Art. 15. To multiply a number by 12: Multiply of the number by 100. How may a number be multiplied by 13? By 141? By 111? (0) Find the sum of the nine products. Art. 16. To multiply a number by 125: Multiply of the number by 1000. How may a number be multiplied by 126? By 127? 96 × 125 By 124? = 12000 Art. 17. To multiply a number by 250: Multiply † of the number by 1000. How may a number be multiplied by 251? By 252? By 249? 48 x 250 of 48 × 1000 = 12000 48 × 251 250 times 48 = 12000 12000 + 48 12048 Art. 18. To square 21, 31, 41, etc.: Multiply the integer by the integer plus 1, and add to the product. 21 × 21 = 2 times 2 + 2 times 1 + 1 of 2 + 1 of 1 But 2 times = Art. 19. To square 25, 35, 45, etc.: figure by the tens' figure increased by 1; as hundreds, to which add 25. Multiply the tens' regard the product To explain this rule, think of 25 as 2 tens and 1⁄2 of a ten, and apply the explanation given under Art. 18. 25 x 252 x 3 hundred and 25 625 35 × 35 = 3 x 4 hundred and 25 == == 1225 45 x 45 = 4 x 5 hundred and 25 = 2025 (s) Find the sum of the six products. *The author is aware that the expressions "Multiply the tens' figure" and "the tens' figure increased by 1" are tabooed by the hypercritical. But it is believed that neither obscurity nor misconception will arise from this use of the word figure. The word as here used clearly means the form value of the figure-the number which the figure by virtue of its shape represents. Art. 20. To multiply 2 by 23, 3 by 34, etc.: Multiply the integer by the integer plus 1, and to the product add the product of the fractions. Observe that this rule will apply only when the integer of the multiplicand and the integer of the multiplier are the same, and the sum of the fractions is 1. 24 by 23 = 2 times 2 + 2 times of 2 + 3 of 1 = But 2 times of 2 = 1 time 2, and of 3 + 3 Hence, 24 × 23 = 2 × 3+3 of 4 = 67% (t) Find the sum of the six products. Art. 21. To multiply 24 by 26. 33 by 37, etc.: Multiply the tens' figure by the tens' figure increased by 1; regard the product as hundreds, to which add the product of the units' figures. Observe that this rule will apply only when the tens' figure of the multiplicand and the tens' figure of the multiplier are alike, and the sum of the units' figures is 10. 22 × 28 = 2 x 3 hundred and 16 = 616 33 x 37 = 3 x 4 hundred and 21 = 1221 Art. 22. To multiply a number by 15: Multiply the number by 10, and to the product add of the product. 64 x 15 10 times 64 = 640 = 640320 960 (v) Find the sum of the twelve products. Art. 23. To multiply a number by 99: Take 100 times the number, from which subtract the number itself. How may a number be multiplied by 98? (w) Find the sum of the nine products. Art. 24. To multiply a number by 75: Multiply number by 100. How may a number be multiplied by 66%? of the (x) Find the sum of the twelve products. Art. 25. To divide a number by 25; by 331; by 121; by 16; by 20; by 50. (See pp. 212, 213, and 214, of this book.) (y) Find the sum of the six quotients. NOTE. Without a pencil, tell the integral quotient and the remainder resulting from the incomplete division of 1584 by 25. Art. 26. To divide a number by 125; by 250: Observe that 125 is contained in each thousand of a number, 8 times; that 250 is contained in each thousand of a number, 4 times. (z) Find the sum of the twelve quotients. NOTE. Without a pencil, tell the integral quotient and the remainder resulting from the incomplete division of 15450 by 250, |