to place the multiplier under the multiplicand, and to proceed as follows: : 2347 multiplicand 14082 product 6 times 7 units are 42 units, or 4 tens and 2 units, set down 2 in the place of the units, and reserve 4 tens for the next place; 6 times 4 tens are 24 tens, and the 4 tens reserved are 28 tens, or 2 hundreds and 8 tens, set down the 8 tens and carry the 2 hundreds to the next place; 6 times 3 hundreds are 18 hundreds, and 2 are 20 hundreds, or 2 thousands and 0 hundreds, set down 0 in the hundreds' place, and reserve 2 thousands; 6×2 thousands=12 thousands, and 2 thousands are 14 thousands, set down 14. The product is 14082. Therefore, this operation consists in multiplying each figure of the multiplicand by the multiplier, beginning with the units, and setting down the results under the figure multiplied by, reserving, as in addition. 72. Let it now be required to multiply 4598 by 365. The meaning of which is to repeat 4598, 5 times+60 times +300 times, and to add the results. Place the multiplier under the multiplicand, so that the units of the same order be under each other : 4598 The product of 4598 and 5 is found as in (§ 71); to multiply 4598 by 6 tens, we proceed as if we multiplied by 6 units, taking care to place the first figure obtained in the tens' place, because 6 tens x 8 units are 48 tens, or 4 hundreds +8 tens; 6 tens × 9 tens 54 hundreds, and 4 hundreds reserved=58 hundreds, or 5 thousands +8 hundreds, &c.; lastly, to multiply 4598 by 3 hundreds, multiply as if it were by 3 units, writing the first figure in the column of the hundreds, because 3 hundreds × 8 units are 24 hundreds, or 2 thousands+4 hundreds, &c. Add the partial products, and the sum is the product required. From which it is inferred that we must multiply one of the factors by each figure of the other, placing the unit of each line in the place under the figure of the multiplier from which it came, and add the several lines together. 73. It has been explained in numeration, that the respective values of figures increase ten-fold as we proceed from the units' place towards the left hand. Let us take any number, 24, for instance, put a zero at the right hand, it becomes 240, that is to say, the units have become tens and the tens hundreds, or the number has acquired ten times its former value. If two zeros had been placed at the end of 24, the number would have one hundred times its previous value, and so on. Therefore, to multiply a number by 10, 100, 1000, &c., place at the right hand of that number 1, 2, 3, &c., zeros. 74. Let us multiply 30 by 20. If we were to omit the zeros, and say 3×2=6, this result would be evidently 10 x 10, or 100 times too small, then affix both zeros at the end of the product and we have 600 for the answer. Therefore, when one or both factors have ciphers on the right hand multiply them together, omitting the ciphers, and then place on the right hand of the product as many ciphers as there are on the right hand of the factors. 75. Since 5x7=7×5, or 14 × 25=25 × 14, &c., we shall be able to ascertain that a multiplication is right by transposing the factors; and if the product thus obtained agree with the product found previously, the operations have been correctly performed in both cases. 76. EXERCISES. 1. How many lines are there in a book containing 247 pages, and each page 18 lines? Since each page contains 18 lines, 247 pages will contain 247 × 18 lines. 2. Sound moves about 1142 feet per second. How many feet will it move in 144 seconds? By the question, we know that in one second sound travels 1142 feet; therefore, in 144 seconds it travels 144 x 1142 feet. Multiplication. Factors{144 multiplier 1142 multiplicand Proof. 144 1142 288 576 144 144 164448 3. In a plantation there are 274 rows of trees, each containing 485 trees. How many trees are there in the plantation? 4. How many times does the hammer of a clock strike in 365 days, or one year, at 156 strokes per day? 5. One mile contains 1760 yards. How many yards are there from the earth to the moon, the distance being 240000 miles? 6. Perform the following multiplications :—24 × 762; 63×196; 5704×487; 7800 × 365; 3600 × 49200. 7. How many hours are there in 365 days, of 24 hours each? 8. The equator, like every other circumference, is supposed to be divided into 360 parts, or degrees, each containing 69 miles. How many miles is it round the earth? 9. Light travels with a velocity of 192000 miles per second, and the sun's light reaches us in 8 minutes 13 seconds, or in 493 seconds. What is our distance from the sun? 10. Two persons start at the same time, one from London and the other from Worksop; the first travels 18 miles per day, and the other 19 miles; they meet after travelling four days. How far are the two places from each other? 11. A person owns 79 horses, worth £26 each, on an average; and 347 head of cattle, valued at £13 each. What is the value of all? 12. A draper bought 347 yards of cloth, at 27 shillings per yard, and 276 yards at 18 shillings, for which he paid 12337 shillings. How much has he yet to pay? 13. Reckoning every year of 365 days 6 hours, how many hours is it since the birth of our Lord, which took place 1853 years ago? 14. A boy who is 14 years 8 months and 14 days old, wishes to know how many seconds he has lived; three years of his life were leap years; of the 8 months, 5 were of 31 days, 2 of 30 days, and the last of 28 days. 15. A wheel of a locomotive engine makes 25 revolutions in one second. How many revolutions will it make in 7 hours 17 minutes and 24 seconds? Also, how many yards has it run over, the wheel being six yards round? 16. A merchant employs 12 clerks, three at the rate of £240 annually, five at the rate of £180, and four at £80; 14 servants, eight of them at £22 each, on the average, and the six others at £15 each; for house rent, &c., he spends £1400; the house-keeping expenses amount to £2200. In one year he did business to the amount of £224600, upon which he laid out £206700. What were his profits at the year's end? DIVISION. 77. If eight apples are to be parted, or divided equally between two persons, how many will each receive? In this question it is proposed to make as many equal shares, or portions of the apples, as there are persons, or to divide 8 by 2, or to see how many times 2 is contained in 8, and the answer is 4. Other questions similar to this might be proposed by the teacher, and solved mentally. 78. Therefore, division consists in dividing numbers into equal parts, or in finding how many times one number is contained in another. 79. The number which is divided is called the dividend, the number we divide by, the divisor, and the number resulting from the operation is named the quotient. In the previous question, which number is the dividend, which the divisor, and which the quotient? 8 8, divided by 2, equal 4, is expressed thus: 8÷2, or 4. |