SUBTRACTION Subtraction is the process of finding the difference of two quantities. The number subtracted is the subtrahend. The other number is the minuend. The sign of subtraction is, and is read minus. In subtraction, as in addition, the numbers must represent things of the same kind. In performing subtraction, the subtrahend is usually written directly below the minuend, the figures of like orders being placed in the same vertical column, as in addition. To each figure of the subtrahend is then added whatever number is required to produce the figure above it in the minuend, or if this is smaller than that of the subtrahend this figure plus 10. This "difference" is placed directly below: ILLUSTRATIVE EXAMPLE. From 913 take 537. 913 537 376 (a) 7 and 6 are 13 (3+10). Write 6, carry 1. 1 and 3 are 4; 4 and 7 are 11 (1+10). Write 7, carry 1. 1 and 5 are 6; 6 and 3 are 9. Write 3. (b) The usual explanation of subtraction is: 7 from 13 leaves 6; 30 from 100 leaves 70; 500 from 800 leaves 300. 913 376 (c) The work might be performed thus: 7 from 13 Addition gives a check on the work of subtraction; for the minuend is the sum of the difference and the subtrahend. Thus, in the above example, 913376 +537. Any method of subtraction may be used, but the additive method, here given first, is now regarded as the best. EXERCISE 6 The population of the United States in 1900 and in 1910 by states is stated in the following table. Find the increase in each state and territory from 1900 to 1910, and verify the answers. MULTIPLICATION Find, by adding, the value of 7+7+7+7+7, taking The process can be shortened, as follows: 7 x 535. Five 7's are 35. 7 five times. The Multiplication is thus seen to be a short method of addition when the numbers to be added are all the same. number to be repeatedly added is called the multiplicand. The number which indicates how many times the multiplicand is to be added is called the multiplier. The result of multiplication is a product. The multiplicand and multiplier are factors of the product. A number used with names of objects is a concrete number; thus, $6, 7 apples. A number that is not concrete is called a pure number, or an abstract number; thus, 6, 7, 12. Since the multiplier denotes "number of times," it must always be an abstract number. The multiplicand may be either concrete or abstract. Thus, $6 can be multiplied by 7, or added to itself 7 times, but $6 cannot be multiplied by 7 apples, or added to itself 7 apples times. The product is concrete if the multiplicand is concrete. The product is abstract if the multiplicand is abstract. In multiplication as in addition, as the sum is of the same kind as the addends, so the product is of the same kind as the multiplicand. Thus, 5 x 7 apples = 35 apples. The sign of multiplication is x, and is read multiplied by, or times. Thus, $34. x 7 means $34. is to be multiplied by 7. 7 x $34. is read 7 times $34. In each case, $34. is the multiplicand, and 7 is the multiplier. A change in the order of the factors does not change the product. MULTIPLICATION BY NUMBERS CONTAINING MORE THAN ONE DIGIT In multiplying by a number which contains two or more figures, we multiply each figure of the multiplicand by the units' figure of the multiplier, then by the tens' figure, and so on until all the figures are used, the work being arranged as shown in the following example. These separate results are called partial products. The multiplier is placed below the multiplicand, and the right-hand figure of each partial product is placed directly below the figure of the multiplier which was used in obtaining that partial product. Then the partial products are added. Example 1. Multiply 3562 by 249. 3562 249 32058 14248 7124 886938 EXPLANATION. 9 x 2 = 18; write 8 and carry 1. 9 x 654, 54+1 (which was carried from 18) 55; write 5 and carry 5. 9×5=45; 45 plus 550; write 0 and carry 5. 9 × 3 = 27; 27 plus 5 = 32; write 32. Then multiply by 4; = 4 x 2 = 8; write 8 in the tens' place because 4 is in the tens' place; 4 x 6 = 24; write 4 in the next place and carry 2; 4 x 5 = 20, 20 + 2 = 22; write 2 and carry 2; 4 × 3 = 12, 12 +2=14; write 14. Then multiply by the 2 of the hundreds' place, and add the three partial products. As the result of multiplying any figure by zero, or of multiplying zero by any number, is zero, we may omit the zeros of the multiplier, if we remember to place in their proper positions the products obtained by using the other figures of the multiplier; that is, to place each figure under the figure of its own order in the multiplier, as hundreds under hundreds, and so forth. C Example 2. Multiply 45073 by 2007. 45073 2007 315511 90146 90461511 = EXPLANATION. 45073 multiplied by 7 = 315511, the first partial product; 45073 × 2 90146, the last partial product. Its right-hand figure 6 is placed under the 2 of the multiplier, just as it would be placed, if instead of ciphers there had been two digits in the multiplier. EXERCISE 7 1. An office desk costs $25. How much will 3 such desks cost? 8 desks? 36 desks? 49 desks? 2. Eggs sell for 26 per dozen. Find the cost of 8 dozen; 18 dozen; 94 dozen. 3. There are 5,280 feet in a mile. How many feet in 19 miles? in 76 miles? 4. How many days in 39 weeks? 5. A contractor pays in wages $78 a day. How much will he pay in 78 days? 6. How many hours in 89 days? 7. A train travels at the rate of 34 miles an hour. How far will it run in 47 hours? 8. How many acres in a ranch containing 98 sections of land? (1 section 640 acres.) = 9. A degree on a meridian of the earth is about 69 miles. How many miles in 17 degrees? 10. A cubic foot of rock weighs 148 pounds. How many pounds do 3,297 cubic feet weigh? 11. The rent of a dwelling is $28 per month. Find the rent for 3 years. 12. A gallon of water contains 231 cubic inches. How many cubic inches in 368 gallons? |