CANCELLATION Cancellation shortens division by rejecting equal factors from both divisor and dividend. Dividing divisor and dividend by the same number does not alter the quotient. To reject a factor of any number is to divide the number by that factor. When a factor is canceled, the unit 1 is considered to take its place. One factor in the dividend will cancel only one equal factor in the divisor. When all the factors or numbers in the dividend are canceled, the factor 1 should be retained. We remove the factor 12 from the 36 in the dividend, and from the 60 in the divisor. Dividing by 5 the quotient, 5, in the divisor and the factor 210 in the dividend, we have left in the dividend the factors 42 and 3. Their only factor left in the divisor. 10 x 15 x 16 product is 126. 1 is the The quotient is 126 = 126. 18 × 20 × 21 8. What is the value of 36 × 42 × 52 divided by 2 × 3 × 4? divided by 9 × 21? divided by 21 × 13? 9. Which of the following numbers are prime? Find the prime factors of those which are composite: 227. 207, 469. α. 37, 4393. 93, 101, 7771, 111, b. 107, 9959, c. 239, 7111, d. 507, 6539, 113, 110, 1793, 10. Divide the product of 45 times 84 by 9. 11. 15 × 7 × 12 by 5 × 3 × 7 x 2. 12. 75 × 15 × 24 by 25 × 3 × 6 × 4 × 5. 1363. 13. What is the value of 99 x 360 x 365 divided by 11 × 73? divided by 33 × 18? divided by 44 × 5? 14. What is the value of 9 × 40 × 100 divided by 6 × 8 x 10? divided by 3 × 6 × 25? 31. 11 doz. suits were sold at $9.75 a suit in equal lots to 22 different men. How much in all did each pay? QUESTIONS 1. A besieged town containing 15,000 inhabitants has provisions sufficient to last 5 weeks. How long will the provisions last if 7000 of the inhabitants be sent away? 2. How far can a man travel in 27 da. at the rate of 149 mi. in 4 da. ? 3. If a person steps 4 yd. in 5 paces, to how many yards will 729 of his paces be equal? 4. @ 29 per lb., what is the cost of 174 lb. of coffee? 5. The numbers 24, 28, 32, 49, 81, are to be multiplied together and the product is to be divided by 8 x + x 7 x 9 x 6. What will be the quotient? 6. Divide 16 × 20 × 8 × 3 by 30 × 8 × 6. 7. How many pounds of raisins at 15¢ a pound are equal in value to 25 pounds of pepper at 48¢ a pound? 8. How much muslin at 25¢ a yard is equal in value to 100 yd. of silk at 87 a yard? 9. How many yards of serge at 46 a yard should be given for 23 bu. of wheat at 92 a bushel? 10. How long will it take a train to go 100 mi. at the rate of 42 mi. per hr.? SURFACE MEASURE When two lines meet at a point, their difference in direction is called the angle of the two lines. The point where they meet is called the vertex of the angle. The lines BA and BC form an angle whose vertex is at B. In reading an angle, the letter at the vertex is always made the middle one. The angle may be A ACUTE ANGLE read either as the angle ABC or as the angle CBA. In this figure there are two angles, DBA and ABC. The first is the difference in direction of the two lines AB and BD, and the second is the difference of the two lines AB and BC. Which angle is the larger? C OBTUSE ANGLE ACUTE ANGLE Α B B D When the two angles formed by one straight line meeting another straight line are equal to each other, they are called right angles, and the two lines are said to be perpendicular to each other. When a straight line BD meets a straight line AC so as to make the adjacent angles ABD and BDC equal to each other, then ABD and DBC are each right angles, and DB and AC perpendicular to each other. An angle greater than a right angle is called an obtuse angle, RIGHT ANGLE D C B and one less than a right angle is called an acute angle. SURFACES A plane has two dimensions, length and breadth. A four-sided plane figure, with all its angles right angles, is called a rectangle. A rectangle, with all its sides equal, is called a square. A square has four equal sides and four equal angles. If a square measures a foot on each side, it is called a square foot; if it measures a yard on each side, it is called a square yard. A square unit is any surface equivalent to a square one unit long and one unit wide. All surfaces are measured by square measure, that is, by the number of squares of a given size to which the surfaces are equivalent. The sum of the lengths of all the sides of any figure is its perimeter. 1. Measure the lengths of the perimeters of the two figures on this page. 2. Measure the length of the sides of the cover of this book. 3. How many sq. in. are in a square whose perimeter is 16 in. long? Draw the square and prove the answer. 4. How many sq. in. are in a rectangle whose perimeter is 24 in. long? Draw the figure on the blackboard. |