In like manner we find A's tax for personal estate to be $8.6175 3.75 Total NOTE. A Tax is a sum of money required of individuals by government, for the use of the nation: or by a town, county, society, or corporation, for the purpose of defraying expenses. A tax is sometimes imposed by government, in small sums, upon every male citizen above a certain age; and this being the same to all, not varying with the amount of an individual's property, is called a tax of so much a head, or so much a poll: the word poll meaning head. Or, the more concise expression is poll-tax. In some states no poll-tax is allowed. Other taxes are usually rated or assessed on property. Property is of two kinds, real and personal: real property, or real estate, consists of possessions which are fixed and immoveable; as lands and buildings. Personal property comprehends all other possessions, which of course are moveable, 2. A town is valued at $298,769.50, and is taxed $6,177.89. There are 675 polls taxed $.30 each: what is the tax on a dollar? Ans. $.02 on $1. 3. Suppose a tax of $755 be laid on a town, and the inventory of all the estates in the town amounts to $9345: what must Bp whose estate is valued at $149? Ans. $12.037+ pay, 4. A certain town is taxed $2140; the whole property of the town is valued at $500000; there are 200 polls, which are taxed $.70 each; A's property is valued at $1400, and he pays but one poll: what will be the tax on $1, and what will be A's tax? Ans. $.004 on a dollar, and A's tax $6.30. REMARK. The scholar will bear in mind, that the table in the preceding page was computed at 7 mills on the dollar; therefore it will not answer for any other rate. 5. A certain town, valued at $64530, raises a tax of $2259.90; there are 540 polls, which are taxed $.60 each: what is the tax on a dollar, and what will be A's tax, whose real estate is valued at $1340, his personal property at $874, and who pays for 2 polls? Ans. 67.62. Questions. What is a tax? What is the meaning of the word poll?-real estate?-personal? What is (first) taken out of the whole tax remainder how assessed? How assessed on real estate? How on personal estate? d VULGAR FRACTIONS. FRACTIONS are either proper, improper, compound, or mixed. A proper fraction, is a fraction whose numerator is less than its denominator; as, 2, 3, 13, &c. An improper fraction, is a fraction whose numerator is larger than its denominator; as, 5, 3, 16, 5 7 17 &c. 19 A compound fraction, is a fraction expressed in a compound form, being a fraction of a fraction, or two or more fractions connected together, as of of of 5 of 18, which are read thus one-half of three-fourths of two-sevenths of five-elevenths of nineteen-twentieths, &c. Any whole number may be made an improper fraction, by drawing a line under it, and putting a unit, or 1, for a denominator, as, 6 may be expressed fractionwise, thus, f, and 12 thus, 12, &c. A prime number is that which can only be measured (that is, divided) by itself or a unit, as 5, 7, &c. That number which is produced by multiplying several numbers together, is called a composite number; thus, 360 is a composite number produced by 3×4×6×5=360, &c. 1. If an apple be cut into four parts, by what fraction is 1 part expressed?-2 parts?-3 parts?-4 parts?-how many parts make a whole apple? 2. What part of an orange is a part of 3 oranges? 3. Reduce to a decimal. 4. Reduce 1qr. 2na. to a decimal of a yard. 5. What is the value of .9375 of an acre? 6. What is the largest common measure of 18 7. Reduce 1776 to its lowest terms? 1887 8. Reduce 896 to its lowest terms? 1152 Ans. .375. Ans. 3R. 30P. Ans. 16 The former rule given to reduce a fraction to its lowest terms, was to divide both terms of the fraction by its largest common measure; but it can be done thus,-Divide both of the terms of the fraction by any number that will divide them without a remainder, and the quotient again in the same manner, and so on as long as they are susceptible of division by a common measure. 1. Reduce 288 to its lowest terms. 480 Thus, 8)28836=93 the answer. 2. Reduce 3. Reduce 4. Reduce to its lowest term. to its lowest term. Ans.. Questions. How many kinds of vulgar fractions are there? What are they called? What is a proper fraction?-Improper?-Compound ?-Mixed? How is a whole made an improper fraction? What is called a composite number? CASE 2. To reduce a mixed number to its equivalent improper fraction. -Multiply the whole number by the denominator of the fraction, and add the numerator of the fraction to the product for the numerator, under which write the given denominator. To reduce a whole number to an equivalent fraction, having a given denominator. RULE. Multiply the whole number by the given denominator; place the product over the said denominator, and it will form the fraction required. EXAMPLES. 1. Reduce 6 to a fraction, whose denominator shall be 8. Thus, 6×8 48 and 48 the ans. Proof. 48=48÷8=6, 2. Reduce 15 to a fraction, whose denominator shall be 12. Ans. CASE 4.† To reduce an improper fraction to its equivalent whole or mixed number. *All fractions represent a division of a numerator by the denominator, and are taken together as proper and adequate expressions of the quotient. Thus, denotes 3 to be divided by 4, and =.75. †This Case is evidently the reverse of Case 2. RULE. Divide the numerator by the denominator, the quotient will be the whole number, and the remainder, if any, will be the numerator to the given denominator. EXAMPLES. 1. Reduce 23 to its equivalent, whole, or mixed number. 8)293* 36 Ans. 2. Reduce 213 to its equivalent, whole, or mixed number. Ans. 127. 3. Reduce to its equivalent whole number. Ans. 9. CASE 5. To reduce a compound fraction to an equivalent simple one. RULE. Multiply all the numerators continually together for a new numerator, and all the denominators for a new denominator, and they will form the simple fraction required. If part of the compound fraction be a whole or mixed number, it must be reduced to an improper fraction, by Case 2d or 3d. If the denominator of any number of a compound fraction be equal to the numerator of another number thereof, these equal numerators and denominators may be expunged, and the other numbers continually multiplied (as by the rule) will produce the fractions required in lower terms. EXAMPLES. 1. Reduce of of of to a single fraction. Or, by expunging the equal numerators and denominators, it will give as before. 2. Reduce of of of 11 to a single fraction. Ans. 3. Reduce of of 3 of 12 to a single fraction. Ans.훅=11. 4. Reduce 1 of 17 of to a single fraction. CASE 6. To reduce fractions of different denominations to equivalent fractions having a common denominator. RULE Multiply each numerator into all the denominators, except its own, for a new numerator, and all the denominators into each other continually, for a common denominator. EXAMPLES. 1. Reduce 1, 2, and §, to equivalent fractions, having a common denominator. 1 × 5 × 8=40, the new numerator for 4× 5 × 8=160, the common denominator. Therefore the new equivalent fractions are 4,, and 100, the answer. 1609 2. Reduce,, 3, 4, and 1, to fractions having a common denominator. Ans. 57, 78, set, see, ist. 864 11527 3. Reduce, of 5, 71, and, to a common denomiAns. 12, 1872 1812 nator. 1872 18727 14508 432 4. Reduce, of 14, 75, and §, to a common denomi nator. Ans. 8448 6720 7200 11520 11520 11520 11520 CASE 7. To reduce any given fractions to others, which shall have the least common denominator. RULE. Find the least common multiple (already taught) of all the denominators of the given fractions, and it will be the common denominator required. Divide the common denominator by the denominator of each fraction, and multiply the quotient by the numerator, and the product will be the numerator of the fraction required. EXAMPLES. 1. Reduce, 4, and 7, to fractions having the least common denominator possible. K2 |