18. Multiply 700630.0003 by 1006.07, and prove by dividing the product by the multiplier.

19. Find the continued product of 18, 13, and 11; obtain the square root of the product to two decimal places.

20. Divide 126819 by 21, using factors.

21. What is the least number of times that 315 must be added to 1594 that the sum may exceed a million ? 22. Multiply 67412 by 9997 as shortly as you can. 23. Divide 789 by .10063 to 3 decimal places.

24. Find the H.C.F. of 10481 and 17617.

25. Four men can walk 30, 35, 40, and 45 miles a day, respectively; what is the least distance they can all walk in an exact number of days?

26. Find the L.C.M. of 12, 64, 80, 96, 120, 160.

27. Find the prime factors of 1176 and 19404, and hence write down their G.C.M. and L.C.M..

28. The quotient is twice the divisor, and the remainder which is 50 is one-fifth part of the quotient. Find the dividend.

29. Simplify

forms.

80 × 51 125 × 219

; obtain the answer in two

30. Find the least number which can be divided by 7, 20, 28, and 35, and leave 3 as remainder in each case.

31. What number is that which after being subtracted 19 times from 1000 leaves a remainder of 12?

32. Multiply three thousand eighty-seven by seventytwo thousand nine hundred thirty. What numbers less than 12 will exactly divide the product?

33. (a) Simplify 650 × 1.25÷.5.

(b) The answer is a multiple of which of the following numbers: 5, 15, 25, 65, 105, 125?

Obtain (b) by first obtaining primes of the answer.

34. Find 19 × 17 × 11 × 2.5 × 1.25.

35. Find 652 × .11.

36. Simplify (a) 12 × 4 −2+6 (18 – 14).

(b) 12 × (4 − 2) + 6 (18 – 14).

(c) 2(72 × 4) – {2 + 6 (18 − 14)}.

37. If a number when divided by 35 give a remainder 27, what remainder will it give when divided by 7?

38. What is the greatest and what is the least number of four digits which is exactly divisible by 73?

39. Find the H.C.F. and the L.C.M. of 21, 22, 24, 28, 32, 33; also of 16, 18, 20, 24, 30, 36.

40. Find the number nearest to 1000 and exactly divisible by 39.

41. Multiply 7863 by 999, and see if the product is divisible by 3.

42. Find √4912.888464.

43. Find √36.

(a) Divide the following numbers by 2.

(b) Prove your answers by first simplifying the numbers, and then dividing by 2.

44. 3(6+8); 3(6 × 8).

45. 4 (68); 4(6 x 8).

46. 4 (18+6); 4(18÷6).

47. (6 x 2) (8+10).

48. 6 (123) +86 +4 +2.

49. 28 [7 (3 + 2)].

CHAPTER IV.

FRACTIONS.

108. If a unit be divided into 2, 3, 4, 5, etc., equal parts, these parts are called halves, third-parts, fourth-parts, fifth-parts, etc., or more shortly and more generally, halves, thirds, fourths, fifths, etc.

If the unit quantity be divided into any number of equal parts, one or more of these parts is called a Fraction of the unit.

For example, if a unit quantity, as one apple, be divided into sevenths, three of these parts constitute three sevenths, and the three sevenths is a fraction of seven sevenths, the unit quantity.

109. The number which indicates how many parts of the unit quantity are to be used is called the Numerator. The number which indicates into how many parts the unit quantity is divided is called the Denominator.

110. The expression formed by writing a numerator just above a denominator with a line between is called a Common Fraction.

Thus,, (eight-thirteenths), (one twenty-third), are common fractions (called briefly fractions).

Common fractions are also called vulgar fractions.

NOTE. A fraction is an expression of division, the numerator and denominator corresponding to the dividend and divisor respectively. What is true of dividend and divisor is true of numerator and denominator. When the indicated division is performed, the quotient is generally a decimal.

111. If we have 3 units, and divide each of them into 5 equal parts, and then take one of the parts from each divided unit, we shall take one part out of every five, that is, one-fifth of the whole three units; but each of the parts is one-fifth of a single unit and we take 3 of them: we therefore take 3 fifths of one unit.

Thus, 3 fifths of 1 unit is the same as 1 fifth of 3 units. Hence,, which by definition denotes 3 fifths of 1 unit, may also be considered to stand for 1 fifth of 3 units. The same holds good for all other fractions; for example,

1. Write in figures the following fractions: fiveninths, six-elevenths, eleven twenty-thirds, sixteen twentysevenths, seventeen ninety-firsts, ninety-five one hundred fourths.

239

5
I'

90

2. Write in words:,,, HI, 219, 2000.

112. The numerator and denominator of a fraction are called its Terms.

When the numerator is less than the denominator, the fraction is called a Proper Fraction; and when the numerator is equal to or greater than the denominator, the fraction is called an Improper Fraction.

A number made up of an integer and a fraction is called a Mixed Number.

Thus, 24 (2 and 4), which means 2 + †, is a mixed number.

Changing the form of an expression, or changing the units in terms of which any quantity is expressed, is called Reduction.

113. Reducing a mixed number to an improper fraction. Consider, for example, 32.

Each unit contains 7 sevenths, therefore 3 units contain 3 times 7 sevenths.

Hence, 32 = 3 times 7 sevenths + 2 sevenths =

21+ 2 23
7

=

7

From the above it will be seen that a mixed number is equivalent to an improper fraction whose denominator is the denominator of the fractional part, and whose numerator is obtained by multiplying the integral part by the denominator of the fraction and adding its numerator.

It should be noticed that a whole number can be expressed as a fraction with any given denominator. For example,

66 x 7 sevenths = 42; also, 6 = 6 × 13 thirteenths = }}.

114. Conversely, reducing an improper fraction to a whole or mixed number.

Consider, for example, 23.

Since 7 sevenths make 1 unit,

23 = 3 × 7 sevenths + 2 sevenths = 3+2 sevenths = 34.

=

6, since 4 fourths

1.

Again, 246 × 4 fourths: From the above it will be seen that an improper fraction is reduced to a mixed number by dividing its numerator by its denominator; the quotient will form the integral part, while the remainder, if any, will form the numerator of the fractional part, whose denominator must be the denominator of the improper fraction.