297 8 2376 3 2379 Test. Or thus, the multiplication and addition being performed mentally : 8 × 7 = 56, + 3 = 59 8×972, +5=77 8x216, +7=23 2379 The remainder is a part of the dividend upon which the operation of division has not been performed, and so we express it with the divisor, using the sign of division, and put it with the quotient as part of the general result. Thus, 2973. NOTE. For the present and until otherwise explained, the expression is to be read three divided by eight, and similar expressions in a similar manner. When, as in the above example, only the final results are recorded, the process is called Short Division. 429 hundreds = 4290 tens, + 2 tens = 4292 tens. 436 in 4292, 9 times. 9 × 436 3924. PROBLEMS IN ADDITION, SUBTRACTION, MULTIPLICATION, AND DIVISION. The following problems are given as illustrations of the way to use the Practice Tables. Proficiency in figuring can be attained only by constant practice. The kind of practice which the tables furnish is what is required for the purpose. If one can perform such exercises readily, everything else in arithmetic follows easily and simply. If one cannot do these things, neither can he do the others. The tables present a purely chance arrangement of the digits, having been made up as follows: Of 100 small squares of cardboard, ten were marked 1; ten, 2; etc. The 100 squares were well shuffled in a hat, and the number first drawn taken for the first digit of the first horizontal row; the number next drawn, for the second digit of the first hori zontal row; and so on. After each drawing, the card drawn was put back into the hat, and the whole re-shuffled. In using the tables, if a dividend, a divisor, or a factor, is found to be 0, substitute 9 for 0. FROM PRACTICE TABLE NO. 1. (PAGE 52.) 13. Add columns a to h, lines 1 to 5 (i.e. 45459686, and the four numbers just below it). Test the result by adding in the reverse direction. 14. Add columns a-h, lines 6-10; 11-15; 16-20; 21-25. To separate the digits wanted from the rest of the table, first put a sheet of paper over the table so as to cover all the columns after h. Then, for the first of the above problems, put slate or paper just below line 5. For the second problem, put a book, ruler, or some other object over the first five lines, and slate or paper under line 10. In a similar manner, any figures wanted may be brought to view. 15. Add 1-13; a-e, f-j, k-o, p-t, u-z. 16. Add 1-25; a-c, d-f, g-i, j-l, m-o, p-r, s-u, v-z. 17. Add a-z; 1-4, 5-8, 9-12, 13-16, 17-20, 21-25. FROM PRACTICE TABLE No. 1. 18. Find the difference between a-j, 1 and 2. Add the remainder to the subtrahend and see if the sum equals the minuend. 19. Find the difference between a-j, 2 and 3. (As 3 is the larger, take 2 from 3. Don't rewrite, but subtract the upper number from the lower, as they stand.) 20. Find the difference between a-j; 3 and 4, 4 and 5, 5 and 6, etc. 21. Multiply each line of a-d by e; a-e by f-h; i-p by q-w. 22. Divide each line of a-i by j; a-g by h-i; r−z by o-q. Prove the correctness of each of the above divisions by multiplying quotient by divisor, and adding remainder. DIVISIBILITY OF NUMBERS. A number is said to be divisible by another if it contains the other some number of times, without a remainder. Any number ending with 0 is a multiple of 10. It is, therefore, divisible by 10 and also by 2, a factor of 10. Every number is made up of tens + its units. Hence, A number is divisible by 2 if its units are divisible by 2. In other words, we separate the number into two parts, tens and units. One part, the tens, must of necessity be divisible by 2. If, then, the other part, the units, be divisible by 2, both parts are divisible by 2, and hence the whole number is divisible by 2. NOTE. It is mathematically exact to say that any number, 3 for instance, is made up of tens and units. There are 0 tens, 3 units. O is divisible by any number. 0 ÷ any number 0. Any number ending with 0 is a multiple of 10. It is, therefore, divisible by 10 and also by 5, a factor of 10. Every number is made up of tens + its units. Hence, A number is divisible by 5 if its units are divisible by 5. 2468 2460 +8. 8 is divisible by 2, so 2468 is divisible by 2. 8 is not divisible by 5, so 2468 is not divisible by 5. It is, Every Any number ending with 00 is a multiple of 100. therefore, divisible by 100, also by 4, a factor of 100. number is made up of hundreds + its tens and units. Hence, A number is divisible by 4 if its tens and units, taken together as one number, are divisible by 4. 28 is divisible by 4, so 8228 is divisible by 4. 42 is not divisible by 4, so 16142 is not divisible by 4. Any number ending with 000 is a multiple of 1000. It is, therefore, divisible by 1000 and also by 8, a factor of 1000. Every number is made up of thousands + its hundreds, tens, and units. Hence, A number is divisible by 8 if its hundreds, tens, and units, taken together as one number, are divisible by 8. 184 is divisible by 8, so 37184 is divisible by 8. Thus, any number is made up of 9's (in this case there are 555 +88+2 of them), and the sum of its digits (in this case, 5+8+2+3 = 18). Hence, A number is divisible by 9 if the sum of its digits is divisible by 9. 5823 is divisible by 9, the sum of its digits being 18, a number divisible by 9. 3746 is not divisible by 9, the sum of its digits being 20, a number not divisible by 9. COROLLARY. If the sum of the digits of a number be divided by 9, the remainder will be the same as the remainder after dividing the number by 9. 9 is divisible by 3. Any number of 9's are divisible by 3. So, from the previous demonstration, A number is divisible by 3 if the sum of its digits is divisible by 3. |