Since every ratio may be expressed under the form of i fraction, and since the numerator and denominator may tiplied or divided by the same number, without atteng tar value (Arts. 143 and 144), it follows that, If both terms of a ratio be multiplied or discard i same number, the ratio will not be changed. 232. A SIMPLE RATIO is when both terus an 11111***.. bers; thus, 7:12, is a simp 12.. 233. A COMPOUND RATIO is one wer aus for her mos tiplication of two simple ratios: tuus tu empat kavan 5: 10, and 3:12 if we multiply the corresponding t RATIO AND PROPORTION. 229. Two numbers, of the same kind, may be compared in two ways: 1st. By considering how much one is greater or less than the other, which is shown by their difference; and, 24. By considering how many times one number is greater or less than another, which is shown by their quotient. In comparing two numbers, by means of their difference, the less is always taken from the greater. In comparing two numbers by their quotient, one is regarded as a standard which measures the other; hence, to measure a number, is to find how many times it contains the standard. 230. A RATIO is the quotient arising from dividing one number by another. The Terms of a ratio are the divisor and dividend; hence, every ratio has two terms. The Divisor is called the ANTECEDENT; and the Dividend is called the CONSEQUENT. The ANTECEDENT and CONSEQUENT, taken together, are called a COUPLET. 231. The ratio of one number to another is expressed in two ways: 1st. By a colon; thus, 4: 16; and is read, 4 is to 16; or, 16 divided by 4. 2d. In a fractional form, as 16; or, 16 divided by 4. 229. In how many ways may two numbers of the same kind le compared with each other? If you compare by their difference, what do you do? If you compare by the quotient, how do you regard one of the numbers? How do you measure a number? 230. What is a ratio? What are its terms? How many terms has every ratio? What is the divisor called? What the dividend?-231. In how many ways is the ratio expressed? What are they? How is it read? Since every ratio may be expressed under the form of a fraction, and since the numerator and denominator may be mul tiplied or divided by the same number, without altering the value (Arts. 143 and 144), it follows that, If both terms of a ratio be multiplied or divided by the same number, the ratio will not be changed. 232. A SIMPLE RATIO is when both terms are simple numbers; thus, 7:12, is a simple ratio. 233. A COMPOUND RATIO is one which arises from the multiplication of two simple ratios: thus, in the simple ratios 5 10, and 3 : 12, if we multiply the corresponding terms together, we have 5 X 3 : 10 X 12, which is compounded of the ratios of 5 to 10, and of 3 to 12. 234. The ELEMENTS of a term are its factors: thus, 5 and 3 are the elements of the first term, and 10 and 12 of the second. These elements are generally written in a column, thus, and read, 5 multiplied by 3, to 10 multiplied by 12 NOTE.-A compound ratio may be reduced to a simple ratio, by multiplying the elements; thus the last ratio is that of 15 to 120 235. To find the ratio of one number to another. When the antecedent is less than the consequent, the ratio shows how many times the consequent is greater than the antecedent. When the antecedent is greater than the consequent, the ratio shows what part the consequent is of the antecedent The phrase," what part," implies the quotient of a less number divided by a greater. 232. What is a simple ratio?-233. What is a compound ratio ?— 234. What are the elements of a term? Examples. 1. What is the ratio of 9 tons to 15 tons ? ANALYSIS. In this example the antecedent is 9 tons, and the consequent is 15 tons; the ratio is therefore expressed by the fraction 13. = = 2. What is the ratio of 6 inches to 24 inches? 3. What is the ratio of 7 feet to 35 feet? 4. What is the ratio of fifteen dollars to 6 dollars? 5. What is the compound ratio of 5: 6 and 4 : 10? 6. What is the compound ratio of 6 : 9 and 3:4? 7. What is the compound ratio 8. What part of 6 is 4? 9. What part of 10 is 5? 10. What part of 34 is 17? 11. What part of 450 is 300? 12. What part of 96 is 16? 14. 15. of 4 : 5, 9: 8, and 3:5? 13. 8 is what part of 12? 16 is what part of 48? 18 is what part of 90? 16. 15 is what part of 165? 17. 9 is what part of 11? 236. To find the antecedent or consequent, when the ratio and one of the terms are given. 1. The ratio of two numbers is 5; and the antecedent is 4 dollars : what is the consequent ? ANALYSIS.-Since the ratio is equal to the quotient of the consequent divided by the antecedent, it follows: 1st. That the consequent is equal to the antecedent multiplied by the ratio: 24. That the antecedent is equal to the consequent divided by the ratio. Examples. OPERATION. Ratio 5 = consequent. antecedent. 5 X ant. cons. $4 × 5 = $20=cons. 1. The ratio of two numbers is 7, and the antecedent is 16 cwt.: what is the consequent ? 2. The consequent is 30 tons, and the ratio is 6: what is the antecedent ? 3. The antecedent is 15, and the ratio is 4: what is the consequent ? 7: 4. The ratio of two numbers is 13, and the consequent is what is the antecedent? 5. The ratio of two numbers is, and the antecedent is : what is the consequent ? 6. The ratio of the monthly wages of two men is 8: the greater wages of one is $256: what is the wages of the other? 7. The ratio is 25, and the consequent is 14 X5 X 10: what is the antecedent? 8. The value of a horse is 2 times that of an OX : the value of the horse is $143: what is the value of the ox? SIMPLE PROPORTION. 237. A SIMPLE PROPORTION is an expression of equality between two simple and equal ratios. Thus, the two couplets, 20 and 1 : 5, 4 : having the same ratio 5, form a proportion, and are written, : 20 :: 1 : 5, by simply placing a double colon between the couplets. The terms are read, 4 is to 20 as 1 is to 5, and taken together, they are called a proportion. 238. The 1st and 4th terms of a proportion are called the extremes; the 2d and 3d terms, the means. Thus, in the pro portion, 6 : 24 8 :: : 32, 6 and 32 are the extremes, and 24 and 8 the means: |