6. What part of 100 is 30? or what is the ratio of 100 to 30? 7. What are the ratios of the proportions Ans. 1. 3 Q. When four numbers are in proportion, what is the second term divided by the first equal to? What is this quotient called? What does the ratio of two numbers express? What is it equal to? What is the ratio of 1 to 5? Of 2 to 8 ? Of 3 to 27 ? Of 6 to 36? Of 12 to 144? Of 9 to 81? Of 10 to 100? Of 10 to 1? Of 12 to 2? Of 8 to 2? Of 8 to 1? In every proportion what is the ratio of the 1st term to the 2nd equal to ? § 140. Ex. 2. If 4lb. of tea cost $8 what will 1276. cost at the same rate? It is evident that the 4th term, or cost of 127b. of tea, must be as many times greater than $8 as 127b. is greater than 47b. But since the quotient of 12 divided by 4 expresses how many times 12 is greater than 4, it follows that the fourth term will be equal to $8 multiplied by this quotient: that is, equal to $8 multiplied by 3, or equal to $24. But we obtain the same result whether we multiply the 3rd term $8 by the quotient 3, or first multiply it by the 2nd term and then divide the product by the 1st term; and the same may be shown for every proportion. Hence we conclude, That the 4th term of every proportion may be found by multiplying the 2nd and 3rd terms together, and dividing their product by the 1st term. Q. How do you find the fourth term of a proportion, when the first three terms are known? EXAMPLES. 1. The first three terms of a proportion are 1, 2, and 3: what is the 4th ? Ans. 6. 2. The first three terms are 6, 2, and 1: what is the 4th? Ans.. 3. The first three terms are 10, 3, and 1: what is the 4th? Ans. § 141. The 1st and 4th terms of a propor Mon are called the two extremes, and the 2nd and 3rd terms are called the two means. Now, since the 4th term is obtained by dividing the product of the 2nd and 3rd terms by the 1st term, and since the product of the divisor by the quotient is equal to the dividend, it follows, That in every proportion the product of the two extremes is equal to the product of the two means. Thus, in the first example, 162 12 we have, 1x12=6×2=12 and in 4×24=8x12=96. the proportion, 4 12 8: 24 Q. What is the product of the extremes equal to? If the product of the extremes be divided by one of them what will the quotient be? If it be divided by one of the means, what will the quotient be? § 142. The Rule of Three takes its name from the circumstance that three numbers are always given to find a fourth, which shall bear the same proportion to one of the given numbers as exist between the other two. We have, for finding the 4th term, the following GENERAL RULE. I. Reduce the two numbers which have different names from the answer sought, to the lowest denomination named in either of them. II. Set the number which is of the same kind with the answer sought in the third place, and then consider from the nature of the question whether the answer will be greater or less than the third term. III. When the answer is greater than the third term, write the least of the remaining numbers in the first place, but when it is less place the greater there. IV. Then multiply the second and third terms together and divide the product by the first term: the quotient will be the fourth term or answer sought, and will be of the same denomination as the third term. Ex. 3. If 48 yards of cloth cost $67,25, what will 144 yards cost at the same rate? In this example, the answer is to be dollars, we place the $67,25 as Then, in the 3rd term. for the cost of 144 yards of cloth. 6725 48)9684,00($201,75 96 84 48 360 336 240 240 Q. From what does the Rule of Three take its name? What is the first thing to be done in stating a question? Which number do you make the third term? How do you determine which to put in the first? After stating the question, how do you find the 4th term? What will be its denomination? Ex. 4. If 6 men can dig a certain ditch in 40 days, how many days would 30 men be employed in digging it? less than the third: therefore, 30 men, the greater of the remaining numbers, is written in the first term. Besides, it is plain that the fourth term must be just so many times less than 40, as 6 is less than 30. Ex. 5. If 25 yards of cloth cost £2 38 4d, what will 5 yards cost at the same rate? When we come to divide the product of the 2nd and 3rd terms by the first, it is found the £10 does not contain 25. We then reduce to the next lower denomination and divide as in divivision of denominate numbers. OPERATION. yd. yd. £ S. d. 25: 5 :: 2 3 4: Ans. 5 25) £10 16s 8d 20 25)216(88 200 16 12 25)200(8d 200 Ex. 6. If 3cwt. of sugar cost £9 2s Od, what will 4cwt. 3qr. 261b. cost at the same rate? 4cwt. 3qr. 261b. 3cwt. 4 4 £9 2s Od 20 1828 12 2184 PROOF. $143. The product of the two means is equal to the product of the extremes (see § 141). Hence, if either of these equal products be divided by one of the mean terms the quotient will be the other. Therefore, Divide the product of the extremes by one of the mean terms, and if the work is right the quotient will be the other mean term. EXAMPLES. 1. The 1st term is 4, the 2d 8, the 3d 12, and the answer 24 is the answer true? The product of the extremes is 96. If this be divided by 8 the quotient is 12; if by 12 the quotient is 8: hence, the an swer was true. OPERATION OF PROOF. RULE OF THREE BY CANCELLING. If two numbers are to be multiplied together and their product divided by a third, the operation may be abridged by striking out or cancelling any factor which is common to the divisor and either of the other numbers. For example, if 6 is to be multiplied by 8 and the product divided by 4, we have in the latter case we cancelled the factor 4 in the numerator and denominator, and multiplied 6 by the quotient 2. It is found most convenient to draw a vertical line and to place the numbers to be multiplied together on the right, and the divisors on the left. Then, OPERATION. 18 18 ... 6 2 Ans. 12. I. If there be two equal numbers, one on each side of the line, omit them. II. If any number on the left has a common divisor with a number on the right, divide those numbers by their |