Explanation.—I annex as many ciphers to the multiplicand as there are ciphers in the multiplier. Analysis: 237 × 100 = 237 hundreds = 23700. Find the following products: Operation 23700 71. There are 100 cigars in a box; how many are there in 13 boxes? 72. One cubic foot of water weighs 1000 ounces; what is the weight of 137 cubic feet? 73. Ten mills make a cent, and 100 cents make a dollar; how many mills are equal to $17. 69. To multiply when ciphers are on the right in one or both factors. (74) Multiply 7600 by 30. Explanation.—I multiply as if there were no ciphers on the right in the numbers; then annex to the product as many ciphers as there are on the right in both factors. ANALYSIS: 7600 X 30 =76 hundreds X 3 tens = 228000. 75. Multiply 2100 by 400. 76. Multiply 3650 by 1200. 77. Multiply 402000 by 350000. Operation. 7600 30 228000 = 228 thousands == 78. The salary of the President of the United States is $50,000 a year; how much will it amount to in 120 years? 79. Sound travels about 1140 feet per second; how far does it travel in 5 minutes? 80. If a man make 2100 steps in walking a mile, how many steps will he make in walking a distance of 24900 miles? 70. To multiply by an integer which is a little less than 10, 100, 1000, etc. 376 by 3 and subtract the product, 1128, from 37600 to obtain the true result. Multiply: 82. 377 by 9. 83. 4375 by 997. 84. 593 by 98. 85. 64013 by 9999. 86. If an engine travel at an average speed of 34 miles an hour, how far can it travel in 99 hours? 87. 24 sheets of paper are a quire, and 20 quires are a How many sheets are there in 998 reams? ream. 88. Find the value of 16642 × 996 × 99. 71. To multiply two numbers when one part taken as units, in the multiplier, is a factor of another part so taken. (89) Multiply 570372 by 120324. Operation. 570372 120324 1711116 13688928 68444640 68629440528 Explanation. In the multiplier I select a part, expressed by one or more figures, which is a factor of another part, represented by one or more figures. Thus, I select 3, 24 and 120, since 8X3 = 24, and 5X 24: = 120. I now multiply 570372 by 3, and obtain the first partial product, 1711116; then multiply this product by 8, (= 570372 × 24), writing this right hand figure under the 4. Next, I multiply the second partial product (13688928), by 5, (= 570372 × 120), and write the right hand figure of this product under the 0 of the multiplier; and, finally, add to obtain the total product. 72. To multiply by an integer more than 10 and less than 20. (96) Multiply 7638 by 16. Explanation. I consider a cipher to be placed before the multiplicand; then multiply by the unit's figure of the multiplier, and in addition Operation. 07638 122208 Ans. to carrying as usual, I also carry the figure last multiplied. Thus: 6 X 8=48, write 8 and carry 4 and 8, or 12; 6 × 3 = 18, and 1230, write 0 and carry 3 and 3, or 6; 6 X 6 = 36, and 6 are 42, write 2 and carry 4 and 6, or 10; 6 X 7 = 42, and 10 = 2 and carry 5 and 7, or 12; 6 × 0 = C, and 12 12. Find the following products: 97. 456 X 16. 99. 345 X 15. 98. 5086 X 17. 100. 7348 X 13. 52, write 101. A man buys 14 horses, for which he pays on an average $175 each; how much do they all cost? 102. A farmer plants on each of 19 acres 8945 hills of corn; how many does he plant in all? For other contractions, see Appendix. QUESTIONS. What is multiplication? The multiplicand? The multiplier? The product? Sign of multiplication? Factors? Can a concrete number be multiplied by a concrete number? Give the general rule? How is multiplication proved? How do you multiply when the multiplier is some power of 10? When it has Os on the right? When it is a little less than some power of 10? When it is between 12 and 20? DIVISION. 73. (1) I have 12 apples which I wish to divide equally among 4 boys; how many apples can I give to each? If I give each boy one apple, it will require 4 apples, and 8 apples will be left. If I give each of them another apple, 4 apples will be left. Again, if I give each of them an apple a third time, there will be none left. Hence, I can give each boy one apple three times, or one apple taken as many times as 4 is contained in 12, which is three apples. Again, since 12 apples are to be divided into 4 equal parts, each part will be one-fourth of 12 apples, which is 3 apples. This is Division. 12 ཐཱཁ***° g。 74. Division is a short method of finding how many times one number is contained in another of the same kind; or the process of finding one of the equal parts of a number. 75. The number to be divided is the dividend; the number by which it is divided, the divisor; and the result, the quotient. 76. When a part of the dividend is left after the division is performed, it is the remainder, and must always be less than the divisor. When there is no remainder, the division is said to be exact. 77. The sign of division is÷, which is read divided by, or contains, and indicates that the number before it is to be divided by the number after it. NOTES.-1. When the dividend is a concrete number and the divisor an abstract number, should be read divided by. (1) 24 days 64 days, is read, 24 days divided by 6 equals 4 days. 2. When the dividend and divisor are both concrete numbers, + should be read contains; for, really, there is no such thing as divid ing by a concrete number, except as a measure. (2) 16 pints 8 pints=2, is read 16 pints contains 8 pints 2 times. In such cases it is better to use the colon (:). See Ratio. 3. When the dividend and divisor are both abstract numbers, ÷ may be read divided by, or contains. 4. Division is also denoted by writing the divisor under the dividend, with a line between them, or by placing the divisor on the left of the dividend, with a curved line between them. 5. Division is the reverse of multiplication; the latter unites equal parts into one number, and the former separates a number into equal parts. 78. PRINCIPLES.-1°. When the dividend and divisor are like numbers, the quotient is an abstract number. 2°. When the divisor is an abstract number, the dividend and quotient are like numbers. 3°. The dividend is equal to the product of the divisor and quotient, plus the remainder. Oral Drill in Matter and Method. 79. Name, in order, the numbers exactly divisible 1st. 2d. 3d. 4th. 5th. 6th. 7th. 8th. 9th. |