figure be equal to the sum of the other two, as 462, 792, 682, 594, 385, &c., and the division is performed by striking out the middle figure. 11-Will measure any number, if the sum of its periods of two figures can be measured by 11, thus, 29,46,24, is divisible by 11, because the sum of its periods of two figures 29+46+24=99, a multiple of 11. Or if the sum of the figures in the odd places be equal to the sum of those in the even places, or their difference be 11, 11 will measure, as 485672, here (4+5+7) are the figures in the odd places, and (8+6+2) those of the even, the sums being equal, 11 will therefore measure 485682. Remark.-If a number can be measured by two other numbers prime to each other, (that is, numbers which have no common factor,) it can be measured by the product of those numbers, thus, 192 can be measured by 4, and also by 3, it can therefore be measured by 12, their product. 13-Will measure a number of 3 or 4 figures, if the two right-hand figures be 4 times the left-hand one or two, 936, 624, 1768, 1976, 2392, &c. Or 13 will measure a number of 6 figures, if the alternate periods of 3 figures when subtracted leave a remainder of 0 or 13, or a multiple of 13, as 246, 207, 764, 673, 476,476, &c. 14-Will measure any even number which 7 measures. 15-Will measure any number which 3 and 5 both measure. 17—Will measure any number of 2 or 3 figures, if the right-hand figure be of the left-hand figure or figures, or a number of 3 or 4 figures, if the two right-hand figures be double the left, thus, 51, 255, 459, 357, &c., and 612, 918, 4386, 2754, &c. Also in numbers of 4 or 5 figures, if the 3 right-hand figures be 3 times the left-hand figure or figures, as 39117, 24072, 74222, 87261, &c. Again, we may cut off 2 or 3 figures to the right of a number, and divide the number thus cut off by the number of its figures, and if the difference between this quotient and the other part of the given number be either 0, 17, or a multiple of 17, 17 will measure, thus, 297,636, here cut off 636÷÷3= 212, which subtracted from the other part of the given num ber, 297-212 leaves 85, a multiple of 17, therefore 297,636 can be measured by 17; again 63478 can be measured by 17, because if we cut off 78 and take of it from 634 the remainder will be 595, which is a multiple of 17. If the figures cut off be not divisible by the required figure, add 17 to make it divisible, thus, 749,853 cut off three figures, 853, this is not divisible by 3, but by adding 17 it becomes so, thus, 853+17=870÷÷3=290, this taken from 749 leaves 459, which can be measured by 17, therefore 749,853 can be measured by 17. 18-Will measure any even number which 9 measures. 19-Will measure any number, if twice the right-hand figure added to the left make 19, or a multiple of 19, as 76, 95, 133, 171, 266, 456, 779, &c., we will examine if 19 will measure 874, twice 4, the right-hand figure added to 87 make 95, a multiple of 19, therefore 19 will measure 874; again, suppose we try 56943, here twice 3 or 6 added to 4 make 10, twice 10 or 20 added to 9 make 29, take 19 from 29 leaves 10, then twice 10 added to 6 make 26, take 19 from this leaves 7, then twice 7 added to 5 make 19, hence 56943 can be measured by 19. 21-Will measure any number which 3 and 7 will measure. 22-Will measure any even number which 11 measures. 37-Has a peculiar property, it will measure any number of three figures when all the figures are the same, and the quotient will be the sum of the figures, thus, 37 will measure 444, 777, or 888, and the quotient in the first will be 12, in the second 21, and the third 24, that is, the first is three times 4, the second 3 times 7, the third 3 times 8. To divide several periods, as 333,888,777, &c., write a cipher between the sums, thus, 333,888,777÷÷÷37=9024021. 43-Will measure numbers of three, four, or five figures, if the one or two right-hand figures be of the left, as 11137, 17759, 19264, 26187, &c., and the quotient will be 7 times the right-hand period; thus, to divide 14749 by 43 we multiply 49 (the right-hand period) by 7, and we have the quotient 343,-7 will measure all such numbers. 59-Will measure numbers of 4 or 5 figures, if the three right-hand figures be 3 times the left, as 74222, 38114, 69207, 87261, here we perceive 17 will also measure. 67-Will measure numbers of 3, 4, or 5 figures, if the two right-hand figures be of the left, as 16683, 7839, 13869 19497, 17889, &c. The quotient will always be 3 times the right-hand period, thus, if we wish to divide 17889 by 67 we have only to multiply 89 by 3=267. MENTAL EXERCISES. 1. Multiply 14 by 12, subtract 18 from the product, and divide the remainder by 5, and tell the quotient. 2. If I divide 192 by 12, and multiply the quotient by 9, and then divide the product by 8, what will be the last quotient? 3. If of 48 be multiplied by 4, and the product be divided by 18, what will be the quotient? 4. If of 72 be divided by 16, and the quotient be multiplied by 14, and this product divided by 6, what will be the last quotient. 5. If 7 times 24 be divided by 8, what will be the square of the quotient? 6. If the square of 18 be divided by 12, and the quotient be multiplied by 8, and this product divided by 18, what will be the last quotient? 7. If the square of 24 be divided by 16, and the quotient be multiplied by 13, what will be the product? 8. If 27 be multiplied by 23, and the product by 25, what I will be the result? 9. If 91 be divided by 7, and the square of the quotient be multiplied by 125, what will be the product? 10. If the square of 76 be divided by 19 what will the quotient be? 11. If 888 be divided by 37, what will be the square of the quotient? 12. What is the continued product of 24, 28, 7, 11, 13? 13. If 17889 be divided by 67, and the quotient be multiplied by 7x11×13, what will be the result? 14. What is the result of (34×38)÷÷÷4? 15. What is the result of 26×27×13? 16. What is the result of 53×57×7? 17. What is the result of 357÷÷÷17×19? PROPORTION. 1. If 17 cords of wood be worth $85, what are 13 cords worth? Here we may say, if 17 cords cost $85, 1 cord will cost 1 of $85, or $5, and 13 cords will cost 13 times 5 or $65; or we may say, if 17 cords cost $85, 13 cords will cost a sum bearing the same relation to $85, which 13 cords bear to 17 cords. $3 Thus, 185 Ans. 13 65 Ratio is the relation which one quantity bears to another of the same kind with respect to magnitude; or the ratio of two numbers is the quotient resulting from the division of the first by the second. Thus the ratio of 13 to 17 is 1, and of 65 to 85 it is §5=13. Proportion is equality of ratio. Four numbers are said to be proportionals when the ratio of the first to the second is the same as the ratio of the third to the fourth; hence, 17, 13, 85, 65, are proportionals. The first and fourth are called the extreme terms, and the second and third are called the means. If four numbers be in proportion, the product of the extremes is equal to that of the Thus, 4, 7, 12, 21, are proportionals, and means. Extremes. Means. If we have three terms of a proportion given, the fourth is readily found. Thus, if we have given the three terms, 3, 12, and 8, we observe that the second term is 4 times the first, and in order that the same relation may exist between the third and fourth as that of the first and second, the fourth must be 4 times the third; now if we multiply the second and third together and divide the product by the first, we get (12×8)=96÷÷÷3=32, which is 4 times the third term. This principle is one of the most important agents in the science of calculation. In simple proportion, three terms are given and a fourth required to be found. 2. If 8 bushels of corn cost $5.60, what will 13 bushels cost? 13 bushels will cost a sum of money, bearing the same relation to $5.60, which 13 bears to 8, the relation or ratio of 13 to 8 is 13, and that of $9.10 to $5.60 is 910-13 13. 560 Thus 85.60 70 13 Ans. $9.10 RULE FOR THE STATEMENT AND SOLUTION OF QUESTIONS IN SIMPLE PROPORTION. Place on the right of a vertical line that number which is of the same name or kind as the required answer, then ascertain from the reading of the question, whether the answer must be greater or less than this number (of its own name.) If it must be greater, place the greater of the two remaining terms or numbers on the right of the line, and the other on the left; but if the answer must be less than this first mentioned number, then place the less of the two remaining terms or numbers on the right of the line, and the other on the left. Now divide any number on one side of the line by any number on the opposite side, (if they measure each other,) then multiply all the numbers remaining on the right together for a dividend, or for the answer to the question, if no number remains on the left for a divisor; but if a divisor remains on the left the quotient will be the answer. NOTE. The two numbers on opposite sides of the line (differing in name from the answer) must be expressed in the same denomination. 3. If 18 tons of hay cost $324 what will 23 tons cost at the same rate? Operation. Ans. 23 × 18 $414 The required answer is evidently money; therefore the $324 is the number of the same name as the required answer, and as this $324 is the price of 18 tons, and we wish to know the price of 23 tons, the answer must be greater than $324; therefore we place the 23, the greater of the two remaining numbers on the right of the line, and the other (18) on the left; now since we know 324 to be |