Art. 18. To square 21, 31, 41, etc.: Multiply the integer by the integer plus 1, and add to the product. 2 × 21 = 2 times 2 + 2 times 1 + 1 of 2 + 1 of 1 But 2 times+of 2 = 1 time 2; and of = { Art. 19. To square 25, 35, 45, etc.: Multiply the tens' figure by the tens' figure increased by 1; regard the product as hundreds, to which add 25. To explain this rule, think of 25 as 2 tens and of a ten, and apply the explanation given under Art. 18. 45 x 45 = 4 × 5 hundred and 25 = 2025 = 1225 *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 24 by 24, 31 by 31, 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. 2 by 24 = 2 times 2 + 2 times + ✯ of 2 + ✯ of ‡ of ↓ = √ 3 × 4+ of } = 12 2 × 23 = 2 × 3 + } of } = 6% (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 × 3 hundred and 16 = = 616 33 × 37 = 3 × 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. (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 twelve products. Art. 24. To multiply a number by 75: Multiply number by 100. How may a number be multiplied by 663 ? of the (x) Find the sum of the twelve products. Art. 25. To divide a number by 25; by 33; by 121; by 16 by 20; by 50. (See pp. 72, 73, and 74, 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 remain. der resulting from the incomplete division of 15450 by 250. Art. 27. When the same factor occurs in a dividend and in its divisor, it may be omitted from both without changing their ratio. Hence all the factors that are common to a dividend and its divisor may be stricken out (cancelled) and the quotient (ratio) be unchanged. NOTE. In this discussion and in the solution of problems in this connection, all the numbers are regarded as abstract. 12 2 427 Operation No. 2. 2x3x7 2 = 4 7 Observe that the striking out of the factors 2 and 3 from the dividend and its divisor does not change their ratio-the quotient. 70 70 Observe that if all the factors of one of the numbers are cancelled the number becomes 1 and not 0. The factor 5 is 5 times 1; the factor 7, 7 times 1. Hence in the above problem there really remain in the divisor, after the cancellation, the factors 1 and 1=1X1=1. III. Divide 48 × 8 × 4 = 1536 by 8 × 4 × 4 = 128. Operation No. 1. 128)1536 (12 256 Observe that it is not necessary to obtain the prime factors of a dividend and its di visor to employ cancellation in finding the quotient. In the above the composite factor 8 is stricken out of the divisor and out of the 48 of the dividend. |