Multiplication -- Decimals. 108. Find the product of 728.37 and .6. Operation. Explanation. 72'8.37 To multiply by .6 means to take 6 times 1 tenth of .6 the number. One tenth of 728.37 is 72.837. Six times 72.837 equals 437.022. 437.022 NOTE 1. — The separatrix is used to indicate the place of the decimal point in the number denoting one tenth of the multiplicand. NOTE 2.— The decimal point should be written in the product when, in the process of multiplication, the place is reached where it belongs. Do not multiply all the figures and then attempt to determine the place of the point. 109. Find the product of 746.2 and .25. Operation. Explanation. 7'46.2 To multiply by .25 means to take 25 times 1 hun dredth of the number. .25 One hundredth of 746.2 is 7.462. Five times 7.462 37.310 equals 37.310. Twenty times 7.462 equals 149.24. 149.24 37.310 + 149.24 = 186.550. 186.550 NOTE. — For direction as to the use of the separa trix, see Werner Arithmetic, Book II., p. 133, note. 110. Observe that when a multiplication of decimals is complete, the number of decimal places in the product is equal to the number of decimal places in the multiplicand and multiplier. 111. PROBLEMS. .48 (a) Find the sum of the ten products. (P. 324.) Multiplication - United States Money. 112. Find the product of $34.50 and 53.4. Operation. Explanation. $314.50 To multiply by 53.4, means to take 53 times the 53.4 multiplicand plus 4 tenths of the multiplicand. $13.800 One tenth of $34.50 is $3.45. $103.50 Four tenths of $34.50 is $13.80. $1725.0 Three times $34.50 is $103.50. $1842.300 Fifty times $34.50 is $ 1725. The sum of these three partial products is $1842.30. Practical application of the foregoing. If one acre of land costs $34.50, how much will 53.4 acres cost? 53.4 acres cost $1842.30. 113. Complete the following bill, and find the amount of it. INST. FOR THE BLIND, 1897. To GEO. E. SYBRANT, Dr. TO THE PUPIL. - Remember that any inaccuracy in solving business problems makes the work valueless. Accuracy ranks next in importance to integrity in the selection of an accountant. (P. 325.) Multiplication — Denominate Numbers. 114. Find the product of 3 tons 850 lb. and 8. Operation. Explanation. 3 tons 850 lb. Eight times 850 lb. equals 6800 lb. 8 6800 lb. equals 3 tons 800 lb. 27 tons 800 lb. Write the 800 lb. and add the 3 tons to the next partial product. Eight times 3 tons equals 24 tons; 24 tons plus 3 tons equals 27 tons. 3 tons 800 lb. multiplied by 8 equals 27 tons 800 lb. 115. PROBLEMS. 1. If one side of a square garden measures 6 rd. 8 ft., what is the perimeter of the garden? 2. The circumference of a certain bicycle track is 13 rd. 12 ft. How far does the rider travel who goes around it 12 times? 3. The length of a rectangular field is 15 rd. 10 ft. and the width 9 rd. 8 ft. What is the perimeter of the field ? 4. There is a walk 5 feet wide around a rectangular grass plat 3 rd. 6 ft. by 2 rd. 10 ft. What is the outside perimeter of the walk ? 5. How far does the person travel who walks once around the grass plat described in problem 4, if he keeps his track in the center of the walk ? (a) Find the sum of the five answers. 116. PROBLEMS. 1. If a train moves on the average at the rate of a mile in 1 min. 25 sec., in how long a time will it move 325 miles ? 2. If each of 56 grain bags contains 2 bush. 1 pk. 4 qt., how much grain in all ? 3. If the circumference of a wagon wheel is 15 feet 6 inches, how far will the wagon move while the wheel revolves 1000 times ? (P. 326.) Algebraic Multiplication. 117. EXAMPLES. No. 1. No. 2. 8 +4-3 a +36-40 4 4 32 + 16 – 12 4a + 12b - 16c No. 3. No. 4. 3a-2b + c 2a + 3b-5c d 2 d 3 ad — 2 bd +cd 4 ad + 6 bd - 10 cd 1. Observe that in the above examples we multiply each term of the multiplicand by the multiplier. As a matter of convenience it is best to begin with the term of the multiplicand on the left side and proceed from left to right. 2. Prove example No. 1 by uniting the terms of the multiplicand and comparing 4 times the number thus obtained with the number obtained by uniting the terms of the product. 3. Verify Example No. 2 by letting a=5, b=3, and c=2. 4. Verify Example No. 3, by giving the following values to the letters: a=7, b=4, c=3, d=5. 5. Verify Example No. 4 by giving any values you may choose to each letter. 118. PROBLEMS. 1. Multiply 3 ab — 2 bc + 5c by 2 d. 2. Multiply 2 ax + 4 bx – y by 5. 3. Multiply 3 bc + ab- bc by 3 d. 4. Multiply x-y+by 3 ab. 5. Multiply ax + bx – cx by 2 y. 6. Verify each of the above problems by giving the following values to the letters: a = 3, b=2,c=4, d=5, x= 7, y = 6,2=8. Algebraic Multiplication. 119. EXPONENT. 1. a xa, or aa which means a multiplied by a, is usually written a?. This is read a square or a second power. 2. 63 (to be read b cube or b third power) means b taken three times as a factor. It is b xb xb. 3. a4 (to be read a fourth power, or simply a fourth) means that a is taken four times as a factor. It is a xa x. аха. 4. The small figure at the right of a letter tells the number of times the letter is to be used as a factor. The figure so used is called an exponent. When the exponent is 1, it is not usually expressed; thus a means ał. 120. PROBLEMS. On the supposition that a= 2, b=3, and c= 4, find the numerical value of each of the following expressions : 1. a2 + 2 ab + 62 6. 5 alb – 2 bc 2. 3 ab2 + 5 be 7. a313 - 3 3. 4a4b2 + 3 622 8. ab2 + 2 4. 2 a2b + 2 ab2 9. a252c2-ab3 5. 3 62c2 + 5 ab 10. 2 a2622 (a) Find the sum of the numerical values of the above. 121. EXAMPLES. No. 1. No. 2. 4 ax + 2 by + c - 3 box + 2 by-C a2 2 62 4 aʼx + 2 a+by + a-c 6 84x + 4b3y - 2 boc 1. Observe that in the above examples the product of the coefficients and the sum of the exponents is taken. 2. Verify each of the above examples by letting a = 2, b=3, c=4, x = 5, y=6. |