3. The following are the names of the 12 calendar months into which the civil year is divided, with the number of days in each. The number of days in each month may be easily remembered from the following lines: "Thirty days hath September, Except in Leap year, then is the time, When February has twenty-nine." 48. How many days in 3 weeks? In 4 wks.? In 5 wks. In 7 wks. ? In 9 wks.? 49. How many weeks in 14 days? In 21 days? In 32 days? In 35 days? Ip 40 days? CIRCULAR MEASURE OR MOTION. 159. Circular Measure is applied to the divisions of the circle, to latitude and longitude, and the motion of the heavenly bodies. 60 seconds (") make 1 minute, marked ' 1 degree, 1 sign, 12 signs, or 360° QUEST.-159, What is Circular Measure? Repeat the Table. OBS. 1. The circumference of every circle is divided, or supposed to be divided, into 360 equal parts, called degrees, as in the subjoined figure. 2. The minute is the same as a geographical mile, 60 of which make a degree. (Art. 151.) 360° 90° 180° 270° 50. In 2 degrees, how many minutes? In 3 degrees? 51. In 2 signs, how many degrees? In 3 signs, how many? In 4 signs, how many? 52. How many signs in 60 degrees? In 90 degrees? 20 quires 1 ream. 24 sheets of paper A sheet folded in two leaves, is called a folio. A sheet folded in four leaves, is called a quarto, or 4to. A sheet folded in eight leaves, is called an octavo, or 8vo. • A sheet folded in twelve leaves, is called a duodecimo, or 12mo. A sheet folded in eighteen leaves, is called an 18mo. QUEST. Obs. How is the circumference of every circle divided? To what does the minute answer? REDUCTION OF COMPOUND NUMBERS. 160. The process of changing compound numbers from one denomination into another, without altering their value, is called Reduction. EXERCISES FOR THE SLATE. Ex. 1. Reduce £3 to farthings. £3 20s. in 1£. 60 shillings. 720 pence. 4 far. in 1d. Ans. 2880 far. We first reduce the given pounds to shillings. This is done by multiplying them by 20, because 20s. make £1. (Art. 147.) That is, since there are 20s. in £1, in £3 there are 3 times 20s. or 60s. We now reduce the 60s. to pence, by multiplying them by 12, because 12d. make 1s. Finally, we reduce the 720d. to farthings, by multiplying them by 4, because 4 far. make 1d. The last product, 2880 far., is the answer, that is, £3=2880 far. 2. Reduce £2, 3s. 6d. and 2 far. to farthings. Operation. £. S. d. far. 2 3 6 2 20s. in £1. 43 shillings. 12d. in 1s. 522 pence. 2090 far. Ans. In this example there are shillings, pence, and farthings.Hence, when the pounds are reduced to shillings, the given shillings (3) must be added mentally to the product. In like manner, when the shillings are reduced to pence, the given pence (6) must be added; and when the pence are reduced to farthings, the given farthings (2) must be added. OBS. 1. In these examples it is required to reduce higher denominations to lower; as pounds to shillings, shillings to pence, &c., which is done by successive multiplications. QUEST.-160. What is Reduction? How are pounds reduced to shillings? Why multiply by 20? How are shillings reduced to pence? Why? How, pence to farthings? Why? 2. But it often happens that we wish to reduce lower denominations to higher, as farthings to pence, pence to shillings, and shillings to pounds. Thus, 3. In 2880 farthings, how many pounds? Operation. 4)2880 far. 12)720d. 20)60s. First, we reduce the given farthings to pence, which is the next higher denomination. This is done by dividing them by 4. For, since 4 far. make 1d., (Art. 147,) in 2880 far. there are as many pence as 4 is contained times in 2880. And 4 is contained in 2880, 720 times. We now reduce the pence (720) to shillings, by dividing them by 12, because 12d. make 1s. Finally, we reduce the shillings (60) to pounds, by dividing by 20, because 20s. make £1. Thus, 2880 far.£3, which is the answer required. £3 Ans. 4. How many pounds in Operation. 4)2090 far. £2, 3s. over. Ans. £2, 3s. 6d. 2 far. 2090 farthings? In dividing by 4 there is a remainder of 2 far. ; in dividing by 12, there is a remainder of 6d.; in dividing by 20, the quotient is £2 and 3s. over. The answer, therefore, is £2, 3s. 6d. 2 far. That is, 2090 far.= £2, 3s. 6d. 2 far. OBS. 1. The last two examples are exactly the reverse of the first two; that is, lower denominations are required to be reduced to higher, which is done by successive divisions. 2. Reducing compound numbers to lower denominations is usually called Reduction Descending; reducing them to higher denominations, Reduction Ascending. The former employs multiplication; the latter division. They mutually prove each other. QUEST. Obs. Which of the fundamental rules is employed in reducing higher denominations to lower? Ex. 3. How are farthings reduced to pence? Why divide by 4? How reduce pence to shillings? Why? How, shillings to pounds? Why? Obs. Which of the fundamental rules is employed in reducing lower denominations to higher? What is re- . ducing compound numbers to lower denominations usually called? To higher denominations? What rule is employed by the former? By the latter? 161. From the preceding illustrations we derive the following GENERAL RULE FOR REDUCTION. I. To reduce compound Nos. to lower denominations. Multiply the highest denomination given, by that number which it takes of the next lower denomination to make ONE of this higher; to the product, add the number expressed in this lower denomination in the given example. Proceed in this manner with each successive denomination, till you come to the one required. II. To reduce compound Nos. to higher denominations. Divide the given denomination by that number which it takes of this denomination to make ONE of the next higher. Proceed in this manner with each successive denomination, till you come to the one required. The last quotient, with the several remainders, will be the answer sought. 162. PROOF.-Reverse the operation; that is, reduce back the answer to the original denominations, and if the result correspond with the numbers given, the work is right. OBS. Each remainder is of the same denomination as the dividend from which it arose. (Art. 66. Obs. 2.) STERLING MONEY. (ART. 147.) 5. In £35, 4s. 6d. how many pence? QUEST.-161. How are compound numbers reduced to lower denominations? How reduced to higher denominations? Obs. Of what denomination is each remainder? 162. How is Reduction proved? |