OPERATION. 8. If sound moves 1142 feet in a second, how far will it move in one minute ? Ans. 68520 feet. 43. When the multiplier is 1 with one or more ciphers annexed, as 10, 100, &c. Ex. 1. In 1 day there are 24 hours ; how many hours in 10 days ? In 100 days ? Answers 240, 2400. The removal of a figure one Multiplicand 24 24 place to the left makes the valMultiplier 10 100 ue expressed tenfold. (Art. 7.) Therefore, by annexing one Product 2 40 2400 cipher to 24, the multiplicand, Or thus, each figure is removed one 2400. place to the left, and the value expressed made tenfold, or multiplied by 10; and by annexing two ciphers, each figure is removed two places to the left, and the value expressed made one hundred-fold, or multiplied by 100. RULE. Annex to the multiplicand as many ciphers as has the multiplier. The number thus formed will be the product required. EXAMPLES FOR PRACTICE. 2. Multiply 2356 by 10. Ans. 23560. 3. Multiply 5873 by 100. Ans. 587300. 4. Multiply 7964 by 1000. Ans. 7964000. 5. Multiply 98725 by 100000. Ans. 9872500000. 240, OPERATION. 44. When there are ciphers on the right hand of tho multiplier or multiplicand, or both. Ex. 1. What will 600 acres of land cost at 20 dollars per acre ? Ans. 12000 dollars. The multiplicand may be resolved into Multiplicand 600 the factors 6 and 100, and the multiplier Multiplier 20 into the factors 2 and 10. Now, it is evi dent (Art. 42), if these several factors be Product 1 2000 multiplied together, they will produce the same product as the original factors 600 and 20. Thus 6 X 2 12, and 12 X 100=1200, and 1200 X 10 12000. 43. What is the effect of removing a figure one place to the left? What is the effect of annexing a cipher? Two ciphers ? &c. The rule when the multiplier is 1 with ciphers annexed ? - 44. How do you arrange the figures for multiplication, when there are ciphers on the right hand of either the multiplier or multiplicand, or both? Why does multiplying the significant figures and annexing the ciphers produce the true product ? RULE. — Write the significant figures of the multiplier under those of the multiplicand, and multiply them together. To their product annex as many ciphers as there are on the right of both multiplicand and multiplier. 4. Multiply 8010700 by 9000909. Ans. 72103581726300. 5. Multiply 700110000 by 700110000. Ans. 490154012100000000. 6. Multiply 4070607 by 7007000. Ans. 28522743249000. 7. Multiply 4110000 by 1017010. Ans. 4179911100000. 8. Multiply twenty-nine millions two thousand nine hundred and nine by four hundred and four thousand. Ans. 11717175236000. 9. Multiply eighty-seven millions by eight hundred thousand seven hundred. Ans. 69660900000000. 10. Multiply one million one thousand one hundred by nine hundred nine thousand and ninety. Ans. 910089999000. 11. Multiply forty-nine millions and forty-nine by four hundred and ninety thousand. Ans. 24010024010000. 12. Multiply two hundred millions two hundred by two millions two thousand and two. Ans. 400400800400400. 13. Multiply four millions forty thousand four hundred by three hundred three thousand. Ans. 1224241200000. 14. Multiply three hundred thousand thirty by forty-seven thousand seventy. Ans. 14122412100. 15. Multiply fifteen millions one hundred by two thousand two hundred. Ans. 33000220000. 16. Multiply one billion twenty thousand by one thousand one hundred. Ans. 1100022000000. 44. What is the rule ? DIVISION. 45. When it is required to find how many times one number contains another, the process is called Division. Ex. 1. A boy has 32 cents, which he wishes to give to 8 of his companions, to each an equal number; how many must each receive? ILLUSTRATION. Each must receive as many cents as 8, the number of companions, is contained times in 32, the number of cents. We therefore inquire what number 8 must be multiplied by to make 32. By trial, we find that 4 is the number; because 4 times 8 are 32. Hence 8 is contained in 32 4 times, and each of his companions must receive 4 cents. 66 88 66 110 66 121 10 in 10 10 66 20 10 30 10 40 10 50 10 60 10 70 10 80 10 10 1 time 10 in 110 11 times 11 in 77 7 times 12 in 36 3 times 8 66 12 48 11 99 9 12 60 5 4 11 10 12 72 6 11 12 84 11 22 2 times) 11 « 132 12 " 12 96 8 12 9 12 « 120 10 12 in 12 1 time 12 66 132 11 16 108 66 90 12 24 2 times | 12 “ 144 12" 66 100 10 2. A farmer received 8 dollars for 2 sheep; what was the price of each? ILLUSTRATION. Since he received 8 dollars for 2 sheep, for 1 sheep he must receive as many dollars as 2 is contained times in 8. 2 is contained in 8 4 times, because 4 times 2 are 8; hence 4 dollars was the price of each sheep. 3. A man gave 15 dollars for 3 barrels of flour; what was the cost of each barrel ? 4. A lady divided 20 oranges among her 5 daughters ; how many did each receive ? 5. If 4 casks of lime cost 12 dollars, what costs 1 cask ? 6. A laborer earned 48 shillings in 6 days; what did he receive per day? 7. A man can perform a certain piece of labor in 30 days; how long will it take five men to do the same ? 8. When 72 dollars are paid for 8 acres of land, what costs 1 acre ? What cost 3 acres ? 9. If 21 pounds of flour can be obtained for 3 dollars, how much can be obtained for 1 dollar ? How much for 8 dollars ? How much for 9 dollars ? 10. Gave 56 cents for 8 pounds of raisins; what costs 1 pound? What cost 7 pounds ? 11. If a man walk 24 miles in 6 hours, how far will he walk in 1 hour ? How far in 10 hours ? 12. Paid 56 dollars for 7 hundred weight of sugar; what costs 1 hundred weight? What cost 10 hundred weight ? 13. If 5 horses will eat a load of hay in 1 week, how long would it last 1 horse ? 14. In 20, how many times 2? How many times 4? How many times 5 ? How many times 10? 15. In 24, how many times 3 ? How many times 4? How many times 6 ? How many times 8? 16. How many times 7 in 21 ? In 28 ? In 56 ? In 35 ? In 14 ? In 63 ? In 77 ? In 70 ? In 84 ? 17. How many times 6 in 12? In 36 ? In 18? In 54 ? In 60 ? In 42 ? In 48 ? In 72 ? In 66 ? 18. How many times 9 in 27 ? In 45 ? In 63 ? In 81 ? In 99 ? In 108 ? 19. How many times 11 in 22? In 55 ? In 77 ? In 88? In 110 ? In 132 ? 20. How many times 12 in 36? In 60 ? In 72 ? In 84 ? In 120 ? In 144 ? 46. Division is the process of finding how many times one number is contained in another; or the process of separating a number into a proposed number of equal parts. In division there are three principal terms: the Dividend, the Divisor, and the Quotient, or Answer. The Dividend is the number to be divided. The Quotient is the number of times the divisor is contained in the dividend; or one of the equal parts into which it is divided. When the dividend does not contain the divisor an exact number of times, the excess is called a Remainder, and may be regarded as a fourth term in the division. The remainder, being part of the dividend, will always be of the same denomination or kind as the dividend, and must always be less than the divisor. Note. — When the divisor and dividend are of the same kind, the quotient will be an abstract number; and when they are not of the same kind, the quotient will be of the same kind as the dividend. 47. The Sign of Division is a short horizontal line, with a dot above it and another below; thus, •. It shows that the number before it is to be divided by the number after it. Thus 6 • 2 = 3 is read, 6 divided by 2 is equal to 3. Division is also indicated by writing the dividend above a short horizontal line and the divisor below ; thus, = 3 is read, 6 divided by 2 is equal to 3. There is a third method of indicating division, by a curved line placed between the divisor and dividend. Thus, the expression 6 ) 12 shows that 12 is to be divided by 6. 48. Short Division, or when the divisor does not exceed 12. Ex. la Divide 8574 dollars equally among 6 men. Ans. 1429 dollars. 46. What is division? What are the three principal terins in division ? What is the dividend ? The divisor? What is the quotient? The remainder? What will be the denomination of the remainder? How does it compare with the divisor? — 47: What is the first sign of division, and what does it show? What is the second, and what does it show? What is the third, and what does it show? — 48. What is short division ? |