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 1, 1, 1, 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 is found by dividing 1 by 4, which (ART. 46,) is 1÷3=1ק=3.

In the same way we find the reciprocal of 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 multiplication. 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. 1,,, to, tr.

2. What are the reciprocals of 18, 23, and 41 ?

Ans. 1, 2, tr.

Ans. 1, 1, 4, 8.

3. What are the reciprocals of †, 4, 4, k?
4. What are the reciprocals of 1, 2, 34?
5. What are the reciprocals of 4 of 4 of 7?

Ans. †,†4, 15.

Ans. 101 Ans. Fo

Ans. 1.

Ans. 7.

Ans. 8..

Ans. 3.

Ans. 2557.
Ans. 17

Ans. .

Ans. 20

Ans. 5.

Ans. If. Ans. 1741.

16. Reduce 31 to an improper fraction. 17. Reduce 1511 to an improper fraction. 18. Reduce 37 to an improper fraction. 19. Reduce 1 to an improper fraction. 20. Reduce 100++ to an improper fraction. 21. Reduce of of to its simplest form. 22. Reduce of off to its simplest form. Ans. †. 23. Reduce of off of 3 to its simplest form. Ans. 1%. 24. Reduce of 1 of 3 to its simplest form.

of

Ans. 4.

Ans..

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

Ans. 200.

Ans. 1, 1, 1.

26. Reduce,,, to equivalent fractions having a common denominator. 27. Reduce,,,,, to equivalent fractions having a common denominator. Ans. 38, 20, 15, 18, 18.

28. Reduce 3, 3, 4, , to equivalent fractions having a common denominator.

Ans. 18, 18, 15, 28.

29. Reduce,,,, to equivalent fractions having a common denominator. Ans. : 78, 79.

30. Reduce,,,, to equivalent fractions having

Ans. 2003, 3575, 3085, 1835.

50

a common denominator.
31. What is the sum of 1, 1, 1?

32. What is the sum of 3, 4, 5 ?
33. From a piece of cloth and

5005

Ans. 11.

Ans. 14-223.

of the whole was cut

off. What part of the whole was thus taken away?

broken off in a storm.

Ans. 30 feet.

37. A tree 150 feet high had

What was the length broken off?

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

owns

39. A owns

to B

of his share. 33

of a ship, valued at $15422; he sells What is the value of what A has

left; also, what is the value of B's part?

40. A cotton mill is sold for $30000, of which A owns of the whole, B and C each ownofof the whole, How many dollars does each one claim?

41. A and B have a melon, of which A owns 3, and 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?

Ans.

A must have of it.

{ +

42. A farmer had of his sheep in one field, in a

100

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

14

43. If I divide 616 dollars between A, B, C, and D, by giving A of the whole, B of the remainder, C § of what then remained, and D the balance, how much will each receive? A had 154 dollars.

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

Thus, TOO, 15%, TOU, TOUU, TOOOO, &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: 01, 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 hundredths; 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.

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. is read two tenths."

0.2

twenty-five hundredths.

three hundred and sixty-five thousandths.

one hundred and five thousandths.

three hundredths.