EXAMPLES FOR PRACTICE. 1. If 1300 men receive 460 dollars apiece, how many dollars will they all receive? OPERATION. 460 1300 138 46 Ans. 598000 dollars. The ciphers in the multiplicand and multiplier, counted together, are three. Disregarding these, we write the significant figures of the multiplier under the significant figures of the multiplicand, and multiply; after which we annex three ciphers to the right hand of the product, which gives the true answer. 2. The number of distinct buildings in New England, appropriated to the spinning, weaving, and printing of cotton goods, was estimated, in 1826, at 400, running, on an average, 700 spindles each; what was the whole number of spindles ? 3. Multiply 357 by 6300. 1512 000 756 77112 In the operation it will be seen, that multiplying by ciphers produces nothing. Therefore, III. When there are ciphers between the significant figures of the multiplier, we may omit the ciphers, multiplying by the significant figures only, placing the first figure of each product directly under the figure by which we multiply. EXAMPLES FOR PRACTICE. 8. Multiply 154326 by 3007. 1. What is multiplication? 2. What is the number to be multiplied called? 3. to multiply by called? 4. What is the result or answer called? 5. Taken collectively, what are the multiplicand and multiplier called? 6. What is the sign of multiplication? 7. What does it show? 8. In what order must the given number be placed for multiplication? 9. How do you proceed when the multiplier is less than 12? 10. When it exceeds 12, what is the method of procedure? 11. What is a composite number? 12. What is to be understood by the component parts, or factors, of any number? 13. How may you proceed when the multiplier is a composite number? 14. To multiply by 10, 100, 1000, &c., what suffices? 15. Why? 16. When there are ciphers on the right hand of the multiplicand, multiplier, either or both, how may we proceed? 17. When there are ciphers between the significant figures of the multiplier, how are they to be treated? EXERCISES. 1. An army of 10700 men, having plundered a city, took so much money, that, when it was shared among them, each man received 46 dollars; what was the sum of money taken? Proceeding to the next column, we say, 1 (hundred) from 2, (hundreds,) and there remains 1, (hundred,) which we set down in hundred's place, and the work is done. It now ap pears, that the number of sheep left was 123; that is, 237-114123. After the same manner are performed the following examples: 15. There are two farms; one is valued at 3750, and the other at 1500 dollars; what is the difference in the value of the two farms? 16. A man's property is worth 8560 dollars, but he has debts to the amount of 3500 dollars; what will remain after paying his debts? 17. James, having 15 cents, bought a pen-knife, for which he gave 7 cents; how many cents had he left? OPERATION. 15 cents. 7 cents. 8 cents left. A difficulty presents itself here; for we cannot take 7 from 5; but we can take 7 from 15, and there will remain 8. 18. A man bought a horse for 85 dollars, and a cow for 27 dollars; what did the horse cost him more than the cow? OPERATION. 27 The The same difficulty meets us here as in Horse, 85 the last example; we cannot take 7 from Cow, 5; but in the last example the larger number consisted of 1 ten ar 5 units, which Difference, 58 together make 15; we therefore took 7 from 15. Here we have 8 tens and 5 units. We can now, in the mind, suppose I ten taken from the 8 tens, which would leave 7 tens, and this 1 ten we can suppose joined to the 5 units, making 15. We can now take 7 from 15, as before, and there will remain 8, which we set down. taking of 1 ten out of 8 tens, and joining it with the 5 units, is called borrowing ten. Proceeding to the next higher or der, or tens, we must consider the upper figure, 8, from which we borrowed, 1 less, calling it 7; then, taking 2 (tens) from 7, (tens,) there will remain 5, (tens,) which we set down, making the difference 58 dollars. Or, instead of making the upper figure 1. less, calling it 7, we may make the lower figure one more, calling it 3, and the result will be the same; for 3 from 8 leaves 5, the same as 2 from 7, 19. A man borrowed 713 dollars, and paid 471 dollars; how many dollars did he then owe? many? 20. 1612-465 how many? 21. 43751-6782 how many? = 713-471 how Ans. 242 dollars. Ans. 1147. Ans. 36969. 8. The pupil will readily perceive, that subtraction is the reverse of addition. 22. A man bought 40 sheep, and sold 13 many had he left? 40-18 how many? 23. A man sold 18 sheep, and had 22 left; he at first? 18+22= how many? of them; how Ans. 22 sheep. how many had Ans. 40. 24. A man bought a horse for 75 dollars, and a cow for 16 dollars; what was the difference of the costs? 75-16 how many? Reversed, 59+16== how many? 25. 114-103 how many? Reversed, 11 + 103 = how many? 26. 14376 how many? Reversed, 67+76 = how Dany? Hence, subtraction may be proved by addition, as in the oregoing examples, and addition by subtraction. To prove subtraction, we may add the remainder to the subtruhend, and, if the work is right, the amount will be equal to the minuend. To prove addition, we may subtract, successively, from the amount, the several numbers which were added to produce it, and, if the work is right, there will be no remainder. Thus 7+ 8+ 6 = 21; proof, 21 — 6 = 15, and 1587, and 7—7— 0. From the remarks and illustrations now given, we deduce the following RULE. I. Write down the numbers, the less under the greater, placing units under units, tens under tens, &c. and draw a line under them. II. Beginning with units, take successively each figure in the lower number from the figure over it, and write the remainder directly below. III. When the figure in the lower number exceeds the figure over it, suppose 10 to be added to the upper figure; but in this case we must add 1 to the lower figure in the next column, before subtracting. This is called borrowing 10. EXAMPLES FOR PRACTICE. 27. If a farm and the buildings on it be valued at 10000, and the buildings alone be valued at 4567 dollars, what is the value of the land? 28. The population of New England, at the census in 1809, was 1,232,454; in 1820 it was 1,659,854; what was the increase in 20 years? 29. What is the difference between 7,648,203 and 928,671 ? 30. How much must you add to 358,642 to make 1,487,945 ? 31. A man bought an estate for 13,682 dollars, and sold it again for 15,293 dollars; did he gain or lose by it? and how much? 32. From 364,710,825,193 take 27,940,386,574. SUPPLEMENT TO SUBTRACTION. QUESTIONS. 1. What is subtraction? 2. What is the greater number called? 3. the less number? 4. What is the result 5. What is the sign of subtraction? or answer called? 7. What is understood by borrowing 6. What is the rule? ten? 8. Of what is subtraction the reverse? 9. How is subtraction proved? 10. How is addition proved by subtraction ? EXERCISES, 1. How long from the discovery of America by Columbus, in 1492, to the commencement of the Revolutionary war in 1775, which gained our Independence? 2. Supposing a man to have been born in the year 1773, now old was he in 1827? 3. Supposing a man to have been 80 years old in the year 1826, in what year was he born? 4. There are two numbers, whose difference is 8764; the greater number is 15687; I demand the less? |