C. m. Reduce 124 guilders, 14 stivers, into federal money. Guil. cts. Guil. $ d. mills. G. mills. As 390 : 1 :: 48633 : 124,7 Proof. V.-OF HAMBURGH, IN GERMANY. kvcounts are kept in Hamburgh in marks, sous and de1 sd-lubs, and by some in rix dollars. 12 deniers-lubs make I sous-lubs. 1 mark-lubs. EXAMPLES. Keduce 641 marks, 8 sous, to federal money, 3)641,5 $213,833 Ans. Bul to reduce federal money into marks, multiply the given sum by 3, &c. EXAMPLES. Reduce 121 dollars, 90 cts. into marks banco. 3 305,70=365 marks, 11 sous, 2,4 den. Ans. VI.-OF SPAIN. 34 marvadies of plate make 1 rial of plate. 1 piastre or picce of 8. EXAMPLES. 435 rials« 485 dimeses 48 dols. 50 cts. &c. But to reduce cents into rials of plate, divide by 10, Thus, 8-15 cents +10=84,5=84 rials, 17 marvadies, &c. VII.-OF PORTUGAL. Accounts are kept throughout this kingdom in milreas, and reas, reckoning 1000 reas to a milea. NOTE.--A milrea is = 124 cents; therefore to reduce milreas into federal money, multiply by 124, and the product will be cents, and decimals of a cent. EXAMPLES. 1. In 340 milreas how many cents ? 340 x 124=42160 cents=$421, 60 cts. Ans. 2. In 211 milreas, 48 reas, how many cents ? Note. When the reas are less than 100, place a cipher before them. Thus, 211,048 X 124=26169,952 cts. or 261 dols. 69 cts. 9 mills. + Ans. But to reduce cents into milreas, divide them by J24; and if decimals arise you must carry on the quotient as far as three decimal places; then the whole numbers thereol will be the milreas, and the decimals will be the reas. EXAMPLES. 1. In 4195 cents, how many milreas ? 4195=124=33,830+or 33 milreas, 830 reas. Ans. 2. In 24 dols. 92 cents, how many milreas of Portual ? Ans. 20 milreas, 096 reas. 194 55/ EXAMPLES, 1. In 641 Tales of China, how many cents ? Ans. 94868 2. In 50 Pagodas of India, how many cents ? Ans. 9700 3. In 98 Rupees of Bengal, how many cents ? Ans. 5439 VULGAR FRACTIONS. 2 8 HAVING briefly introduced Vulgar Fractions iumediately after reduction of whole numbers, and given some general definitions, and a few such problems therein as were necessary to prepare and lead the scholar immediately to decimals; the learner is therefore requested to read those general definitions in page 69. Vulgar Fractions are either proper, improper, single, compound, or mixed. 1. A single, simple, or proper fraction, is when the nu. merator is less than the denominator, as , , , , &c. 2. An Improper Fraction, is when the numerator ex reeds the denominator, as ģ, , &c. 3. A Compound Fraction, is the fraction of a fraction, coupled by the word of, thus, of ', of of, &c. 4. A Mixed Number, is composed of a whole number and fraction, thus, 81, 141, &c. 5. Any whole number may be expressed like a fraction: by drawing a line under it, and putting 1 for denominator, chus, 8=1, and 12 thus, Y, &c. 6. The common measure of two or more numbers, is that number which will divide each of them without a remainder; thus, 3 is the common measure of 12, 24, and 30; and the greatest number which will do this is called the greatest common measure. 7. A number, which can be measured by two or more numbers, is called their common multiple : and if it be the least number that can be so measured, it is called the leas common multiple : thus 24 is the common multiple 2, 3 ana but their least common multiple is 12. To find the least common multiple of two or more numbers. ROLE: 1. Divide by any number that will divide two or more of the given numbers without a remainder, and set the quotients, together with the undivided numbers, in a line beneath. 2. Divide the second lines as before, and so on till there are no two numbers that can be divided; then the continued product of the dio wors and quotients, will give the multiple required. EXAMPLES. 1. What is the least common multiple of 4, 5, 6 and 101 Operation, x 5)4 5 6 10 5x2x2x3=60 Ans. 2. What is the common multiple of 6 and 8? Ans. 24. 3 What is the least number that 3, 5, 8 and 12 wil measure? Ans. 120. 4. What is the least number that can be divided by the 9 digits separately, without a remainder ? Ans. 2520. REDUCTION OF VULGAR FRACTIONS, IS the bringing them out of one form into another, in or der to prepare them for the operation of Addition, Sub traction, &c. CASE I. To abbreviate or reduce fractions to their lowest terms. RULE.-1. Find a common measure, by dividing the greater term by the less, and this divisor by the remainder, and so on, always dividing the last divisor by the last remainder, till nothing remains; the last divisor is the common measure.* 2. Divide both of the terms of the fraction by the common measure, and the quotients will make the fraction required. * To find the greatest common measure of more than two numbers, you must find the greatest common measure of two of them as per rule above; then, of that common measure and one of the other unmbers, and so on through all the numbers to the last; then will the greatest common measure fast found be the answer. Or, if you choose, you may take that easy method in Problem I. (pago 09.) EXAMPLES. 1. Reduce to its lowest terms. Operation. common measure, 8);=, Ans, 3) Rem.. 00 2. Reduoe :0 its lowest terms. Ans. Ans. 11 Ans. į 55 CASE II. To reduce a mixed number to its equivalent improper fraction. Rule.-Multiply the whole number by the denominator of the gi. in fraction, and to the product add the numerator, this sumr writion thove the denominator will forin the fraction required EXAMPLES. 18 1. Reduce 45% to its equivalent improper fraction 45 X8+7=397 Ans. 2. Reduce 191% to its equivalent improper fraction, Ans. 3. Reduce 1670 to an improper fraction. Ans. 1908 4 Reduce 61}}; to its equivalent improper fraction, Ans. 27035 CASE III. To find the value of an improper fraction. RJE.--Divide the numerator by the denominator, and the quo Tent will be the value sought. 360 3 60 EXAMPLES ANSWERS 5)48(93 1913 1. Find the value of $ 84,4 61131 3.80 |