Ex. 5. Approximate to the ratio of 1520: 1518. (1520): (1518)=1518+ × 2 : 1518 nearly; =1519 1518 nearly. Ex. LVI. Examples for practice. 1. Which is the greater (1) 7: 9 or 10: 3; (2) 18: 13 or 162: 117; (3) a+2b: a+b or a+3b: a+2b? 2. Is the ratio of 9: 16 increased or diminished by subtracting 2 from each of its terms? How is its value affected if each of its terms be multiplied by 2? 3. Shew that the ratio of a2-2: a2+x2 is greater than the ratio of a-x: a+x, unless x=a. 4. How is the ratio a: a-2b affected by adding d to both terms? 5. Find the ratio compounded of 14: 10, 6: 7, and 5: 4; the ratio compounded of a2-x2: a2, a+x: c2, and c : a-x; and the ratio compounded of 16: 5, the triplicate ratio of 5: 4, and the subduplicate ratio of 9 4. y: b, and b: a , 6. (a) If the ratios of x+y: a, x— be compounded together, shew that the resulting ratio is a ratio of equality. (8) If the ratios of 3a+2: 6a+ 1, and of 2a +3: a+2 be compounded together, is the resulting ratio a ratio of greater or lesser inequality? 7. Approximate to the value of the ratio of (1) (28)3 : (27)3; (2) (1002)*: (1000) *. 8. Two vessels, A and B, each contain a mixture of wine and water, A in the ratio of 3: 2, and B in the ratio of 7: 3. What quantity must be taken from each, in order to make a third mixture which shall contain 5 gallons of water and 11 gallons of wine? PROPORTION. 124. DEF. PROPORTION is the relation of equality subsisting between two ratios. Thus if the ratios a : b and c :d be equal; that is, if the relation which a has to b in respect of magnitude be the same as that which c has to d, this equality of the ratios is called PROPORTION; it is expressed by saying that a is to b as c is to d; and it is thus represented: a: b :: c: d, or sometimes a : b=c:d; also the four quantities a, b, c, d, are said to be proportionals. The terms a and d are called the EXTREMES, and the terms b and C, the or product of extremes = product of means. Note. If the student has difficulty in remembering that he has to multiply each term by bd, the following considerations will assist him: he has the fraction a b ; he wants to make it ad; he must therefore remove b from the denominator; if he multiply the numerator by b, he will effect this; he also wants d in the numerator; this will be done by multiplying the numerator by d: and similarly, in all examples in proportion, the student must consider what must be done with the given fractions in order to produce the required fractions. 126. If ad=bc, then a b c : d. the numerator, and bring b into the denominator; divide therefore by bd, a we require a = b с d ; that is, we must remove b from the ator of and bring e into its place; multiply therefore by, rator of the first fraction, which can be done by adding from a in the numerator of the first fraction, which can be done by 131. If a b c d, then a+b: a−b :: c+d: c-d. 132. a+b c+d a -b c-d b a-b d If a b c d, and c d e f, then a be: f. 133. If a b c d, and be :: df, then a e::c: f. 4. If a, b, c, d be in continued proportion, i. e. if a b b c :: c ad: a3 b3. : d, If a b c d, and e f :: g: h, then ae: bf :: cg: dh. If four quantities be proportionals, the greatest and least of them re greater than the other two together. b, c, d be proportionals, and let a be the greatest quantity, and d the least. |