L. To reduce Fractions of a higher Denomination into lower. We have seen (T XXXVIII.) that fractions are multiplied by multiplying their numerators, or dividing their denominators. 1. Reduce £ to the fraction of a penny. In this example, we multiply the 1, in as in Reduction of whole numbers, viz., pounds by what makes a pound, shillings by what makes a shilling, &c. But this operation may be expressed differently, thus; 40 X 20 X 12=48=d.; or, by dividing the denominators thus;÷20=24÷ 12=d., Ans., as before, in its lowest terms. RULE. How, then, would you proceed? A. Multiply the fraction as in Reduction of whole numbers More Exercises for the Slate. 2. Reduce of a pound to the fraction of a shilling. A. 12s. 3. Reduce 1920 of a pound to the fraction of a farthing. A. qr. 4. Reduce Tobs of a hogshead to the fraction of a gallon. A. 16 gal. 5. Reduce of a bushel to the fraction of a quart. 6. Reduce T44T of a day to the fraction of a minute. 7. Reduce Toog of a cwt. to the fraction of a pound. 8 Reduce 20 of a hhd. to the fraction of a pint. A. lb. A. pt. 9. Reduce of a pound to the fraction of a shilling. A. s. II. To reduce Fractions of a lower Denomination into a higher. We have seen, that, to divide a fraction, (TXL.) we must multiply the denominator, or divide the numerator. This rule is the reverse of the last, (¶ L.), and proves it. 1. Reduce of a penny to the fraction of a pound. OPERATION. 12 24 20 New denom. 480 Then, o, Ans. In this example, we divide as in Reẻduction, (XXIX), viz. pence by pence, shillings by shillings; but, in order for this, we must either multiply the denominator or divide the numerator by the same numbers that we should divide by in Reduction of whole numbers. The same result will be obtained d. 8. if performed thus: × 12×20= To £, Ans. Hence the following RULE. 1. How do you proceed? A. Divide as in Reduction of whole numbers. More Exercises for the Slate. 2. Reduce of a shilling to the fraction of a pound. 3. Reduce of a farthing to the fraction of a pound. A. 1920 £. 4. Reduce of a gallon to the fraction of a hogshead. A. Toog hhd. 5. Reduce of a quart to the fraction of a bushel. A. TT bu 6. Reduce of a minute to the fraction of a day. A. 14T 7. Reduce of a pound to the fraction of a cwt. A. T008 8. Reduce of a pint to the fraction of a hhd. A. 2520=630. 9. Reduce of a shilling to the fraction of a pound. DECIMAL FRACTIONS. ¶ LII. Q. When such fractions as these occur, viz. To, 180, ooo, how is a unit supposed to be divided? A. Into 10 equal parts, called tenths; and each tenth into 10 other equal parts, called hundredths; and each hundredth into 10 more equal parts, called thousandths, &c. Q. How is it customary to write such expressions? A. By taking away the denominator, and placing a comma before the numerator. 25 65 Let me see you write down, in this manner, fo, foo, 100, 1000 525 Q. What name do you give to fractions written in this man ner? A. Decimal Fractions. Q. Why called decimal? A. From the Latin word decem, signifying ten; because they increase and decrease in a ten fold proportion, like whole numbers. Q. What are all other fractions called? A. Vulgar, or com mon fractions. Q. In whole numbers, we are accustomed to call the right hand figure, units, from which we begin to reckon, or numerate; hence it was found convenient to make the same place a starting point in decimals; and, to do this, we make use of a comma; what, then, is the use of this comma? A. It merely shows where the units' place is. Q. What are the figures on the left of the comma called? A. Whole numbers. Q. What are the figures on the right of the comma called? A. Decimals. Q. What, then, may the comma properly be called? A. Separatrix. Q. Why? A. Because it separates the decimals from the whole numbers. Q. What is the first figure at the right of the separatrix called? A. 10ths. Q. What is the second, third, fourth, &c.? A. The second is hundredths, the third thousandths, the fourth ten thousandths, and so on, as in the numeration of whole numbers. Let me see you write down again fo in the form of a deci mal. Q. As the first figure at the right of the separatrix is tenths, in writing down To, then, where must a cipher be placed' A. In the tenths' place. Let me see you write down in the form of a decimal A.,05, Write down Too, 180, 180. Q. How would you write down in decimals Tooo? A. By placing 2 ciphers at the right of the separatrix, that is, before the 7. Let me see you write it down? A.,007. Let me see you write down Too? A.,002. Q. Why do you write 2 down with 2 ciphers before it? A. Be cause in Too, the 2 is thousandths; consequently, the 2 must be thousandths when written down in decimals. Q. What does,5 signify? A. . Q. What does,05 signify? A. 180. Q. Now, as o, and as multiplying To by 10 produces which is also equal to, how much less in value is,05 than,5? A. Ten times. Q. Why? A. Because the parts in are ten times smaller than in ; and, as the numerator is the same in both expres sions, consequently, the value is lessened 10 times. Q. How, then, do decimal figures decrease in value from the left towards the right? A. In a tenfold proportion. Q. What does ,50 mean. A. 5 tenths, and no hundredths. Q. What, then, is the value of a cipher at the right of deci mals? A. No value. Q. We have seen that,5 is 10 times as much in value as ,05, or To; what effect, then, does a cipher have placed at the left of decimals? A. It decreases their value in a tenfold propor tion. Q. Since decimals decrease from the left to the right in a tenfold proportion, how, then, must they increase from the right to the left? A. In the same proportion. Q. Since it was shown, that ,5; 25, what, then, will always be the denominator of any decimal expression? A. The figure 1, with as many ciphers placed at the right of it as there are decimal places. Let me see you write down the following decimals on your slate, and change them into a common, or vulgar fraction, by placing their proper denominators under each, viz.,5,05,005,62 ,0225,37. Q.,25 is, and,5 is in value, ,25 or ,5? Q. By what, then, is the value of any decimal figures determined? A. By their distance from the units' place, or sepa ratrix. Q. When a whole number and decimal are joined together, thus; 2,5, what is the expression called? A. Å mixed number. Q. As any whole number may be reduced to tenths, hun dredths, thousandths, &c. by annexing ciphers, (for multiplying by 10, 100, &c.) thus, 5 is 50 tenths, 500 hundredths, &c.; how, then, may any mixed number be read, as 25,4? tenths, giving the name of the decimal to all the figures. Q. How is 25,36 read? A. 2536 hundredths. Q. How is 5,125 read? A. 5125 thousandths. A. 254 Q. What would 5125 thousandths be, written in the form of a vulgar or common fraction? A. 5185. This is evident from the fact, that 135 (an improper fraction), reduced to a mixed number again, is equal to 5,125. The pupil may learn the names of any decimal expression, as far as ten-millionths, also how to read or write decimals, from the following 26-25.26,25 read 26, and 25 Hundredths. 310000000..3,0000008 read 3, and 8 Ten-Millionths. 365 365, 0000 000 read 365. Exercises for the Slate. Write in decimal form 7 tenths, 42 hundredths, 62 and 25 hundredths, 7 and 426 thousandths, 24 thousandths, 3 ten-thousandths, 4 hundredths, 2 ten-thousandths, 3 millionths. Write the fractional part of the following numbers in the form |