« ΠροηγούμενηΣυνέχεια »
108. How many guineas are equal in value to 2050 Spanish dollars, at 4s 6d each 2
109. How many cwt. long weight in 145ct 14r 141bs common weight?
111. In 245 ells English, 34r 2n; how many yards, qrs and nails?
portions of 2a 3r each ; how many such will it make, and what will remain Po
118. From the 19th December 1819, till the 29th of September in the following year, both days inclusive; how many days and weeks?
119. If a sailor serve from 1st of February, 'till the 20th of October following; how many months, each month 28 days?
120. Divide 351 between 2 men, and give one 5 guineas more than the other?
121. If in the City of Dublin there are 250,000 individuals, and each consume + fö of flour when made into bread, each day; what quantity would be sufficient to last them 7 days?
124. In 45ct 34r 15Ib of oats; how many barrels, stones and pounds?
* 117. Bring the given number of acres, to roods, which, divide by 11 the number of roods in 2a 3r. the remainder, if any, will be roods.
120. Divide 351 by 2, also divide 5 guineas } 2; the two quotients added together, will give the greater share, and this subtracted from the given sum, will give the less share.
If any whole thing, a foot, a yard, a pound, &c. is divided into three equal parts, any of those parts is one-third of the whole, written thus, #. If two of them are taken, they are two-thirds, or #: such expressions are called fractions : the number above the line is called the numerator of the fraction, and the number below the line the denominator.
A proper fraction is that whose denominator is greater than its numerator, If the numerator be equal to the denominator, or greater, the fraction is called improper.
The denominator always denotes the number of equal parts into which the whole thing or integer is conceived to be divided, The numerator denotes the number of those parts which are taken in the fraction; thus, the fraction + intimates that the integer is divided into 7 equal parts, and that we take three of those parts in the fraction.
Any improper fraction, whose numerator and denominator are equal, such as #, #, #, &c. is equal to one integer, which we suppose to be divided into equal parts, Thus, if a pound is divided into 7 equal parts, and that 7 of those parts are taken, the whole pound is taken, neither more nor less. It is therefore, manifest, that % or any proper fraction is less than the whole, and that +, or any improper fraction, whose numerator is greater than its denominator, is greater than the whole. Here it is to be understood, that one whole thing divided as one integer is meant, whether a pound, a 'foot, a yard, &c. or ever so many pounds, feet, yards, &c.
When we consider any fraction, such as #, we conceive it ‘to be two-thirds of one. But there is another view, to which it will be proper to attend. It may be considered as the third part of 2, for as two-thirds are twice as great as one-third, and 2 twice as great as one, it is evident that two-thirds of 1 and one-third of 2, are equivalent expressions. In like manner, the fraction 3 may be considered, either as three-sevenths of one, or as the seventh part of 3; the latter being three times greater than the seventh-part of one. Thus, any fraction may be considered as a quotient, arising from the diviision of its numerator by its denominator, and hence fractional notation is commonly employed to express division.
The value of a fraction increases or lessens, directly as the numerator is increased or lessened, (the denominator remaining unchanged) or it increases or lessens as the denominator is lessened or increased, (the numerator remaining unchanged) thus # of any one whole thing, is evidently twice as much as # of the same, but + is also evidently twice as great as or of the same whole thing. The value of a fraction remains unaltered, if both its terms are multiplied by the same number; that value depending altogether on the comparative magnitude of its terms, and not on their absolute greatness or smallness. Thus, the fraction # is equal to the fraction # or +, or #3 respectively, for " comparing any one with the other, as the fractions # and #3 in the latter, the whole thing is conceived to be divided into 10 times as many equal parts as in the former, each of which is, therefore, ten' times less than the former; consequently, if we take ten times as many of them as of the former, we shall take just the same quantity of the whole; for example, as a twelfth part of a foot being one inch, or of a foot is 6inches, but that is just half a foot, or the fraction #, or one-half; and if both the terms of a fraction be divided by a common divisor, the value of the fraction remains still unaltered, as in the former case. A simple fraction contains one or many parts of some one whole, as one-fourth of a yard, three-fourths of a pound, five. sixths of a foot, &c. o - & compound fraction is otherwise called a fraction of a fraction, as two-thirds of three-fourths of a pound, or twenty shillings; which is the same as to say two-thirds of fifteen ` shillings. A complex fraction is that which has a fraction either' in its numerator or denominator, or in each of them. Thus, 3. #. # and # are complex fractions. The same operations can be performed on fractional as on integral quantities. Before entering on these operations, it is proper to show how such quantities may be modified, without changing their value, so as to fit them for the several operations to be performed on them, for the different uses to which they are to be applied. This will constitute reduction of fractions. For a demonstration of this rule, and other observations on frattions, see John Walker's Philosophy of Arithmetic, Rages 41 to 44. L
- REDUCTION OF FRACTIONS. First preparatory problem.
To find the greatest common measure of two numbers, that
is, the greatest number which will evenly divide each of them. IRU Alo, o
Divide the greatest number by the least, if there is no remainder, the least is the greatest common measure, as no number greater than itself can be a measure of the same number; but if there is a remainder after the first division, then divide the divisor by that remainder; and so on, the last divisor by the last remainder, until a remainder is found, which will measure, or exactly divide the last divisor; this will be the greatest common measure required. But if l is the ilast remainder, one is the greatest common measure, and the numbers are said to be prime to each other, and are already in their lowest terms. -
€ramples. . s What is the greatest common measure of 64 and 144. 64)144 * 2...... 16)64 4
Here 16 is the greatest common measure, being the last remainder which will measure the former divisor.
Find the greatest common measure of the following pairs of numbers, each pair respectively.
1. 27 and 135 2. ' 14 and 98
11. 522 and 5436 12. 843 and 6414 , 13. 405 and 3875 14. . 327 and 693 15, 81 and 405 16. 432 and 1224 17. , 64 and 384 18. 135 and 1240 . 19. 132 and 1728 20. 415 and 1045 21. 672 and 1236 22. 3164 and 8162 23. 7314 and 9999 24. 2857.14 and 999999
For a demonstration of this rule, see John Walker's Philosophy of 4rithmetic, page 87.
Second preparatory problem.--To find the least common multiple of two or more numbers, that is the least number into which each of the given numbers may be evenly divided.
Place the numbers in a row, after each other, with a point or comma between each; then see what divisor will measure the greatest number of them; divide the numbers thereby, and place the quotients respectively under each number so divided, and bring down any number or numbers which the first divisor did not measure orderly in a row, with the quotients of the numbers already divided; divide this row as before, by the divisor, which will measure the greatest number of them, and proceed as before ; and so on, until there be no two numbers which can be divided by a common divisor, then the continual product of all the divisors, and the last quotients, will give the least common multiple required.
What is the least common multiple of the numbers, 4, 6, and 8, and 3, 5, 6, 10 and 12, each set respectively?
Find the least common multiple of the following sets of numbers, each set respectively.
1, 3, 9, . and 12 2. 5, 9, 13 and 17 3. 2, 4, and 8 4. 4, 6, 8, 10 and 12 5. 5, 7, 42 and 10 6. 1, 3, 5, 7 and 9 7. 1, 9, 5 and 7 8. 2, 5, 11, 14 and 17 9. 3, 4, 5 and 6 10. 5, 9, 13, 17 and 21 11. 6, 12, 18 and 24 12. 2, 6, 10, 14 and 21 13, 2, 4, 8 and 16 14. 2, 7, 12, 17 and 22 15. 6, 5, 4 and 10 16. 1, 4, 9, 16 and 25 17. 4, 8, 16 and 32 18. 1, 8, 27, 64 and 125 19. 6, 12, 24 and 48 20. 1, 4, 7, 10, 13 and 16 21. The 9 digits. 22. 3, 7, 11, 15, 19 and 23
23. 1, 7, 21, 47 and 61 24. 1, 3, 5, 7, 9, ll and 13
Norr.--When no two of the given numbers can be divided by a common divisor, ths continual product of the given numbers will be the least common multiple,