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RECIPROCALS OF NUMBERS.

47. The reciprocal of a number is the result obtained by dividing 1 by the number. Thus, the reciprocals of 2 3, 4, and 5, are }, }, 4, and . From this we discover that the reciprocal of an integer, or whole number, is equal to a vulgar fraction whose numerator is 1, and whose denominator is the given number.

The reciprocal of it is foui.d by dividing 1 by t; which (Art. 46,) is 1--=1x=.

In the same way we find the reciprocal of 1 to be 4, and in general, the reciprocal of a vulgar fraction is the value of the fraction when inverted.

Note. From this, we see that dividing by any number is in effect the same as multiplying by the reciprocal of that number. So that operations of division may be included under those of muitiplication. A practical application of this principle may be seen under Reduction of Denominate Fractions. (Art. 89.)

EXAMPLES.

1. What are the reciprocals of 7, 8, 9, 10, 11 ?

Ans. 7, }, }, to, te 2. What are the reciprocals of 18, 23, and 41 ?

Ans. tor ta, itt 3. What are the reciprocals of 3, 4, 5,? Ans. 1, 4,4,. 4. What are the reciprocals of 14, 25, 37? Ans. 5, 4, 4s 5. What are the reciprocals of 4 of 4, § of } ?

Ans. 4 of , of 4. 48. EXERCISES IN VULGAR FRACTIONS. 1. Reduce to its lowest terms.

Ans. H. 2. Reduce to its lowest terms.

Ans. B. 3. Reduce 1 to its lowest terms.

Ans.

4. Reduce by to its lowest terns.

Ans. . 5. Reduce 317 to its lowest terms. Ans. 4. 6. Reduce 772 to its lowest terms. Ans. 11 7. Reduce 15636 to its lowest terms. Ans. 8. Reduce me to its lowest terms. Ans. jos: 9. Reduce 1999. to its lowest terms. Ans. 10. Reduce 515 to a mixed number. Ans. 11őt 11. Reduce 3,7 to a mixed number. Ans. 7 12. Reduce 4.8 to a whole number.

Ans. 8. 13. Reduce il to a mixed number. Ans. 377. 14. Reduce ?,377 to a mixed number. Ans. 2557. 15. Keduce 107 to a mixed number. Ans. 1947. 16. Reduce 3 to an improper fraction. Ans. 7. 17. Reduce 151} to an improper fraction. Ans. 29.6. 18. Reduce 317 to an improper fraction. Ans. 19. 19. Reduce 1 Tot to an improper fraction. Ans. H. 20. Reduce 10017 to an improper fraction. Ans. 474. 21. Reduce 1 off of to its simplest form. Ans. 1. 22. Reduce o of of il to its simplest form. Ans. to 23. Reduce 7 of 1 of of 3 to its simplest form. Ans. 16. 24. Reduce to off of it of 3} to its simplest form.

Ans. 25. Reduce 4 of 14 of of 100 to its simplest form.

Ans. 200. 26. Reduced, , 4, to equivalent fractions having a com. mon denominator.

Ans. 19, tr, . 27. Reduce , }, }, }, }, to equivalent fractions having a common denominator.

Ans. 39, é!, 16, 1%, 1%. 28. Reduce 37, \, \, , to equivalent fractions having a common denominator.

Ans. 7, ži, L., zm. 29. Reduce }, }, }, tr, to equivalent fractions having a common denominator. Ans. 136, 14:10: Lions

30. Reduce }, , tu ti, to equivalent fractions having a common denominator. Ans. 2003, 3675, 3485, $335.

31. What is the sum of }, }, { ? Ans. 13=11. 32. What is the sum of , , ? Ans. 143=273.

33. From a piece of cloth } and į of the whole was cut auf. What part of the whole was thus taken away?

Ans. 34. From 1 subtract ļ.

Ans. . 35. From 'o subtract 1:

Ans. ila. 36 From subtract 6.

Ans, no 37. A tree 150 feet high had broken off in a storm. What was the length broken off ? Ans. 30 feet.

38. A and B together possess 1477 sheep, of which A owns and B 4. How many belong to each man?

Ans SA's, 844..

B's, 633. 39. A owns of a ship, valued at $15422; he sells to B of his share. What is the value of what A has left; also, what is the value of B's part?

Ans.
SA's remaining part is $1402.

B's part is $2804. 40. A cotton mill is sold for $30000, of which A owns of the whole, B and C each own } of of the whole. How many dollars does each one claim ?

[A claims $6000. Ans. B claims $5000.

iC claims $5000. 41. A and B have a melon, of which A owns , and B b; C offers them one shilling, to partake equally with them of the melon, which was agreed to. How must the shilling be divided between A and B ? A S A must have } of it.

* B must have of it. 42 A farmer had į of his sheep in one field, 1 in a

second field, and the residue, which was 779, in a third field How many sheep had he in all ? Ans. 1230 sheep.

43. If I divide 616 dollars between A, B, C, and D, by giving A of the whole, B t of the remainder, C of what then remained, and D the balance, how much will each receive ?

SA had 154 dollars

B 66 165 66
Ans. c 6 264 bit

(D 6 33 :

DECIMAL FRACTIONS.

49. A Decimal Fraction is that particular form of Fraction, whose denominator consists of a unit, followed by one or more ciphers.

Thus : to, Ho, TOO, PO, Tod, Too, Totoo, &c., are Decimal Fractions.

In practice, the denominators of Decimal Fractions are not written, but always understood.

The above Decimal Fractions are usually written as follows: 0.1, 0.3, 0·04, 0.37, 0·08, 0·003, 0·0047, &c.

The period, or decimal point, serves to separate the decimals from the whole numbers.

The first figure on the right of the decimal point, is in the place of tenths; the second figure is in the place of hurdredths; the third figure in the place of thousandths, and so on; the value of the units of the successive figures decreasing from the left towards the right, in a tenfold, ratio, as in whole numbers. The following table will exhibit this.

NUMERATION TABLE OF WHOLE NUMBERS AND DECIMALS.

&c.
e Tens of Billions.
w Hundreds of Millions.

&c.
w Hundreds of Thousands.
w Tens of Millions.
W Tens of Thousands.
w Billions.
w Thousands.
w Hundreds.
w Millions.
• Decimal point.
e Hundred Thousandths.
e Ten Thousandths.
w Thousandths.
w Hundredths.
w Hundred Millionths.
w Tenths.
w Units.
e Ten Millionths.

Millionths.
e Tens.
e Ten Billionths.
e Billionths.

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3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

Ascending. 2 K Descending. This table is in accordance with the French method of numeration (Art. 6,) where each period of three figures changes its name and value.

Since decimals, like whole numbers, decrease from the left towards the right in a ten-fold ratio, they may be connected together by means of the decimal point, and then operated upon by precisely the same rules as for whole numbers, provided we are careful to keep the decimal point always in the right place.

Annexing a cipher to a decimal does not change its value, because it is the same as multiplying its numerator and denominator by 10. Thus : 0.3=0·30=0.300=&c. But prefixing a cipher is the same as removing the decimal figures one place farther to the right, and therefore each cipher, thus prefixed, reduces the value in a ten-fold ratio. Thus: 0-3 is ten times 0:03, or a hundred times 0:003.

0-2 is read two tenths.
0.25 6 twenty-five hundredths.
0-365 6 three hundred and sixty-five thou:

sandths.
0-105 66 one hundred and five thousandths.
0.03 • three hundredths.

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