64 = 10+6=16. &c. But the best method of solution is this: 4 is contained in 6 tens, 1 tens, and 2 tens added to the 4 units, 2 tens+4 units=24 units. in 24 units 6 units times. Divide 16422 by 6. 4 over, which are 4 is contained Solution 6 is contained in 16 thousands 2 thousand times, and 4 thousands over, 4 thousands +4 hundreds = 44 hundreds; 6 is contained in 44 hundreds 7 hundred times, and 2 hundreds over, 2 hundreds+2 tens=22 tens; 6 is contained in 22 tens 3 tens, and 4 over, 4 tens+2 units=42 units; 6 is contained in 42 units 7 times; therefore, the quotient is 2737. 84. The process is rather longer when the divisor is larger than 12. Example: Divide 6164 by 46. 46)6164(134 46 156 138 184 184 The divisor is placed to the left of the dividend, separated by a small curved line; on the right, draw another line. Take the least number of figures on the left of the dividend, that will contain the divisor, find how many times they contain it, and place the result on the right as the first figure of the quotient; and the product arising from the multiplication of the divisor by this figure being subtracted from the dividend, annex to the remainder the next figure of the dividend, and divide the result as before; and proceed thus till all the figures of the dividend have been brought down or used. If, after having annexed a figure to the remainder, the divisor is not contained in it, we place zero in the quotient, annex the next figure of the dividend to the remainder, and proceed as before. 86. It very often occurs that the divisor is not contained exactly in the dividend, then the last remainder may be joined to the quotient, under this form: +remainder...... (or see § 92). 87. Since the quotient is the number of times the divisor is contained in the dividend, it follows that the quotient multiplied by the divisor must necessarily be equal to the dividend, minus the remainder, if there be any. This affords a method of ascertaining whether a divisor has been correctly performed. 88. EXERCISES. 1. The annual income of a person is £38360. How much is that per day? The income must be divided into 365 equal parts, and one of them is the daily income; therefore, £38360 must be divided by 365. 2. 648 chests of tea cost £7128. What is the price of one chest? 3. What number is 25 times less than 3675 ? 4. £8400 are to be divided equally amongst six persons. How much would each individual's share be? 5. The dividend is 156970, and the divisor 2854. Find the quotient. 6. The dividend is 8760, and the quotient 24. Find the divisor. 7. What number, multiplied by 54, gives 9990 as product? 8. In how many days will a traveller go from London to Geneva if he travel at the rate of 7 miles per hour? The distance between the two cities is 1176 miles. 9. The earth moves in her orbit 583416000 miles in one year. At what rate does she go in one minute? 10. A gentleman's annual income is £2920; he wishes to lay by £1 every day. What must be his daily expenditure? PART III. FRACTIONS. 89. In the previous parts we have only considered whole numbers, or integers, viz., numbers containing once or several times unity; but, were we to suppose unity divided, or broken into several parts, we should obtain other quantities, called fractions; we are, therefore, led to the consideration of that most important part of arithmetic named Vulgar Fractions. 90. If we suppose an apple to be divided into two equal parts, one of the parts is called one-half; if it be divided into three equal parts, each part is one-third; if into four equal parts, each part is one-quarter; into five equal parts, one-fifth, &c. If a rod, a line, &c., was also cut into 2, 3, 4, 5, &c., equal parts we should obtain 2-halves, 3-thirds, 4-fourths, 5-fifths, &c. Therefore, one-half signifies 1 portion of a whole, or unit, which has been divided into 2 equal portions. It follows that 2 halves are equal to the whole. Also, 1-third is one part of a whole, which has been divided into 3 equal parts; hence, 3-thirds are equal to one whole. Apply similar reasoning to the other divisions of unity. 91. When two or more equal parts are taken together, we have quantities, as the following:-2-thirds, 3-fourths, 5-eighths, &c., meaning that unity has been divided into 3 parts, and 2 of them taken, or that two units have been divided into 3 equal parts, and 1 of these taken. In 3-fourths, the whole has been divided into 4 equal parts, and 3 of them taken, or 3 wholes have been divided into 4 equal parts, and 1 of these parts taken, &c. 92. Hence it follows that every numerator is supposed to be divided by its denominator. Thus, we perceive that fractions are expressed with two numbers one the denominator, which shows into how many equal parts unity is divided; and the other the numerator, denoting how many such equal parts are taken. The denominator and numerator are called terms of the fraction; they are thus written : ,,, &c., and read three-fourths, five-eighths, seven-ninths, &c. 94. The pupil must be exercised upon such questions as the following:-What is the meaning of? It implies that unity is divided into 9 equal parts, and 7 of them are taken; 9 is the denominator, and 7 the numerator. He should also represent the value of those fractions by lines, squares, cubes, &c. The line or the square represent unity, which is divided into 9 equal parts, and 7 are marked out. 95. When speaking of fractions in connection with each other, it is evident they must be parts of the same, or equal units; it would, for instance, be too ridiculous to attempt to connect or comparelb. of butter to yard of linen. 96. The greater the number of parts into which unity is divided, the smaller the parts are; thus, is greater than ; greater than ;, greater than ;, greater than 1, &c. Also, the greater the number of parts taken, the greater the fraction is; % is greater than ;, greater than 2; 14, greater than 5, &c. Therefore, the larger the denominator of a fraction is, the smaller is the fraction, if the numerators remain equal; and the larger the numerator is, the larger is the fraction, if the denominators remain the same. 97. If the numerator of a fraction be less than the denominator, the value of the fraction is less than unity, since it does not contain all the parts into which the unit has been divided, it is called a Proper Fraction:, . If the numerator be equal to the denominator, it is unity in the form of a fraction:, g. If the numerator be greater than the denominator, the value of the fraction is more than one unit, and it is called an Improper Fraction, it |