and then write the given numbers underneath, in their appropriate places. EXERCISES IN NOTATION AND NUMERATION. Express the following numbers by figures:1. Four hundred thirty-six. 2. Seven thousand one hundred sixty-four. 4. Fourteen thousand two hundred eighty. 8. Four hundred thirty-three million eight hundred sixteen thousand one hundred forty-nine. 9. Nine hundred thousand ninety. 10. Ten million ten thousand ten hundred ten. 11. Sixty-one billion five million. 12. Five trillion eighty billion nine million one. Point off, numerate, and read the following numbers: 25. Write seven million thirty-six. 26. Write five hundred sixty-three thousand four. 27. Write one million ninety-six thousand. 28. Numerate and read 9004082501. 29. Numerate and read 2584503962047. 30. A certain number contains 3 units of the seventh order, 6 of the fifth, 4 of the fourth, 1 of the third, 5 of the second, and 2 of the first; what is the number? 31. What orders of units are contained in the number 290648 ? 32. What orders of units are contained in the number 1037050 ? ADDITION. MENTAL EXERCISES. 35. 1. Henry gave 5 dollars for a vest, and 7 dollars for a coat; what did he pay for both? ANALYSIS. He gave as many dollars as 5 dollars and 7 dollars, which are 12 dollars. 2. A farmer sold a pig for 3 dollars, and a calf for 8 dollars; what did he receive for both? 3. A drover bought 5 sheep of one man, 9 of another, and 3 of another; how many did he buy in all? 4. How many are 2 and 6? 2 and 7? 2 and 9? 2 and 8? 2 and 10? 5. How many are 4 and 5? 4 and 8? 4 and 7? 4 and 9? 6. How many are 6 and 4? 6 and 6? 6 and 9? 6 and 7? 7. How many are 7 and 7? 7 and 6? 7 and 8? 7 and 10? 7 and 9? 8. How many are 5 and 4 and 6? 7 and 3 and 8? 6 and 9 and 5? 36. From the preceding operations we learn that Addition is the process of uniting several numbers of the same kind into one equivalent number. 37. The Sum or Amount is the result obtained by the process of addition. 38. The sign, +, is called plus, which signifies more. When placed between two numbers, it denotes that they are to be added; thus, 6+4, shows that 6 and 4 are to be added. 39. The sign, =, is called the sign of equality. When placed between two numbers, or sets of numbers, it signifies that they are equal to each other; thus, the expression 6+4=10, is read 6 plus 4 is equal to 10, and denotes that the numbers 6 and 4, taken together, equal the number 10. Define addition. The sum or amount? Sign of addition? Of equality? CASE I. 40. When the amount of each column is less than 10. 1. A farmer sold some hay for 102 dollars, six cows for 162 dollars, and a horse for 125 dollars; what did he receive for all? OPERATION. Hunds. Tens. Units. 102 162 125 Amount, 389 ANALYSIS. Arrange the numbers so that units of like order shall stand in the same column. Then add the columns separately, for convenience commencing at the right hand, and write each result under the column added. Thus, we have 5 and 2 and 2 are 9, the sum of the units; 2 and 6 are 8, the sum of the tens; 1 and 1 and 1 are 3, the sum of the hundreds. Hence, the entire amount is 3 hundreds 8 tens and 9 units, or 389. 6. What is the sum of 144, 321, and 232? Ans. 697. Ans. 856. 8. What is the sum of 42, 103, 321, and 32? 9. A drover bought three droves of sheep. The first contained 230, the second 425, and the third 340; how many sheep did he buy in all? Ans. 995. CASE II. 41. When the amount of any column equals or exceeds 10. 1. A merchant pays 725 dollars a year for the rent of a Case I is what? Give explanation. Case II is what? store, 475 dollars for a clerk, and 367 dollars for other expenses; what is the amount of his expenses? OPERATION. Hunds. 475 367 Sum of the units, Sum of the tens, 15 Sum of the hundreds, 14 Total amount, 17 1567 ANALYSIS. Arranging the numbers as in Case I, we first add the column of units, and find the sum to be 17 units, which is 1 ten and 7 units. Write the 7 units in the place of units, and the 1 ten in the place of tens. The sum of the figures in the column of tens is 15 tens, which is 1 hundred, and 5 tens. Write the 5 tens in the place of tens, and the 1 hundred in the place of hundreds. Next add the column of hundreds, and find the sum to be 14 hundreds, which is 1 thousand and 4 hundreds. Write the 4 hundreds in the place of hundreds, and 1 thousand in the place of thousands. Lastly, by uniting the sum of the units with the sums of the tens and hundreds, we find the total amount to be 1 thousand 5 hundreds 6 tens 7 units, or 1567. This example may be performed by another method, which is the common one in practice. OPERATION. 725 475 Thus: ANALYSIS. Arranging the numbers as before, we add the first column and find the sum to be 17 units; writing the 7 units under the column of units, we add the 1 ten to the column of tens, and find the sum to be 16 tens; writing the 6 tens under the column of tens, we add the 1 hundred to the column of hundreds, and find the sum to be 15 hundreds; as this is the last column, write down its amount, 15; and the whole amount is 1567, as before. 367 1567 1. Units of the same order are written in the same column; and when the sum in any column is 10 or more than 10, it produces one or more units of a higher order, which must be added to the next column. This process is sometimes called "carrying the tens." 2. In adding, learn to pronounce the partial results without naming the numbers separately; thus, instead of saying 7 and 5 are 12, and 5 are 17, simply pronounce the results, 7, 12, 17, etc. Give explanation. Second explanation. What is meant by carry. ing the tens? 42. From the preceding examples and illustrations we deduce the following RULE. I. Write the numbers to be added so that the units of the same order shall stand in the same column; that is, units under units, tens under tens, etc. II. Commencing at units, add each column separately, and write the sum underneath, if it be less than ten. III. If the sum of any column be ten or more than ten, write the unit figure only, and add the ten or tens to the next column. IV. Write the entire sum of the last column. PROOF. 1st. Begin with the right hand or unit column, and add the figures in each column in an opposite direction from that in which they were first added; if the two results agree, the work is supposed to be right. Or, 2d. Separate the numbers added into two sets, by a horizontal line; find the sum of each set separately; add these sums, and if the amount be the same as that first obtained, the work is presumed to be correct. By the methods of proof here given, the numbers are united in new combina. tions, which render it almost impossible for two precisely similar mistakes to occur. The first method is the one commonly used in business. Rule, first step? Second? Third? Fourth? Proof, first method! Second? Upon what principle are these methods of proof founded? |