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 lh 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 at the rate of a mile in 1 min. 25 sec, in how long a time will it move 325 miles? 2. If the circumference of a wagon wheel is 15 feet 6 inches, how far will the wagon move while the wheel revolves 1000 times? , 3atf - 26a" + erf 4ad + 66a" - 10co" 1. Observe that in the above examples we multiply each term of the multiplicand by the multiplier. 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, 6 = 3, and c = 2. 4. Verify example No. 3 by giving the following values to the letters: a = 7, 6 = 4, c = 3, d = 5. 5. Verify example No. 4 by giving any values you may choose to each letter. 118. Problems. 1. Multiply 3a6 - 26c + 5c by 2d. 2. Multiply 2ax + 46a; — y by 5. 3. Multiply 36c + ab - be by 3d. 4. Multiply x - y + z by 3a6. 5. Multiply ax-\-bx — ex by 2y. 6. Verify each of the above problems by giving the following values to the letters: a — 3, 6 = 2, c = 4, d = o, x = 7, y = 6, z = 8. Algebraic Multiplication. 119. Exponent. 1. a x a, ox aa which means a multiplied by a, is usually written a2. This is read a square or a second power. 2. V (to be read b cube or b third power) means b taken three times as a factor. It is b x b x b. 3. a4 (to be read a fourth power, or simply * fourth) means that a is taken four times as a factor. It is a x a x a x a. 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 a\ 120. Problems. On the supposition that a = 2, b = 3, and c = 4, find the numerical value of each of the following expressions: 1. a' + 2ab + b' 6. 5d'b - 2be 2. 3ab2+5bc* 7. a3b3 - c3 3. 4a2&2 + 36V 8. aV + c' 4. 2a2b 4- 2aV 9. a'bY - aW 5. SbV + 5ab 10. 2aW (a) Find the sum of the numerical values of the above 121. Examples. No. 1. No. 2. 4ax + 2by + c Wx -\-2by-c a2 2V 1. Any side of any one of the above figures is parallel to the opposite side of the same figure. Hence the figures are called parallelograms. 2. Each of the above figures has four sides. Hence the figures are called quadrilaterals. 3. If all the sides of a figure are equal, the figure is said to be equilateral. 4. If all-the angles of a parallelogram are right angles (angles of 90°) the figure is said to be rectangular. 5. Which of the above figures are equilateral? 6. Which of the above figures are rectangular? 7. Which of the above figures are not equilateral? 8. Which of the above figures are not rectangular? 9. Which of the above figures are parallelograms? 10. Which of the above figures are quadrilaterals? 11. Can you draw a quadrilateral that is not a parallelogram? 12. Is any one of the above figures an equilateral rectangular parallelogram? 13. In a rhomboid or rhombus two of the angles are less than right angles and two of them are greater than right angles. Convince yourself by cutting a rhomboid from paper and comparing it with rectangular figures that two of the angles of a rhomboid are as much less than two right angles as the other two are greater than two right angles.' 123. Miscellaneous Reviews. 1. If one of the angles of a rhombus is an angle of 80 degrees, what is the number of degrees in each of the other angles? 2. Draw a rhomboid one of whose angles is an angle of 70; give the number of degrees in each of the other angles. 3. An oblong has four right angles. The angles of a rhomboid are together equal to how many right angles? 4. If an oblong is a feet long and b feet wide, the number of square feet in the area is ab.* If the side of a square is a feet, the number of square feet in its area is . 5. If a rectangular solid is a feet long, b feet wide, and c feet thick, the number of cubic feet in its solid contents is abc. If the side of a cube is a feet, the number of cubic feet in its solid contents is . 6. If a man earns b dollars each week and spends c dollars, in one week he will save dollars; in 7 weeks he will save dollars. 7. A framed picture, on the inside of the frame, is 18 in. by 22 in.; the frame is 4 inches wide. How many inches in the outside perimeter of the frame? 8. Think of two fields: one is 9 rd. by 16 rd.; the other is 12 rd. by 12 rd. How do the square rods of the two fields compare? How much more fence would be required to enclose one field than the other? * This means the product of a and b. Observe that it is the number a (not o feet) that we multiply by the number 6 (not 6 feet). While it is probably true (see footnote, p. 181) that the multiplicand always expresses measured quantity, it is also true that we often find the product of two factors mechanically. Indeed this is what we usually do in all multiplication of abstract numbers. In this case we find the product of a and 6 and know from former observations that this number equals the number of square feet in the oblong. |