4 3' Operation, 2} added to 16 and 2} multiplied by 11 = 43' as required ; hence, the ratio of numbers can be proved as above, as well as mixed numbers, whose sum and product may be equal. 2. Find 2 numbers in a given ratio, as 3 to 5, whose sum and product are equal? Ans. 23 and 15. 3. Find 2 numbers in the ratio of 5 to 9, whose sum and product are equal? Ans. 15 and 24. 4. Find 2 numbers in a given ratio as 2 to 3, whose sum and product may be equal? Ans. 1} and 21. 5. Find 2 numbers in the ratio of 3 to 8, whose sum and product are equal? Ans 13 and 344 RULE OF THREE DIRECT IN FRACTIONS. 1. If į of a yard of velvet cost 621 cts. what will 2% yards cost? Operation, * x 145 x 23 = 115200 The Rule for the common method is to invert the first term, then after preparing the fractions, to multiply the numerators together for a new numerator, and the denominators for a new denominator. OR THUS: Let a line be drawn in all statements representing equality, placing multipliers on the right, and divisors on the left, and transpose the numerators when cancelled on both sides. This is a general principle in all proportional operations. 125 125 x 23 = 2875 4) 1 1 7.183. 2 4 4 2 Answer as above. 2. At 183 cts. per pound, what will 631 lbs. come to? Ans. $11.903. 3. If } of a lb. of cinnamon bring of a dollar, what will 13 lbs. cost? Ans. $2.744. 2 3 2 INVERSE PROPORTION BY FRACTIONS. RULE. When the fractions are prepared and the third term inverted for a divisor, as in division of fractions, then agreeably to the Rule in multiplication of fractions, multiply the numerators for a new numerator, and the denominator for a new denominator. 1. How many yards of brown Holland, 5 quarters wide will line 20 yards, that is 3 quarters wide? Ans. 12 yds. 2. How many yards of matting 2 feet, 6 inches broad, will cover a room that is 27 feet long and 20 feet broad? Ans. 72 yds. 3. How much shalloon of a yard wide, will line 44 yards of cloth 13 yards wide? Ans. 9 yds. 4. What quantity of shalloon yard wide, will line 71 yards of cloth 1yards wide? Ans. 15 yds. 5. If 3 men can do a piece of work in 44 hours, in how many hours will 10 men do the same? Ans. 177. 6. How many pieces of cloth at 20 dollars per piece, are equal in value to 2404 pieces at 12į dollars per piece? Ans. 1491174 pieces. PROMISCUOUS EXAMPLES. 1. What will { of 2 cwt. of chocolate come to when 61 lbs. cost i of a dollar? Ans. $10.761 $. 2. If ; of a yard of cloth cost $23, what will 571 yards cost at the same rate? Ans. $16.25. 3. If 4¢ of a yard cost $9.75, what will 134 yards cost? Ans. $29.25. 4. If $1.75 will buy 9 lbs. of loaf sugar, how much will $213.50 buy? Ans. 7 cwt. 2 qrs. 14 lbs. 5. How many yards of carpeting that is half a yard wide, will cover a room that is 30 feet long and 18 feet wide! Ans. 120 yds. ( (Sign ::::) PROPORTION shews the direct relation of one object or thing to another, as to comparison or symmetry, viz: form, size, length, breadth, depth, rate, price, &c. But in relation to Arithmetic or Geometry, it has a determinate meaning. Euclid has proven his Theory of proportions in the fifth book of his Elements by demonstrating its principles, relations, and application to lines; therefore, it is apparent, that we thence derive the name of Geometrical Proportion. In Direct Proportion, the first term is to the second as the third term is to the fourth, that is, as 2 is to 4, so is 8 to 16, written thus: 2 : 4::8:16, these numbers are proportional, because, agreeably to the 16th proposition of Euclid, Lib. 6, the rectangle or product of the extremes is equal to the product of the means; hence, 2 x 16 4 x 8 32 or = 2. From this it is plain, that Proportion is the combination of two equal ratios, and that there are two antecedents and two consequents. Note to Teachers.--Require the learner to recite the answers to the following questions. Q. Why is this called the rule of Proportion? A. Because it shews the combination of two equal ratios. Q. What is the first term of a proportion called! Q. If 2 yards of muslin cost 25 cents, what will 6 yards cost? Ans. 75c. Q. Why is the second term (25 cts.) called the first consequent? A. Because it is the value or cost of the antecedent. Q. How are the terms arranged? A. As the first term (or antecedent) is to the second (its consequent) so is the third, (or antecedent) to the fourth, (or its consequent.) Q. How is the operation performed? A. Multiply the second and third terms together, and divide the product by the first. Q. What is the first antecedent called? A. When the first and third terms can be brought to the same denomination. Q. What is the general rule for stating? A. When a question is written thus: for instance, at 10 cts. per lb., what will 10 lbs. cost? or if 1 lb. cost 10 cts. what will 10 lbs. cost? In either case, the first term is 1 lb., then say as the first antecedent is to its consequent, so is the second antecedent to its consequent. Q. When the commencement of a question is written with the words, what is the value; how much, bought, or sold; what is the rule? A. In either case, the first object or thing mentioned in the question will be the third term. Q. What terms of a proportion can be contracted or reduced? A. The first and third, or first and second. A. Because the means do not shew a proper relation to each other as to cost or value; hence, it is evident, that the second or middle term must be of the same name or denomination with the answer (or “demand.') Q. What is meant by relation as applied to proportion? A. Nothing more than the quotient of a division. 1. If 3 lbs. of sugar cost 24 cents, what will 10 lbs. cost? Ans. 80c. lbs. cts. lbs. As 3 : 24 : : 10 = 80c. or by contraction, As 1': 8: : 10 80c.. 1 lb. and 10 lbs. are the antecedents. 8 cts. and 80 cts. the consequents. The relative proportion of the antecedents and consequents is the same. Thus, i cts. j; cts. = 8. 2. If 3 yards of cloth cost $9, what will 12 yards cost? Ans. $36. 3. What is the value of 9.7 lbs. of silver at $1.5 per ounce? Ans. $174.60. 4. What will 240 bushels of wheat amount to at the rate of $6 for 5 bushels? Ans. $288. 5. How much will 17 cwt. 3 qrs. 14 lbs. of iron cost at $4.75 per cwt.? Ans. $84.905. 6. Sold 120 bushels of corn for $54, how much did it cost per bushel? Ans. 45c. 7. Bought 29 yds. of muslin for $10.87], how much is it per yard? Ans. 37 c. 8. Bought 2 loads of corn, one containing 75 bushels and the other 87 bushels, at 52 cents per bushel, what is the amount? Ans. $84.24. 9. Bought 3 pipes of brandy containing 1204, 1243, and 123gallons at 43} cts. per gallon, how much is the amount? Ans. $161.213. |