Determine the different methods of expressing 24, with the sum of two numbers. Also, 42 and 54. Express 18 with the sum of three numbers, in all the possible ways. Likewise, 27 and 35. What is the sum of 3+4+6= 5+6+7= 8+9+3= 7+6+6= 7+8+9= 5+4+3+6= 7+2+4+9= 8+6+6+7= 6+4+7+9= 3+4+7+9+2= 5+6+6+4+8= 7+8+6+7+9= 5+6+7+4+7+3= 8+6+6+5+5+9= The same exercise ought to be also proposed under the following form, where the pupil is to add up the vertical columns in each square, and afterwards also the horizontal rows : 48. To find the sum of two numbers, 36 and 45, put them under the form 30+6 and 40+5; the required sum is equal to 30+40+6+5. Now, three tens and four tens are seven tens, and six units and five units are eleven units; therefore, the sum is 70+11=70+10+1=81. Whence we perceive that the tens are to be put together separately, and likewise the units. The same method is followed when the numbers are hundreds, thousands, &c. Example: What is the sum of 4631+13524+546+78? Adding up, separately, the units, the tens, the hundreds, &c., we have 17 thousands +16 hundreds + 16 tens+19 units, or 17000 +1600+160+19; the same process being repeated for these numbers, we have 1800+700+70+9, or 18779. This operation is much shortened by the following method: 4631 13524 546 78 18779 The numbers are placed under each other, with the units of the same order in the same vertical column; then adding each number thus: 1+4=5, 5+6=11, 11+8=19 units or one tens +nine units, we set down the nine units, and the one tens we carry mentally to the next or tens column, which sum is 17 tens, or one hundreds and seven tens; the seven is set down in the place of the tens, and the one hundreds are carried to the third column, and so on, which explains why we begin with the right hand column. 49. From the definition of number, it is evident that, to be added up, numbers must be of the same kind, unless they have a common name. Example 5 lions+4 bears+6 foxes=15 animals. 50. To ascertain whether an addition is correct, the operation is repeated, beginning at the bottom of each column; and if the results agree, it is probable the work is right. Or, add the figures of each column, omitting the upper row, then to this sum add the upper row; if the result is equal to the previous answer, the work is probably correct. 51. EXERCISES. 1. On Monday, 7645 persons passed over a bridge; on Tuesday, 8965; on Wednesday, 4905; on Thursday, 3679; on Determine the different methods of expressing 24, with the sum of two numbers. Also, 42 and 54. Express 18 with the sum of three numbers, in all the possible ways. Likewise, 27 and 35. What is the sum of 3+4+6= 5+6+7= 8+9+3= 7+6+6= 7+8+9= 5+4+3+6= 7+2+4+9= 8+6+6+7= 6+4+7+9= 3+4+7+9+2= 5+6+6+4+8= 7+8+6+7+9= 5+6+7+4+7+3= 8+6+6+5+5+9= The same exercise ought to be also proposed under the following form, where the pupil is to add up the vertical columns in each square, and afterwards also the horizontal rows: 48. To find the sum of two numbers, 36 and 45, put them under the form 30+6 and 40+5; the required sum is equal to 30+40+6+5. Now, three tens and four tens are seven tens, and six units and five units are eleven units; therefore, the sum is 70+11=70+10+1=81. Whence we perceive that the tens are to be put together separately, and likewise the units. The same method is followed when the numbers are hundreds, thousands, &c. Example: What is the sum of 4631+13524+546+78? Adding up, separately, the units, the tens, the hundreds, &c., we have 17 thousands +16 hundreds +16 tens +19 units, or 17000 +1600+160+19; the same process being repeated for these numbers, we have 1800+700+70+9, or 18779. This operation is much shortened by the following method:-- 4631 13524 546 78 18779 The numbers are placed under each other, with the units of the same order in the same vertical column; then adding each number thus: 1+4=5,5+6=11, 11+8=19 units or one tens +nine units, we set down the nine units, and the one tens we carry mentally to the next or tens column, which sum is 17 tens, or one hundreds and seven tens; the seven is set down in the place of the tens, and the one hundreds are carried to the third column, and so on, which explains why we begin with the right hand column. 49. From the definition of number, it is evident that, to be added up, numbers must be of the same kind, unless they have a common name. Example 5 lions+4 bears+6 foxes=15 animals. : 50. To ascertain whether an addition is correct, the operation is repeated, beginning at the bottom of each column; and if the results agree, it is probable the work is right. Or, add the figures of each column, omitting the upper row, then to this sum add the upper row; if the result is equal to the previous answer, the work is probably correct. 51. EXERCISES. 1. On Monday, 7645 persons passed over a bridge; on Tuesday, 8965; on Wednesday, 4905; on Thursday, 3679; on Friday, 6844; on Saturday, 7638; on Sunday, 2944. many persons crossed it during the whole time? How It is evident that we have here to find the sum of the 42620 which is equal to the sum required. 2. 86420135+9873546+998739+6978+407369= ? 3. 439+2704+1743206+290704+7680+15647 = ? 4. Find the sum of thirty thousand seven hundred and seven + forty-six thousand nine hundred and seventy-five+four hundred and sixty-four thousand eight hundred and seventy-eight +three thousand and seven+nine millions three hundred and forty-nine thousand three hundred and fifty-six. 5. In a school, the pupils are at study in four rooms; in the first, there are 128 pupils; in the second, 84; in the third, 93; and in the fourth, 65. The number of pupils in the school is required. 6. How many times does a clock (which only strikes the hour) strike, whilst the hour hand goes round the dial? 7. In what year did a man die who was born in 1766, and died at the age of 85 ? 8. A person is born in 1793; in what year will he be 74 years of age? 9. Virgil was born near Mantua, 70 years B.C. years ago is it? 10. Noah's flood occurred 2356 years B.C. How many How long is it since that event? |