in Iowa 192,214, in California 92,597, and in the Territories 92,298. Ans. 6,564,818. ART. 24. Method of adding two or more columns at a single operation. Ex. 1. Washington lived 68 years; John Adams, 91 years; Jefferson, 83 years; Madison, 85 years. What is the sum of the years they all lived? OPERATION. 68 91 83 85 Amount 3 27 Ans. 327. Beginning with the number last written down, we add the units and tens, thus: 85 and 3 equal 88, and 80 equal 168, and 1 equal 169, and 90 equal 259, and 8 equal 267, and 60 equal 327, the sum sought. In like manner may be added more than two columns at one operation. NOTE.The examples that follow can be performed as the above, or by the common method, or by both, as the teacher may advise. ART. 25. § III. SUBTRACTION. MENTAL EXERCISES. WHEN it is required to find the difference between two numbers, the process is called Subtraction. The operation is the reverse of addition. Ex. 1. John has 7 oranges, and his sister but 4; how many more has John than his sister? ILLUSTRATION. will make 7. We first inquire what number added to 4 From addition we learn that 4 and 3 are 7; consequently, if 4 oranges be taken from 7 oranges, 3 will remain Hence John has 3 oranges more than his sister. QUESTIONS.-Art. 25. What does subtraction teach? Of what is it the reverse? 2. Thomas had five oranges, and gave two of them to John; how many had he left? 3. Peter had six marbles, and gave two of them to Samuel; how many had he left? 4. Lydia had four cakes; having lost one, how many had she left? 5. Daniel, having eight cents, gives three to Mary; how many has he left? 6. Benjamin had ten nuts; he gave four to Jane, and three to Emily; how many had he left? 7. Moses gives eleven oranges to John, and eight to Enoch; how many more has John than Enoch? 8. Paid seven dollars for a pair of boots, and two dollars for shoes; how much did the boots cost more than the shoes? 26. How many are 20 less 5? 20 less 8? 20 less 9? 20 less 12? 20 less 15? 20 less 19? 27. Bought a horse for 60 dollars, and sold him for 90 dollars; how much did I gain? ILLUSTRATION. We may divide the two prices of the horse into tens, and subtract the greater from the less. Thus 60 equals 6 tens, and 90 equals 9 tens; 6 tens from 9 tens leave 3 tens, or 30. Therefore I gained 30 dollars. 28. Sold a wagon for 70 dollars, which cost me 100 dollars; how much did I lose? 29. John travels 30 miles a day, and Samuel 90 miles; what is the difference? 30. I have 100 dollars, and after I shall have given 20 to Benjamin, and paid a debt of 30 dollars to J. Smith, how many dollars have I left? 31. John Smith, Jr., had 170 dollars; he gave his oldest daughter, Angeline, 40 dollars, his youngest daughter, Mary, 50 dollars, his oldest son, James, 30, and his youngest son, William, 20 dollars; he also paid 20 dollars for his taxes; how many dollars had he remaining? ART. 26. The pupil, having solved the preceding questions, will perceive, that SUBTRACTION is the taking of one number from another to find the difference. When the two numbers are unequal, the larger is called the Minuend, and the less number the Subtrahend. The answer, or number found by the operation, is called the Difference, or Remainder. NOTE. -The words minuend and subtrahend are derived from two Latin words; the former from minuendum, which signifies to be diminished or made less, and the latter from subtrahendum, which means to be subtracted or taken away. ART. 27. SIGNS. Subtraction is denoted by a short horizontal line, thus, signifying minus, or less. It indicates that the number following is to be taken from the one that precedes it. The expression 6-24 is read, 6 minus, or less, 2 is equal to 4. -- QUESTIONS. Art. 26. What is subtraction? What is the greater number called? What is the less number called? What the answer? - Art. 27. What is the sign of subtraction? What does it signify and indicate? EXERCISES FOR THE SLATE. ART. 28. Method of operation, when the numbers are large, and each figure in the subtrahend is less than the figure above it in the minuend. Ex. 1. Let it be required to take 245 from 468, and to find their difference. Ans. 223. OPERATION. Minuend 468 We place the less number under the greater, units under units, tens under tens, &c., and draw a line below them. We then begin at the right hand, and say, 5 units from 8 units leave 3 units, and write the 3 in units' place below. We then say, 4 tens from 6 tens leave 2 tens, and write the 2 in tens' place below; and proceed with the next figure, and say, 2 hundreds from 4 hundreds leave 2 hundreds, which we write in hundreds' place below. We thus find the difference to be 223. ART. 29. First Method of Proof. — Add the remainder and the subtrahend together, and their sum will be equal to the minuend, if the work is right. This method of proof depends on the principle, That the greater of any two numbers is equal to the less added to the difference between them. EXAMPLES FOR PRACTICE. 2. 2. 3. 3. OPERATION. OPERATION AND PROOF. OPERATION. OPERATION AND PROOF. 8. A farmer paid 539 dollars for a span of fine horses, and sold them for 425 dollars; how much did he lose? Ans. 114 dollars. 9. A farmer raised 896 bushels of wheat, and sold 675 bushels of it; how much did he reserve for his own use? QUESTIONS. Ans. 221 bushels. Where Art. 28. How are numbers arranged for subtraction? do you begin to subtract? Why? Where do you write the difference? Art. 29. What is the first method of proving subtraction? What is the reason of this proof, or on what principle does it depend? |