MULTIPLICATION OF VULGAR FRACTIONS. Rule. Multiply all the numerators of the given fraction together, for a new numerator, and all the denominators, for a new denominator. Note. It will frequently be necessary to prepare the given terms for the operation by the rules of reduction. Questions. Repeat the rule for performing multiplication of vulgar fractions. What is to be noted with respect to the preparation of the given terms? Reduce the given fraction to a common denominator; then subtract the less numerator from the grcater, and place the difference over the common denominator. But if the lower numerator be greater, subtract it from the common denominator, adding in the upper numerator, and carry one to the units' place of the whole number. Note. When the fractions are of different denominations, reduce them to their proper value, and take their difference by compound subtraction. Questions. How do you perform subtraction of vulgar fractions? What is to be done, when the lower numerator is the greater? What is to be noted, when the fractions are of different denominations? and 4 × 2x3=24 common denominator. Then 24 8X5=15) numerators. whence 15 4. From of a league take of a mile. Ans. 1M. 2fur. 16po. 5. From 5 take 23. Ans. 3 12' 6. From 3 of 7, take 1 of 3. Ans. 17 120 DIVISION OF VULGAR FRACTIONS. Rule. Prepare the fractions, if necessary; invert the divisor, and multiply the numerators together for a new numerator, and the denominators for a new denominator. 1. Prepare the given terms, if preparation be necessary, by reduction, and state the question as in whole numbers. 2. Then invert the dividing term, and multiply all the numerators together, and all the denominators together, for the answer. Examples. 1. If of a yard cost £3, what will of a yard cost? 3X2X1= 6 1X9X5= 3)=2s. 8d. Ans. 2. When 31 yards cost 93s., what buys 43 yards? 3. How many yards of linen, to line 20 yards of baize, that is 4. How much will pay for 4 yards, at 153s. per yard?' Ans. 14s. 3d. wide, will be sufficient yard wide? Ans. 12 yards. pieces of cloth, each 273 Ans. £86 19s. when 53 cwt. cost Ans. £2 6s. 318d. of a dollar, Ans. $2.742. 5. What will of a cwt. cost, £31. 6. If of a pound of cinnamon what will 13 pounds come to? bring 7. When 10 men can finish a piece of work in 20 days, in how many days can 6 men do the same? Ans. 344 days. 8. What will of 21 cwt. of chocolate come to, when 6 pounds cost of a dollar? Ans. $10.7613. DECIMAL FRACTIONS. A DECIMAL FRACTION is a part of a whole number or unit, denoted by a point placed to the left of a figure or figures; as, .1, .12, .123. M The first figure after the point denotes so many tenths of a unit; the second, so many hundredths; the third, so many thousandths, and so on. Decimal fractions are read in the same manner as vulgar fractions: .1 is equal to and reads, .12 13.123 123 1000 A number consisting partly of whole numbers and partly of decimal fractions, is called a mixed number: as, 1.1, 12.12, 123.123. It has already been understood that whole numbers, counting from the right towards the left, increase in a tenfold proportion; but decimals, on the contrary, counting from the left towards the right, decrease in a tenfold proportion; as will be better exemplified in the following table. TABLE. Whole numbers. Tens. Units. 100 Thous. part. Millionth part. 10 Millionth part. 100 Millth. part. 1000 Millth. part. Thousands. Thousands. Millions. 100 of Millions. 10 of Millions. Note.-Ciphers placed after decimal figures, neither increase nor decrease their value: thus, .1, .10, and .100, all express the same value, namely, But ciphers placed between the decimal point and any other figure, decrease their value in a tenfold proportion; as, .1, .01, .001: and they all express different values, namely, 100, 1000 Questions. What are decimal fractions, and how are they denoted? How are decimal fractions to be read? What is a number called, which consists partly of a whole number, and partly of a decimal? In what manner do whole numbers increase, and in what manner do decimals decrease in value? What do you observe by the inspection of the table? What is to be noted with respect to placing ciphers after decimal figures? What is to be noted with respect to placing ciphers between the decimal point and any other figure? ADDITION OF DECIMALS. Rule. Set down the given numbers under each other; observing to place tenths under tenths, hundredths under hundredths, &c.; and perform the operation in the same manner as addition of whole numbers. Note that all the decimal points stand exactly under each other, and that the decimal point in the product stands exactly under those in the example. Questions. How are decimal numbers, given to be added, to be set down; and how is the operation then to be performed? What is to be noted with respect to placing the decimal point in the sum, and in the sum total? 5. Add 56.12, .7, 1.314, 5837.01, and .15, together. Ans. 5895.294. |