. 1. In Addition and Subtraction, how do you place the figures as regards the orders ? Ans. Like orders must be placed in the same column; that is, directly under each other. 2. Does the relation of the orders in decimals correspond to that of abstract numbers ? Ans. The relation of any two orders in the same direction is the same. 3. How much greater is the value of the same figure in each consecutive column or order going to the left? Ans. Ten times. 4. Do the names of the orders taken in the same direction correspond ? Ans. No, but in opposite directions. 5. From what order, going in opposite directions, do the names of the other orders correspond ? Ans. From the order of units, going to the left in the Integral numbers, and to the right in the decimals. 6. What are Decimal Fractions ? Ans. Fractions whose denominators are 10, 100, 1000, etc. 7. Express the decimal fraction in the common form; thus, to, ito, etc., and decimally, .1, .01, etc.; and these expressions are respectively of the same value. 8. Which do you generally use? Why? Ans. The common fraction. Because it is more easily understood than the decimal, and is so convenient for cancellation. 9. How does the number of decimals in a product correspond to that of the factors ? Ans. The number is equal. 10. How does the number of decimals in a dividend correspond to that of the divisor and quotient ? Ans. It is equal. 11. If the divisor has more decimals than the dividend, what is necessary to be done? Ans. Add decimal zeros to the dividend until it has as many decimals as the divisor. 12. If the divisor and dividend have the same number of decimals, will there be any in the quotient ? Ans. No. 13. If it be necessary that there should be decimals in the quotient, what must be done? Ans. More decimal zeros must be added to the dividend. 14. Do zeros annexed to a decimal change its value ? Ans. No. 15. If zeros are prefixed to a decimal, and the decimal point removed to the left of the zeros, is the value changed? Ans. Each decimal zero thus prefixed renders the value one-tenth, or divides it by ten. 16. In multiplying by more than one figure, where is the first figure of each line or partial product placed, and why? Ans. It is placed in the column of the same order as the multiplier ; that is, directly under it, because every consecutive order to the left is ten times the value of the preceding order. DENOMINATE NUMBERS. All arithmetical numbers may be considered Denominate, even abstract numbers, as every figure in each successive order, beginning at the right and going to the left, is ten times the value of the same figure in the previous order, and may be arranged in a table; thus, 10 units = 1 ten. In the United States currency, the orders have the same relation; thus, 10 mills (m.) = 1 cent (ct.). 10 dollars = 1 eagle. Dimes and eagles are coins, but are not regarded in computation ; but only dollars (8), cents, and mills, the cents holding two places. There is generally a decimal point placed between dollars and cents; thus, $456.295, which is numerated “ four hundred and fifty-six dollars, twenty-nine cents and five mills. It may also be numerated without any change in its value," four hundred and fifty-six thousand, two hundred and ninety-five mills. ADDITION. As the relations of the orders in United States money is the same as in abstract numbers, hence their application is the same; and in addition and subtraction like orders must be placed under each other, and in every other way the same methods are followed. PROBLEMS. $25.365 9.100 1. What is the sum of twenty-five dollars, thirty-six cents and five mills; twelve dollars, eighteen cents and four mills; nine dollars and ten cents; thirty dollars and five mills; fifteen dollars and three cents. Ans., Ninety-one dollars, sixty-eight cents and four mills. 2. Add the following sums of money : Five dollars, thirty cents and four mills. $5.304 3.002 2.030 Seven dollars and three mills. 17.003 Twelve dollars and one cent 12.010 Nine dollars 9.000 . $38.349 (3.) Add $97.548 68.754 97.632 198.564 $462.498 (4.) Add $386.946 5372.875 64759.654 876943.687 $947463.162 (5.) Add 381,642 mills. 548,753 659,864 3,217,634 4,813,893 mills. REM. 1.—The sum of the last example may be numerated thus: Four millions eight hundred and thirteen thousand, eight hundred and ninety-three mills; or, thus : four thousand eight hundred and thirteen dollars, eighty-nine cents and three mills. REM. 2.-Mills are numerated the same as abstract numbers, SUBTRACTION. $287.304 1. From two hundred and eighty-. 194.293 seven dollars, thirty cents and four $93.011 mills, take one hundred and ninety four dollars, twenty-nine cents and three mills. Remainder, Ninety-three dollars, one cent and one mill. (2.) (3.) (4.) $475648.364 $9,486,397.213 $21795.375 387654.875 6,397,423.875 10963.625 $87993.489 $3,088,973.338 $10831.750 (5.) (6.) (7.) (8.) (9.) 100000 100 100 100.00 100.00 99999 99 2.50 1 1 99 98.50 97.50 REM.-As in addition and subtraction, so also in multiplication, the process is the same as that of abstract integers and decimals; hence there is no need of further esemplification. |