Sept. 1, in $1000 more, Y $1500, and Z takes out $500. X takes out $500, Y puts in $1000, and Z $1000. end of the year they settle, having gained $6880; what is each partner's share of the gain? 25. A, B, and C traded in company. A at first put in $1200, B $1500, and C $1600; in 4 months A put in $300 more and B $400, and C took out $500; in 8 months from the com-. mencement of business, A withdrew all his stock but $600, B put in as much as he at first put in, and C withdrew $500 At the end of a year they found they had gained 12% on the largest total stock at any one time in trade. ought each to take if the firm is dissolved? How many dollars 26. A, B, and C form a partnership in which A is a silent partner who puts in $50000, B puts in $20000, and C nothing but his services. B is to receive for services $2000 a year, and C, who is the business manager, is to have $5000 a year. If they make $14000 a year for two years, and then dissolve the partnership, what should each receive? Ans. A, $60000; B, $28000; C, $10000. 27. If of three partners A puts in $75000, B $60000, and C $40000, and A is to draw a salary of $2000 a year, B of $2500, and C of $3000, how ought a gain of $13500 at the end of one year to be divided? Ans. A, $4571.43-; B, $4557.14+; C, $4371.43 28. January 1, A, B, C, and D formed a partnership, each putting in $20000. April 1, A put in $4000 more; June 1, B took out $5000; and October 1, C took out $6000. Each was to have for services $100 a month. A was sick and away from May 15 to July 15, and C took a vacation from June 20 to September 20, but B and D did not lose a day. If they gained during the year $13000, how ought it to be divided? RATIO. 406. Oral Exercises. 1. If John is 16 years old and his father 48, how do their ages compare? 2. What part of 75 is 15? 3. I have $35 and James has $5; how many times as many dollars as James have I? 4. Compare 18 with 9 ; with 6 ; with 3 ; with 2; with 1; with 12; with 15. 407. Ratio is the relation of one quantity to another of the same kind; or, it is the quotient obtained by dividing one quantity by another of the same kind. 408. Ratio is usually indicated by two dots; thus, 8 : 4 expresses the ratio of 8 to 4. The two quantities compared are the terms of the ratio; the first term is the antecedent, the second the consequent, and the two terms, collectively, a couplet. 409. The antecedent is a dividend, and the consequent a divisor. 410. An inverse, or reciprocal, ratio is a ratio inverted; that is, the antecedent becomes the consequent and the consequent the antecedent. Thus, the inverse ratio of 4: 9 is 9:4 411. The antecedent and consequent being a dividend and divisor, it follows that any change in the antecedent causes a like change in the value of the ratio, and any change in the consequent causes an opposite change in the value of the ratio (Art. 81 and 118). Hence, 1. Multiplying the antecedent multiplies the ratio; and dividing the antecedent divides the ratio (Art. 79, a and b). 2. Multiplying the consequent divides the ratio; and dividing the consequent multiplies the ratio (Art. 79, c and d). 3. Multiplying both antecedent and consequent by the same number, or dividing both by the same number, does not affect the ratio (Art. 80, a and b). 412. The antecedent, consequent, and ratio are so related to each other, that if two of them are given the other can be found; thus: in 12 : 3 = 4, we have Consequent x Ratio Antecedent. 413. When there is but one antecedent and one consequent the ratio is said to be simple; thus, 15:5=3, is a simple ratio. 414. When the corresponding terms of two or more simple ratios are multiplied together the resulting ratio is said to be compound; thus, by multiplying together the corresponding terms of the simple ratios, 6:3= 2, 10: 2 = 5, we have the compound ratio, 6 x 8 x 10 : 3 × 2 × 2=40. 5. What is the ratio of 30 to 6? 6. What is the ratio of 4 to 11? 7. What is the inverse ratio of 20 to 4? Ans. 8. What is the inverse ratio of 3 to 7? 9. What is the ratio compounded of 9 to 4 and 6 to 5? 10. Which is the greater, the ratio of 8 to 7, or of 17 to 14? 11. Which is the greater, the ratio of 7 to 4, or of 18 to 13? 12. If 63 is the antecedent and 9 the consequent, what is the ratio? 13. If 72 is the antecedent and 8 the ratio, what is the consequent ? 14. If 14 is the consequent and 4 the ratio, what is the antecedent? PROPORTION. 415. Oral Exercises. 15. How does the ratio of 16 to 8 compare with the ratio of 24 to 12? 16. How does the ratio of 8 to 10 compare with 12 to 15? 416. Proportion is an equality of ratios. Thus, 8:6= 4:3 is a proportion. The equality of two ratios is indicated by the sign of equality (=), or by four dots (::). Thus, 12:3 = 8:2, or 12 : 3: 8:2, reads, 12 to 3 equals 8 to 2, or 12 is to 3 as 8 is to 2. 417. The first and last terms of a proportion are called extremes; the second and third, means. = 418. If the means are the same number, this number is called a mean proportional. Thus, in 9:6 6:4, 6 is a mean proportional between 9 and 4. 419. In a proportion the product of the extremes is equal to the product of the means. As these fractions are equal, and their denominators the same, their numerators must be equal, or 3 x8 = 6 × 4. But 3 and 8 are the extremes, and 6 and 4 the means. 420. It follows that either extreme is equal to the product of the means divided by the other extreme; and either mean equal to the product of the extremes divided by the other mean. That is, if any three terms of a proportion are given, the remaining term can be found. Hence, the name, Rule of Three. A mean proportional between two numbers is equal to the square root (Art. 438) of their product. 25. If 7 pounds of rice cost 63 cents, what will 11 pounds cost? 7:11 = : 63: Ans. Ans. 99 cents. The cost evidently will increase as the quantity increases, that is, 11 pounds will cost as many times 63 cents as 11 pounds is times 7 pounds, or 7 pounds is to 11 pounds as 63 cents is to the required cost. 26. If 8 men can build a certain wall in 25 days, how long I will it take 12 men to build the same wall? |