RATIO. 276. Katio is relation by quotient. The two numbers (magnitudes) of which the ratio is to be found are called the terms of the ratio. The first term is called the antecedent and the second term the consequent. The ratio is the quotient of the antecedent divided by the consequent. The usual sign of ratio is the colon. It indicates that the ratio of the two numbers between which it stands is to be found, the number preceding the colon being the antecedent, and the number following it, the consequent. The expression, 12 :4 = 3, is read, the ratio of 12 to 4 is 3. Exercise. Read and complete the following: 1. 12:4= 4:12 = 12:2 = 2. 18:9 == 9:18 = 18:6 = 3. 15:5 = 5:15 = 15:10 = Note.—It will be observed that the sign of ratio is the sign of division (-=-) with the line omitted. 277. Every integral number is a ratio. The number 4 is the ratio of a magnitude 4 (inches, ounces, bushels) to the measuring unit 1 (inch, ounce, bushel). The number 7 is the ratio of 7 yards to 1 yard; of 7 dollars to 1 dollar, or of 7 seconds to 1 second, etc. Note The ratio aspect of numbers is not the aspect most frequently uppermost in consciousness; neither ought it to be. But the pupil should now see that number is ratio; that while it implies aggregation and often stands in consciousness for magnitude, its essence is relation—ratio. Ratio. 278. Every fractional number is a ratio. The fraction | is the ratio of the magnitude 3 to the magnitude 4. So V, (3), is the ratio of 12 to 4. Observe that in every case the terms of a ratio may be written as the terms of a fraction; the antecedent becoming the numerator and the consequent the denominator of the fraction. The fraction itself is the ratio. Exercise I. Make the terms of the ratio the terms of a fraction; then reduce the fraction to its simplest form. 1. The ratio of 20 to 6 is -»,»- = -V°- = H 2. The ratio of 6 to 20 is /7 = T%. 3. The ratio of 7 to 5 is —; of 5 to 7, —. 4. The ratio of 12 to 1 is —; of 1 to 12, —. Exercise II. 1. | is the ratio of 5 to 7; of 10 to 14; of 15 to 21, etc. 2. \ is the ratio of .— to —; of — to —; of — to —, etc. 3. \ is the ratio of — to —; of — to —; of — to —, etc. 4. 8 is the ratio of 8 to 1; of — to —; of — to —, etc. Exercise III. 1. Of 2 feet to 8 inches. 2. Of 3 yards to 6 inches. 3. Of 6 rods to 3 yards. 4. Of 2 rods 5 yards to 1 yard 1 foot. * The comparison of two magnitudes involves their measurement by the same standard. To compare feet with inches, Hie inches may be changed to feet or the feet to inches, or both may be changed to yards. Ratio. 279. Not only is number itself ratio, but a large part of tbe work in arithmetic is merely the changing of the form of the expression of ratios. Exercise IV. (Reducing fractions to their lowest terms.) 1. Express the ratio of 30 to 40 in its simplest form. 2. Express the ratio of 560 to 720 in its simplest form. 3. Express the ratio of 425 to 875 in its simplest form. 4. Express the ratio of 5 min. to 2 hours in its simplest form. 5. Express the ratio of 1 lb. 4 oz. to 5 lb. 8 oz. in its simplest form. Exercise V. (Reducing improper fractions to integers.) 1. Express the ratio of 400 to 50 in its simplest form. 2. Express the ratio of 375 to 25 in its simplest form. 3. Express the ratio of 256 to 16 in its simplest form. 4. Express the ratio of 3 hours 20 minutes to 50 minutes in its simplest form. Exercise VI. (Reducing complex fractions to simple fractions.) 1. Express the ratio of £ to \ in its simplest form. 2. Express the ratio of \ to \ in its simplest form. 3. Express the ratio of 2\ to 8| in its simplest form. 4. Express the ratio of | of an inch to 1 foot in its simplest form. Note.—Observe that the denominator in fractions corresponds to the consequent in ratio. Ratio. Exercise VII. (Changing common fractions to decimals.) 1. Express the ratio of 3 to 4 (|), in hundredths. 2. Express the ratio of 20 to 50, in tenths. 3. Express the ratio of 30 to 80, in thousandths. 4. Express the ratio of 50 sq. rd. to 1 acre 40 rd., in hundredths. Exercise VIII. (Finding what per cent one number is of another.) 1. Express the ratio of 15 to 20, in hundredths. 2. Express the ratio of 14 to 200, in hundredths. 3. Express the ratio of 17 to 25, in hundredths. 4. Express the ratio of 16 to 33^, in hundredths. 5. Express the ratio of 27 to 500, in hundredths. Exercise IX. (Changing " per oent" to a common fraction in its lowest terms, or to a whole or mixed number.) 1. A's money equals 40% of B's money, (a) Express the ratio of A's money to B's money in the form of a fraction in its lowest terms. (b) Express the ratio of B's money to A's money in its simplest form. 2. One number is 50% more than another number, (a) Express the ratio of the smaller to the larger number in the form of a fraction in its lowest terms. (b) Express the ratio of the larger to the smaller number in its simplest form. Note.—Observe that the base in percentage corresponds to the consequent in ratio. Ratio. Exercise X. The specific gravity of a liquid or solid is the ratio of its weight to the weight of the same bulk of water. 1. A cubic foot of water weighs 62^ lb. A cubic foot of cork weighs 15 lb. What is the ratio of the weight of the cork to the weight of the water? Express the ratio in hundredths. What is the specific gravity of cork? 2. A certain piece of limestone weighs 37 ounces. Water equal in bulk to the piece of limestone weighs 15 ounces. What is the ratio of the weight of the limestone to the weight of the water? What is the specific gravity of the limestone? 3. A certain bottle holds 10 ounces of water or 9^ ounces of oil. What is the ratio of the weight of the oil to the weight of the water? Express the ratio in hundredths. What is the specific gravity of the oil? Note Observe that in specific gravity problems, the weight of water corresponds to the consequent in ratio problems. 280. Miscellaneous Questions. 1. What is the ratio of a unit of the first integral order to a unit of the first decimal order? 2. What is the ratio of a unit of any order to a unit of the next lower order? 3. What ratio corresponds to 6 per cent? 4. What is the ratio of a dollar to a dime? Of a dime to a cent? Of a cent to a mill? 5. What is the ratio of 1 to -j-^? Of 1 tenth to 1 hundredth? 6. What is the ratio of a rod to a yard? Of a yard to a foot? Of a foot to an inch? |