6. The factors of a certain product are the sum of 7741 and 1234, and the difference between 14392 and 8673; what is the product? 7. What was the barley crop of Illinois in 1870 worth, 111600 acres being planted with this grain, the average yield being 20 bushels per acre, and the value per bushel 62 cents? Ans. $1383840. 8. A plank-road company with a capital of $250000 built 42 miles of road, at an expense of $4756 per mile; how much of the capital was left? 96. The Multiplier any number of 9's.—To multiply by any number of 9's, annex to the multiplicand as many ciphers as there are 9's, and subtract the multiplicand from the result. On the same principle, how would you multiply by 98? By 997? By 9998? By 101? By 1002? 97.-EXAMPLES FOR PRACTICE. Use short methods, when practicable. 1. From 9999 times 87654 take 1293 times 9887. 2. Two vessels separate, one going east 45 miles a day, the other west 95 miles in 24 hours. How far apart will they be at the end of 16 days? 3. In the state of New York, 256336 acres were planted with potatoes in 1870. The average yield was 98 bushels to the acre, and the average value 65 cents a bushel. What was the total value of the crop? 4. What was the cost of public-school education in the state of New York in 1870, there being 11750 schools, with an average of 85 pupils each, if the expense for each pupil was $10.047 ? 5. The German Empire, in 1872, contained 210220 square miles; its population was 191 to the square mile. What was its population? 6. What number is that, to which if 4367 × 999 is added, the sum will be ten million and one ten-millionth ? Ans. 5637367.0000001. 7. In 1870, 59488 immigrants arrived in the United States from Europe. If, on an average, they brought with them specie to the value of 100 francs apiece, what was the whole amount in federal money thus brought into the country, rating the franc at $0.186 ? Ans. $1106476.80. 8. The difference between two numbers is 7.6324; the greater is 99999 times 123465; what is the less? 9. Suppose a wagon to go 6 miles an hour, and each of its wheels to turn 420 times in a mile; how many revolutions will all four wheels make in 3 hours? 10. The flow of water in the Mississippi, at Memphis, Tenn., is estimated at 434000 cubic feet a second. What is the weight of the water that passes that point in an hour, allowing each cubic foot to weigh 63 lb. ? Ans. 98431200000 lb. 11. In 1870, 451714 acres were planted with cotton in N. Carolina; 601764, in S. Carolina; 1330491, in Georgia; 140909, in Florida; 1437272, in Alabama; 1644512, in Mississippi; 920700, in Louisiana; 900937, in Texas; 711734, in Arkansas; 526184, in Tennessee; and 218823, in other states. The average yield was about 200 lb. to the acre. What was the entire crop? Ans. 1777008000 lb. 12. Paid for 999 acres of woodland $27 an acre. The cost of clearing was $9241; the wood brought $8257; and the cleared land sold for $32.16 an acre. profit ? What was the Ans. $4170.84. 13. The multiplicand is 428, the multiplier 99. How much do I add to the product, if I increase the multiplicand by 48? How much do I increase the product, if I add 48 to the multiplier? How much, if I multiply the multiplier by 48? How much, if I multiply the multiplicand by 48? Involution. 98. Powers and Roots.-The products arising from multiplying a number by itself, and this product and the succeeding ones by the same multiplier, are called the Powers of that number. The number so multiplied is called the Root of the several powers. 99. Involution is the process of raising a number to one of its powers. 2 X2 X2=8. 8 is a power of 2, and 2 is the root of 8. The process is Involution, and 2 is said to be involved or raised to the 3d power. 100. Powers, how distinguished and indicated.-Every number has an infinite number of Powers, distinguished as the First, Second, Third, Fourth, etc., according to the number of times that the root is used as a factor. They are indicated by a figure, called an Index (plural, indices), or Exponent, placed above the number at the right; as, 23, 24, 25. 101. The First Power is the number itself; its index is never written. The Second Power is also called the Square, and the Third Power the Cube. First power of 2, Second power, or Square, of 2, 2 22 = 2 × 2 = 4 232 X2 X2 =8 24 = 2 × 2 × 2 × 2 = 16, etc. 102. RULE.-To involve a number, multiply it by itself, and the successive products by the same multiplier, as many times, less 1, as there are units in the index of the power required. 3X3=9. There is one multiplication, though 3 is used as a factor twice, and 9 is the second power. In stead of multiplying by the original number each time, powers already found may be used as multipliers. Thus, for the 7th power, the 4th may be multiplied by the 3d. But observe that the resulting power will be that denoted by the sum of the indices of the multipliers, not their PRODUCT. For, 4 x 4 (4 x 4 x 4 × 4) × (4 × 4 × 4) = 4 = 103. The product of like powers of different roots is equal to the like power of their product. 32 X 52 = 152; for, bers. (3 × 3) × (5 × 5) = (3 × 5) × (3 × 5) = 152 104.-EXAMPLES FOR PRACTICE. 1. Repeat the squares of the numbers from 1 to 20. power. 5. Multiply 28 by 25; 63 by 73; 5 by 55. 6. Find the value of the following, using short methods of multiplication: 99; 635; 1824; 93 × 113; 4272; 108123. Sum of ans. 1266393916364. SHORT METHODS OF SQUARING. 105. By inspecting the results of actual multiplication, we obtain the following rules : RULE I.-To square, at sight, a number expressed by two or more 1's, beginning at the units' place, write the Thus, digits 1, 2, 3, 4, etc., in order, up to and including the fig 111212321 1111'1234321 11111'123454321 ure which corresponds with the number of 1's in the root, and then the digits running down from this figure to 1 inclusive. If there are more than nine l's in the root, after writing 9 in the power, go on with the numbers above 9, writing the units' figure only, and carrying the tens' figure to the next number. Thus, 11111111111112 = 123456789(10)(11)(12)(13)(12) (11) (10)987654321 To square any number expressed by a digit repeated (as 222, 7777, etc.), multiply the square of the corresponding number of 1's, as obtained above, by the square of the repeated digit. Thus, 444' (111 × 4) = 111' x 4' = 12321 × 16 = 197136 Ans. = 106. RULE II.-To square, at sight, a number expressed by two or more 9's, beginning at the units' place, write 1, as many ciphers less one as there are 9's in the Thus, 999801 9992 = 998001 999999980001 given number, 8, and as many 9's as ciphers. 107.-EXAMPLES FOR PRACTICE. According to the above rules, square 111111; 99999; 11111111; 333; 55555; 1111111111; 8888; 777777; 666566; 9999999; 999999999. |