5. A man has of an estate, he gives his son of his share; what portion of the estate has he then left? VI. 1. State the rules for addition and subtraction of vulgar fractions; and prove them by means of an example. Explain the 3. Define a proper, mixed, and compound fraction. method of reducing a compound fraction to a simple one. Ex. of of of 13. 4. Shew by means of an example how a fraction is affected if the same number be added to its numerator and denominator. 5. Multiply 3 by 3,5, and divide 20 by 413 3 between the sum and difference of these results. 4 , and find the difference 6. What number added to 3+10 will produce 3227? and what number divided by 2 will produce? VII. 1. Shew from the nature of fractions that +=29; that 3 of 5=11; and that ÷=1}· 2. Simplify 3. Simplify 318,784. of of 35 and take the result from the sum of 10, 81 4. Add together 1, 3, 1, and }, subtract the sum from 2, multiply the result by of 27 of 8, and find what fraction this is of 99. 5. In a match of cricket, a side of 11 men made a certain number of runs, one obtained th of the number, each of two others th, and each of three others th, the rest made up between them 126; which was the remainder of the score, and 4 of these last scored 5 times as many as the other. What was the whole number of runs, and the score of each man? 80. It has been stated that figures in the units' place retain their intrinsic values, while those to the left of the units' place increase tenfold at each step from the units' place; therefore, according to the same notation, as we proceed from the units' place to the right, every successive figure would decrease ten-fold. We can thus represent whole numbers or integers and fractions under a uniform notation by means of figures in the units' place and on each side of it; for instance, in the number 5673·2412, the figures on the left of the dot represent integers, while those on the right of the dot denote fractions. The number written at length would stand thus, 5 × 1000+ 6 × 100+7×10+3+ + 100 + 1000 + 10000. The dot is termed the decimal point, and all digits to the right of it are called DECIMALS, because they are fractions with either 10, 100, or 10 × 10, 1000, or 10 × 10 × 10, &c. as their respective denominators. 81. It may here be observed, that, when a number is multiplied by itself any number of times, the product is called a POWER of that number; being called the second, third, fourth, &c. power, according as the number is multiplied once, twice, three times, &c. by itself, that is, according as it is employed twice, three times, &c., as a factor. 82. It will be seen from what has been said, that DECIMALS are in fact fractions having either 10 or some power of 10, for their denominators. For this reason also they are called DECIMAL FRACTIONS, in contradistinction to VULGAR FRACTIONS, which, as we have seen, are represented by a different notation, and not limited in their denominators to 10, or powers of 10. 83. From the preceding observations, it appears that Now the least common multiple of the denominators of the fractions is 10000: therefore, reducing the several fractions to equivalent ones with their least common denominator, we get 2 1000 3 100 4 10 5 2345= X + + X + 10 1000 100 X 100 1000 10 10000 2000+300+40 +5 Secondly, ⚫00324= + + + + 0 0. 3 10 100 1000 10000 100000 (the least common multiple of the denominators is 100000) 0 10000 0 1000 3 100 = + + X + + Thirdly, 56·816=5 × 10+6+18+100+1000 (the least common multiple of the denominators is 1000) 1000 6 1000 8 100 1 10 6 50000+ 6000+800+ 10+6 1000 56816 = 1000 Hence, we infer that every decimal, and every number composed of integers and decimals, can be put down in the form of a vulgar fraction, with the figures comprising the decimal or those composing the integer and decimal part (the dot being in either case omitted) as a numerator, and with 1 followed by as many zeros as there are decimal places in the given number for the denominator. 56816 84. Conversely, any fraction having 10 or any power of 10 for its denominator, as be represented in the form 56.816. 1000 may 56816 5 × 10000+6 × 1000+8 × 100+1 × 10+6 1000 85. Again, by what has been said above, it appears that We see that '327, 0327, and 3270 are respectively equivalent to fractions which have the same numerator, and the first and third of which have also the same denominator, while the denominator of the second is greater. Consequently, 327 is equal to 3270, but 0327 is less than either. The value of a decimal is therefore not affected by affixing cyphers to the right of it; but its value is decreased by prefixing cyphers: which effect is exactly opposite to that which is produced by affixing and prefixing cyphers to integers. 86. Hence it appears that a decimal is multiplied by 10, if the decimal point be removed one place towards the right hand; by 100, if two places; by 1000, if three places; and so on: and conversely, a decimal is divided by 10, if the point be removed one place to the left hand; by 100, if two places; by 1000, if three places; and so on. Thus 56 × 10=58 × 10=56. 56÷10=56×6=5=56. 87. The advantage arising from the use of decimals consists in this; viz. that the addition, subtraction, multiplication, and division of decimal fractions are much more easily performed than those of vulgar fractions; and although all vulgar fractions cannot be reduced to finite decimals, yet we can find decimals so near their true value, that the error arising from using the decimal instead of the vulgar fraction is not perceptible. Ex. XXIV. 1. Convert the following decimals into vulgar fractions: •1; 3; 31; 311; 31111; 31111111. 2. Convert the following decimals into vulgar fractions in their lowest terms: ·5; 25; 35; '05 ; ·005; ·256; 0256; 000256; "00008125. 3. Express as vulgar fractions in their lowest terms: 075; 848; 302; 3·434; 343′4; 03434; ·050005; 230-409; 2:30409; 2137.2; 91300-0008; 24.000625; 8213-7169125; 00083276; 1·0000009; *000000001. 90 30142 672819 672819 67281900 5203; 10; 10008; 108000; 1000000000; 10000 5. Multiply 7 separately by 10, 100, 1000, and by 100000; *51 separately by 10, 1000, and by 100000; 378-0186 separately by 1000, and by a million. 7. Express according to the decimal notation, five-tenths; seventenths; nineteen hundredths; twenty-eight hundredths; five thousandths; ninety-seven tenths; one millionth; fourteen and four-tenths; two hundred and eighty, and four ten-thousandths; seven and seven-thousandths; one hundred and one hundred-thousandths; one one-thousandth and one ten-millionth; five-billionths. 8. Express the following decimals in words: 4; 25; 75; 745; 1; 001; '00001; 2375; 2·375; 2375; 00002375; 1.000001; 1000001; 00000001. ADDITION OF DECIMALS. 88. RULE. Place the numbers under each other, units under units, tens under tens, &c., one-tenths under one-tenths, &c.; so that the decimals be all under each other: add as in whole numbers, and place the decimal point in the sum under the decimal point above. Ex. Add together 27.5037, 042, 342, and 2.1. 27.5037 ⚫042 342. 2.1 371-6457 Note. The same method of explanation holds for the fundamental |