Art. 17. When it is required to find a single number to express the sum of the units contained in several smaller numbers, the process is called Addition. Ex. 1. James has 3 pears, and his younger brother has 4, how many have both ? ILLUSTRATION. — To solve this question, the 3 pears and 4 pears must be added together; thus, 3 added to 4 makes 7. Therefore James and his brother have 7 pears. QUESTION. — Art. 17. What is the process called by which we find the sum of several numhers ? 2. How many are and 3? 2 and 5 ? 2 and 7 ? 2 and 9 ? 2 and 4? 2 and 2 ? 2 and 8? 2 and 6? 3. How many are 3 and 3? 3 and 5? 3 and 7? 3 and 9? 3 and 4? 3 and 6? 3 and 8? 3 and 3 ? 4. How many are 4 and 3? 4 and 5? 4 and 8? 4 and 9? 4 and 1? 4 and 2? 4 and 4? 4 and 7 ? 5. How many are 5 and 3? 5 and 4 ? 5 and 7? 5 and 8? 5 and 9 ? 5 and 2 ? 5 and 5? 5 and 6? 5 and 1 ? 6. How many are 6 and 2 ? 6 and 4? 6 and 3? 6 and 5 ? 6 and 7 ? 6 and 9? 6 and 1 ? 6 and 6 ? 6 and 8 ? 7. How many are 7 and 3? 7 and 5 ? 7 and 7? 7 and 6? 7 and 8? 7 and 9? 7 and 2 ? 7 and 4? 7 and 10 ? 8. How many are 8 and 2? 8 and 4? 8 and 5 ? 8 and 7? 8 and 9? 8 and 8? 8 and 1? 8 and 3? 8 and 6? 9. How many are 9 and 1? 9 and 3? 9 and 5 ? 9 and 4? 9 and 6? 9 and 8? 9 and 9 ? 9 and 2 ? 10. James had 4 apples, Samuel gave him 5 more, and John gave him 6; how many had he in all ? 11. Gave 7 dollars for a barrel of flour, 5 dollars for a hundred weight of sugar, and 8 dollars for a tub of butter; what did I give for the whole ? 12. Paid 5 dollars for a pair of boots, 12 dollars for a coat, and 6 dollars for a vest; what was the whole cost ? 13. Gave 25 cents for an arithmetic, and 67 for a geography; what was the cost of both ? ILLUSTRATION. We may divide the cents into tens and units. Thus, 25 equals 2 tens and 5 units; 67 equals 6 teng and 7 units; 2 tens and 6 tens are 8 tens; and 5 units and 7 units are 12 units, or 1 ten and 2 units; 1 ten and 2 units added to 8 tens make 9 tens and 2 units, or 92. Therefore the arithmetic and geography cost 92 cents. 14. On the fourth of July 20 cents were given to Emily, 15 cents to Betsey, 10 cents to Benjamin, and none to Lydia ; what did they all receive ? 15. Bought four loads of hay; the first cost 15 dollars, the second 12 dollars, the third 20 dollars, and the fourth 17 dol lars; what was the price of the whole ? 16. Gave 55 dollars for a horse, 40 dollars for a wagon, and 17 dollars for a harness ; what did they all cost ? 17. Sold 3 loads of wood for 17 dollars, 6 tons of timber for 19 dollars, and a pair of oxen for 60 dollars; what sum did I receive ? Art. 18. From the solution of the preceding questions, the learner will perceive, that ADDITION is the process of collecting several numbers into one sum, which is called their amount. Addition is commonly represented by this character, +, which signifies plus, or added to. The expression 7+5 is read, 7 plus 5, or 7 added to 5. This character, =, is called the sign of equality, and signifies equal to. The expression 7+5=12 is read, 7 plus 5, or 7 added to 5, is equal to 12. EXERCISES FOR THE SLATE. Art. 19. The method of operation when the numbers are large, and the sum of each column is less than 10. Ex. 1. A man bought a watch for 42 dollars, a coat for 16 dollars, and a set of maps for 21 dollars; what did he pay for the whole ? Ans. 79 dollars. Having arranged the numbers so that all the units of the same order shall stand 42 in the same column, we first add the col16 umn of units ; thus, 1 and 6 are 7, and 2 21 are 9 (units), and write down the amount under the column of units. We then add Amount 79 the column of tens; thus, 2 and 1 are 3, and 4 are 7 (tens), which we write under the column of tens, and thus find the amount of the whole to be 79 dollars. OPERATION. Art. 20. First Method of Proof.- Begin at the top and add the columns downward in the same manner as they were before added upward, and if the two sums agree the work is presumed to be right. The reason of this proof is, that, by adding downward, the order of the figures is inverted ; and, therefore, any error made in the first addition would probably be detected in the second. NOTE. — This method of proof is generally used in business. QUESTIONS. — Art. 18. What is addition? What is the sign of addition, and what does it signify? What is the sign of equality, and what does it signify ? — Art. 19. How are numbers arranged for addition ? Which column must first be added? Why? Where do you place its sum? Where must the sum of each column be placed? What is the whole sum called ? - Art. 20. How is addition proved? What is the reason for this method of proof? Is this method in common use? 6. What is the sum of 231, 114, and 324 ? Ans. 669. 7. Required the sum of 235, 321, and 142. Ans. 698. 8. What is the sum of 11, 22, 505, and 461 ? Ans. 999. 9. Sold twelve ploughs for 104 dollars, two wagons for 214 dollars, and one chaise for 121 dollars; what was the amount of the whole ? Ans. 439 dollars. 10. A drover bought 125 sheep of one man, 432 of another, and of a third 311; how many did he buy? Ans. 868 sheep. OPERATION. Acres. Art. 21. Method of operation when the sum of any column is equal to or exceeds 10. Ex. 1. I have three lots of wild land; the first contains 246 acres, the second 764 acres, and the third 918 acres. I wish to know how many acres are in the three lots. Ans. 1928 acres. Having arranged the numbers as in the preceding examples, we first add 246 the units ; thus, 8 and 4 are 12, and 6 7 64 are 18 units, equal 1 ten and 8 units. 918 We write the 8 units under the column of units, we carry or add the 1 ten to Amount 1928 the column of tens ; thus, 1 added to 1 makes 2, and 6 are 8, and 4 are 12 (tens), equal to 1 hundred and 2 tens. We write the 2 tens under the column of tens, and add the 1 hundred to the column of hundreds ; thus, 1 added to 9 makes 10, and 7 are 17, and 2 are 19 (hundreds), equal to 1 thousand and 9 hundreds. We write the 9 under the column of hundreds ; and there being no other column to be added, we set down the 1 thousand in thousands' place, and find the amount of the several numbers to be 1928. NOTE. — A more concise way, in practice, is to omit calling the name of each figure as added, and name only results. QUESTIONS. — Art. 21. When the sum of any column exceeds ten, where are the units written? What is done with the tens? Why do you carry, or add, one for every 10 ? How is the sum of the last column written ? Art. 22. From the preceding examples and illustrations in addition, we deduce the following general RULE. — Write the numbers so that all the units of the same order shall stand in the same column. Add, upward, all the figures in the column of units, and, if the amount be less than ten, write it underneath. But, if the amount be ten or more, write down the unit figure only, and add in the figure denoting the ten or tens with the next column. Proceed in this way with each column, until all are added, observing to write under the last column its whole amount. Art. 23. Second Method of Proof. — Separate the numbers to be added into two parts, by drawing a horizontal line between them. Add the numbers below the line, and set down their sum. Then add this sum and the number, or numbers, above the line together; and, if their sum is equal to the first amount, the work is presumed to be right. The reason of this proof depends on the principle, That the sum of all the parts into which any number is divided is equal to the whole. EXAMPLES FOR PRACTICE. 2. 2. 3. OPERATION OPERATION AND PROOF. OPERATION. OPERATION AND PROOF. 5 26 5 26 241 317 5 32 317 5 3 2 132 5 29 2 07 132 913 Ans. 15 04 Ans. 1893 First am't 1893 217 275 17 6 6 2512 2923 2 59 92 QUESTIONS. - Art 22. What is the general rule for addition ?- Art. 23. What is the second method of proving addition? What is the reason of this method of proof? |