferred, as it always gives a denominator which is the least possible. Other methods of finding a common denominator will occur to the student, after further practice.
If any of the fractions to be reduced to a common denominator be compound, they must first be simplified.
37. Reduce á, 12, 14 and i} to a common denominator.
In this example, the least common denominator is found to be 840. Then the several numerators of the common denominator are found as follows. 840; 8=105, and 105 X 5=525. Ans. ģ=525 840 +12= 70, and 70 X11=770. 840+14= 60, and 60 x 9=540.
14=240 840=15= 56, and 56 X 13=723. 1=748 38. Reduce , and to a common denominator. 39. Reduce
and to a common denominator. 40. Reduce 51 and to a common denominator. 41. Reduce and of 4 to a common denominator.
CASE IX. To reduce a complex fraction to a simple fraction.
RULE. If the numerator or denominator, or both, be whole or mixed numbers, reduce them to improper fractions: multiply the denominator of the lower fraction into the numerator of the upper, for a new numerator; and multiply the denominator of the upper fraction into the numerator of the lower, for a new denominator.
42. Reduce iz to a simple fraction. The operation. = =1X%= Ans. et 43. Simplify each of the following complex fractions.
4. . 51. 3. 2. 6. 2.
5 53 417 18
ADDITION OF FRACTIONS.
Fractions are added by merely adding their numerators, but they must be of the same integers; we cannot immediately add together of a yard and of an inch, for the same reasons that we cannot immediately add together 5 yards and 3 inches. They must, also, be of the same denomination; we cannot immediately add together fourths and fifths.
RULE. Reduce compound fractions, (if there be any)., to simple fractions, and reduce all to a common denominator; then add together the numerators, and place their sum over the common denominator. If the result be an improper fraction, reduce it to a whole or mixed number. 44. Add together, 3, 5, 8 3 and
By operations not here de360
noted, we find the common de
nominator to be 360; and also 225
find the several new numerators. 216
The sum of the fractions is 237 270 =23%, which, added to the
whole numbers, gives the total
=2360 sum, 133%.