Example 2.-From 65074 take 4054. 65074 4054 61020 Commencing as before, at the right-hand figure, 4, we find the figure immediately above it is 4 also; and when 4 is taken from 4, nothing remains; a nothing is therefore placed under the line: passing to the next figure, we say 5 from 7, and 2 remains; the next figure being a nothing, and the one above it also a nothing, nothing remains this being placed under the line, we pass to the next figure, and say 4 from 5, and 1 remains; passing to the next, we find no figure under the 6; there is therefore nothing to be taken from it, and 6 is put down under the line; the whole remainder is found to be 61020. These examples, it will be found, are very simple, because all the under figures of the numbers are either less or exactly equal to the figures immediately above them; but it often happens that some of the under figures are greater than those just above them. We have, therefore, to shew what is to be done in that case. Example 3.-From 8432 take 6815. 8432 6815 1617 Here, it will be observed, that 5, the figure with which we begin, stands to be subtracted from 2; but 5 cannot be taken from 2, because 5 is a greater number than 2. We add, therefore, 10 to the 2, making it 12, and then say 5 from 12, and 7 remain. Now, because 10 was added to the 2, we are said to have borrowed one from the figure just before it, which one is to be repaid by carrying, or adding 1 to the next under figure we come to. Passing, therefore, to the next figure in the above example, we add 1 to the 1 making 2, and 2 from 3, the upper figure, 1 remains; next we find 8 from 4; but as we cannot take 8 from 4, we add 10 to it, making it 14, and we say 8 from 14, and 6 remain; and here, again, having borrowed one, we carry it, as before, to the next under figure 6, making 7, and 7 from 8, 1 remains. PROOF.-To prove that any question in subtraction is correctly wrought, add the remainder to the lower row of figures, and if their sum be equal to the upper row, the account is correct. For example, treat the foregoing question as follows: 8432 6815 1617 8432 Here it is seen that, by adding the remainder, 1617, to the lower row, 6815, we bring out 8432, or the upper row, as the result or sum-the account is, consequently, correct. Subtraction, therefore, may always be proved by addition. 27. What is the difference between 37651 and 28430? 28. What is the difference between 7326 and 9718? 29. How much is 3681 less than 5340? " 50473 30. How much is 53160 more than 2687 ? 31. A woman went to market with 346 eggs in her basket; she sold 288. How many did she return with? Ans. 58 32. From a piece of cloth consisting of 153 yards, a merchant sold 27 yards to A, and 48 yards to B. How much of the piece remained? Ans. 78 33. Lucy went to make some purchases with 34 shillings in her purse. She bought a pound of tea for 4 shillings, pounds of sugar for 2 shillings, 6 pounds of soap for 3 shillings, and 6 yards of linen for 14 shillings. How much should she have over? Ans. 11s. 34. America was discovered in 1492. How long is it since? Ans. (in 1854) 362 35. What age now is a person who was born in 1803 ? Ans. (in 1854) 51 SIMPLE MULTIPLICATION. MULTIPLICATION is the method of ascertaining what a given number will amount to, when repeated a certain number of times; as, for example, what 6 repeated 4 times will amount to; or, in other words, how many are 4 times 6? We know how this is to be done by Addition: the figures are placed under each other, and we say, 6 and 6 are 12, and 6 are 18, and 6 are 24. In Multiplication, however, a different plan is adopted: the 6 and 4 are said to be multiplied together-that is, we multiply 6 by 4, and say at once, 4 times 6 are 24. It is usually most convenient in multiplying two numbers together, to multiply the larger number by the smaller. The number to be multiplied is called the multiplicand; the number that multiplies it, the multiplier; and the result of multiplying the two numbers together, the product. A number that is produced by the multiplication of two other numbers, is called a composite number-as, for instance, 30, which is the product of 5 and 6. The 5 and 6, are called the factors of 30, and 30 is also said to be a multiple of either of these numbers. A number which cannot be produced by the multiplication of two other numbers, is called a prime number. Multiplication is denoted by a cross of this shape x; thus, 3×8= 24, signifies, that by multiplying 8 by 3, the product is 24. The process of multiplication is carried on by means of the following Multiplication Table, which shews how much certain numbers amount to, when multiplied together. The table should be carefully committed to memory, as a knowledge of it is of great value in arithmetic, and saves much trouble in after-life. The following is a more condensed form of the Multiplication Table. The calculations are carried as far as 20 times 20: 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 2 4 6 3 6 4 6 12 18 24 7 14 21 28 8 16 24 32 9 18 27 36 10 20 30 40 11 22 33 44 12 24 36 48 13 26 39 52 14 28 42 56 15 30 45 60 16 32 48 64 65 78 70 75 88 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40! 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 30 36 42 48 54 60 66 72 78 84 90 96 102 108 114 120 35 42 49 56 63 70 77 84 91 98 105 112 119 126 133 140 40 48 56 64 72 80 88 96 104 112 120 128 136 144 152 160 45 54 63 72 81 90 99 108 117 126 135 144 153 162 171 180 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 55 66 77 99 110 121 132 143 154 165 176 187 198 209 220 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 91 104 117 130 143 156 169 182 195 208 221 234 247 260 84 98 112 126 140 154 168 182 196 210 224 238 252 266 280 90 105 120 135 150 165 180 195 210 225 240 255 270 285 300 96 112 128 144 160 176 192 208 224 240 256 272 288 304 320 85 102 119 136 153 170 187 204 221 238 255 272 289 306 323 340 90 108 126 144 162 180 198 216 234 252 270 288 306 324 342 360 95 114 133 152 171 190 209 228 247 266 285 304 323 342 361 380 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 17 34 51 68 19 38 57 76 20 40 60 80 80 When any number in the top row of the table is multiplied by any number in the left-hand side row, the product is found in the compartment or square beneath the former, and opposite the latter. Thus -2 times 2 are 4; 5 times 6 are 30; 20 times 20 are 400. RULES FOR MULTIPLICATION. I. WHEN THE MULTIPLIER DOES NOT EXCEED 12. RULE. Write down the number to be multiplied; place the multiplier below it, at the right-hand side, and draw a line under them; then begin at the right hand, and multiply each successive figure in the multiplicand, by the multiplier; mark down the units of each product below the figure multiplied, and carry the tens, as in Addition, to the next figure, when it is multiplied in its turn. When all the figures have been multiplied, the result is the answer required. Example.-Multiply 27 by 9. 27 Multiplicand. 9 Multiplier. 243 Product. Here, beginning with the right-hand figure, we say 9 times 7 are 63; putting down 3, we carry 6 to the next figure, and say, 9 times 2 are 18, and 6 which was carried, make 24; and writing down both of these figures, as there are no more to multiply, the product is found to be 243. In Multiplication, the tens are carried, and the remainders marked down at each stage of the process, for the same reasons as are explained in Simple Addition, page 7. II. WHEN THE MULTIPLIER EXCEEDS 12. RULE. Write the multiplier below the number to be multiplied, placing units under units, tens under tens, and so on. Then, beginning at the right, multiply the number by each figure of the multiplier in succession, placing each new line of products below the previous one, but a place further to the left; so that each line may commence exactly below the figure in the multiplier producing it when a nothing occurs in the multiplier, pass on to the next figure. Then add up all the lines of products, and their sum is the product required. Examples.-Multiply 5463 by 34; and 76843 by 4563. 1. 5463 34 21852 16389 185742 Ans. In example 1, the number is multiplied first by the 4, the product of which being written down, we proceed to multiply by the 3, and place its product below the other, but one place further to the left, the first figure being put immediately below the 3. A line is then drawn under them, and they are added together. 2. 76843 4563 230529 461058 384215 307372 350634609 Ans. THE REASON for writing each successive line of products, a place further to the left, as in the above examples of Multiplication, is, that each figure of the multiplier, counting from right to left, is a place higher in order than the one next to it, and, therefore, in multiplying, each new product will be of a higher order than the one preceding it, and must accordingly be written one place further to the left. |