Let all Centigrade readings be measured off horizontally and the corresponding Fahrenheit readings vertically to any convenient scales. The equation shows that if C = 0°, F = 32°; if C′ = = 20°, F following table: 68°; and so on. C. are readings below NOTE. The numbers preceded by the sign zero, and such numbers for C, must be measured off from 0 toward the left, and for F. they must be measured off downward. Study the points on Fig. 1, p. 278, and note whether they seem to lie on a straight line. With a ruler draw a straight line through these points. For minus (-) values of C, as for C = -20, proceed thus: F = (-20) + 32 = 9 × (-20) + 32 = = -36 + 32. 9 × (−4) + 32 But 36° measured downward and then 32° measured upward is the same as 4° measured downward, or F and similarly for other minus (−) = = -4°, if C = -20°; values of C. 1. Measure off the value C 60°, then measure vertically upward to the line drawn through the points, thus obtaining the corresponding Fahrenheit reading. What is F. for C. = €60°? 2. Similarly, find from the drawing the values of F. corresponding to these values of C.: 120°; 160°; 180°; 220°; -60°; -100°; 110°. 3. Make a similar table and plot of one or more of the other five equations of problem 5, p. 276. METHODS OF SHORTENING AND CHECKING $142. Illustrations. I. Additions, subtractions, multiplications, and divisions with both integers and decimals are conveniently checked by casting out the 9's. II. To check against gross errors (blunders), first think through the problem, making rough mental calculations with numbers that are approximately correct, and decide about what the result must be. III. Check by performing reverse operations. That is, check addition by adding columns in reverse order; check subtraction by adding the subtrahend to the remainder; check division or square and cube roots by multiplication, and so forth. The second rule may be illustrated by a few problems: 1. Find the value of 15 A. of land at $87. Think thus: 16 A. @ $90 would be worth $1440; at $85, 16 A. would be worth $80 less, or $1360. At $87, 15 A. of land is worth of $85 less than $1440 of $80 ($1400), or about $1383. The exact computation gives $1382.50. Or thus: 87 of 100. Therefore, 15 × 87 = $1382.5. Ans. $1382.5. = 1580 × 2. On June 13, 1903, wheat was quoted in Chicago at 763¢ per bushel. Find the cost of 150 bu. at this price. 3. 250 shares of Illinois Central R. R. stock sold at $135a share. For how much did they sell? 4. The diameter of a circular rod is 13". How many inches in the circumference of a right section of the rod? How many square inches are there in the area of a right section of the rod? (For an approximation use π = 34.) 5. The outside diameter of a circular hollow iron tube is 21" and the inside diameter is 2". How many cubic inches are there in a 12′ length of the tube? 6. A distance was measured with a chain 98.75 ft. long and was found to contain 38.75 lengths of the chain. The chain was supposed to be 100' long. How great an error was made in measuring the line by using the supposed length? $143. Shortening and Checking Addition. 1. The noon temperatures on 7 successive days were 66°, 54°, 44°, 62°, 66°, 79°, 88°. Find the average for the week. 66 54 44 164 226 62 66 292 371 79 88 7)459 65.6° Ans. This is called two-column addition. A little practice will make this method useful for 1, 2, or 3 columns of figures. It may be used to advantage to check additions made in the ordinary way. Another method in use by expert accountants is to group the figures into sums of 10, 20, 30, and so on. Thus, in the given problem, Then 7 + 3, 8 + 6 + 6, 6 8 + 2, 6 + 4, 6 + 4, and 9 makes 39. + 4, 5, are 45. Sum, 459. 2. Write a few two and three column addition problems and practice these methods until you can use them rapidly. $144. Making-up Method of Subtraction. 1. A paying-teller in a bank had $5485 in his cash drawer in the morning, and during the day he paid out the following amounts: $37.50; $165.75; $10.25; $3.50; $2.88; $1.76; $65.17; $968.23; $3.67. How much money remained in the drawer? CONVENIENT FORM Total, $5485.00 37.50 165.75 10.25 2.88 1.76 65.17 968.23 3.67 $4226.29, balance = MENTAL WORK.-Add the first column, thinking thus: 10, 23, 31, 41. 41 and 9 make the next larger number than 41 ending in 0 (the first figure in the total). Write the 9 in the result and add the 5 into the second column. Then 11, 21, 34, 48. 48 +2 50, the next number larger than 48 which ends in 0 (the second figure of the total). Write 2 in the result and add 5 into the thrid column. Then, 21, 31, 39. 39 + 6 - 45. Write 6 and add 4 into fourth column. Then, 10, 17, 26. 26 + 2 = = = - 28. Write 2 and add 2 to next 14, and finally, 1+ 4 5. Write the 4. = column. Then, 12. 12 + 2 The advantage of this method is that it foots all the numbers and subtracts their sum, at once, as the numbers stand in the account book, from the total, giving the balance directly. 2. A bank customer's deposit at the beginning of the month was $398.75. During the month he drew out the following amounts: $16.75; $1.75; $5.25; $12.87; $128.32; $40.45; $2.18, $9.16; $1.57; $11.38; $12.62. Find the customer's balance at the end of the month. §145. Shortened Multiplication. 1. Multiply 73 by 67. 73=70+3 67=70-3 73 × 67 = (70 + 3) (70−3)= 702+70 × 3-3X70-32=702-32=4900-9=4891. Algebraic form: (a+b) (a−b) =a2—b3. This applies to finding the product of any two numbers the sum of whose units is 10 and whose tens digits differ by 1. RULE. Find the difference between the square of the tens and the square of the units in the larger number. 2. Find these products by the rule: 382=(30+8)2=302+2×8×30+82=1444 Algebraic form: (a+b)2=a2+2ab+b2. Show the correctness of the following rule: RULE.-Square the tens, double the product of the tens by the units and square the units, then add the three results. The sum is the square of the number. 4. Square these numbers by the rule: NOTE. Call the 12 in (7) 12 tens; also call the 14 in (8) 14 tens. = (a - b)2 = a2 4)2 2 ab - 402 – 2 X 4 X 40 + 42. Algebraic form: + b2. Make a rule for squaring 36 in this form. 5. To find mentally such products as 35 × 353; RULE.-When the integers are the same and the sum of the fractions is 1, multiply one integer by the other, increased by 1, and add the product of the fractional parts. 6. Find, by the rule, the following products: |