Solution.-5% is. The number minus of itself equals of it. Then 1 of the number is 57, 2% of it is 3, and the number is 20×3, or 60. RULE: Divide the amount by 1 plus the rate or divide the difference by 1 minus the rate; the quotient will be the base. 3. 721 is 3% greater than a certain number. Answer, 700. What is the number? 4. 68 is 66% less than what number? Answer, 200. 5. What number increased by 25% of itself equals 2,125? Answer, 1,700. 6. What number diminished by 6% of itself equals 7.52? Answer, 8. 7. The capacity of the Flusser is 270 tons of coal, which is 50% more than the capacity of the Chauncey. What is the capacity of the Chauncey? Answer, 180 tons. 8. The steaming radius of a ship is 7,200 miles. This is 83% more than the steaming radius of another ship. What is the steaming radius of the other ship? Answer, 3,927,3 miles. 9. An engineer officer ran the steam engineering department of a ship on $17,000 for one year, which was 12% more than what his predecessor had run it for. What was the predecessor's expenses? Answer, $15,1114. 10. The revolution counter on a ship registers 270,875 turns during the last cruise of the ship. This was 66% more than it had registered on the previous cruise. What did it register on the previous cruise? Answer, 162,525. 11. A man saved $1,250 on his second enlistment, which was 33% more than he saved on his first enlistment. What did he save on his first enlistment? Answer, $937.50. 12. A man's wages are $77 per month, which is 40% more than his previous wages. What were his previous wages? Answer, $55. 13. A battleship has a complement of 836, which is 83% more than the complement of a cruiser. What is the complement of the cruiser? Answer, 456. 14. The submarine F-2 has a surface speed of 14 knots, which is 40% more than her speed submerged. What is her submerged speed? Answer, 10 knots. RATIO. I. 1. How does $3 compare with $6 $5 with $15? $8 with $24? 2. How does 2 compare with 12? 3 with 18? 5 with 25? 3. What is the relation of 2 to 10? 5 to 25% 6 to 30? Between 4 and 8? 4. What is the relation of 4 to 8. II. The relation of one number to another of the same kind is a ratio. 1. This relation may be expressed in two ways: Thus, when it is asked "What is the relation of 4 to 8" the answer may be 4 is of 8, called the geometrical ratio, or 4 is 4 less than 8, called the arithmetical ratio. 2. When the relation of one number to another is sought, the first number is the dividend and the second is the divisor. 3. When the relation between two numbers is sought, either may be regarded as the dividend or divisor. III. The numbers compared are the terms of the ratio. IV. The first term is the antecedent and the second the consequent. Thus, in the ratio 5 to 10, 5 is the antecedent and 10 is the consequent. V. The colon (:) is the sign of ratio. Thus, the ratio 6 to 15 is expressed 6 15. : Since the ratio of one number to another is expressed by the quotient arising from dividing the antecedent by the consequent, the colon may be regarded as the sign of division without the dividing line. VI. The antecedent and the consequent together form a couplet. Suggestion: The ratio of to is the same as the ratio of 2 to 4. 12 12 Suggestion: 11⁄2 and 2=12; therefore, the ratio of to is the ratio of to or 8 to 9 or 8. What is the ratio of― 12. to ? to ? to ? to ? 13. to ? to? to? to? 2 13 8 7 8 14. 1 to 2615 to 11 to 12? 10 to 11? A ratio may be direct or inverse. The direct ratio of 20 to 4 is expressed 20:4 or 20. The inverse ratio of 20 to 4 is expressed 4:20 or 26. The term vary implies a ratio. When it is said that two numbers vary with two other numbers we mean that the relation between the first two numbers is the same as that between the other two. The value of a ratio is the result obtained by performing the division indicated. Thus, the value of the ratio 20:4 is 5; it is the quotient obtained by dividing the antecedent by the consequent. By expressing the ratio in a fractional form, for example the ratio 20 to 4 as 20, it is easy to see, from the laws of fractions, that if both terms are multiplied or both divided by the same number it will not alter the value of the ratio. It is also evident, from the laws of fractions, that multiplying the antecedent or dividing the consequent multiplies the ratio and dividing the antecedent or multiplying the consequent divides the ratio. When a ratio is expressed in words, as the ratio of 20 to 4, the first number is always regarded as the antecedent and the second as the consequent, without regard to whether the ratio itself is direct or inverse. When not otherwise specified, all ratios are understood to be direct. To express an inverse ratio, the simplest way of doing it is to express it as if it were a direct ratio, with the first number named as the antecedent and the second as the consequent, and then transpose the antecedent to the place occupied by the consequent and the consequent to the place occupied by the antecedent; or, if the ratio is expressed in fractional form, invert the fraction. Thus, to express the inverse ratio of 20 to 4, first write it 20:4 and then transpose the terms as 4:20, or as 20 and then invert as. Or, the reciprocals of the numbers may be taken as above. To invert a ratio in to transpose its terms. I. Name two numbers having the same relation to each other that 5 has to 10. 2. Name two numbers having the same relation to each other that 4 has to 12, 3 to 15. 3. Name two numbers that have the same relation to each other that 6 has to 30, 10 to 40. Name two numbers having the relation of— 8 to 32. 9 to 27. to . the relation to 8 that 5 has to 20? To 15 that has to ? To 25 that has to To 32 that has to To 48 that has to To 70 that has to 6. 7 to 35. 7. What number has What number has the relation8. To 24 that 6 has to 12? 9. To 30 that 7 has to 21? 10. To 48 that 9 has to 36? 11. To 72 that 5 has to 60? 12. To 88 that 8 has to 64? 13. If the earnings of 6 men are $30 in a given time, how will the earnings of 10 men compare with the earnings of 6 men in the same. time? 10 ? 14. Since a ratio is the quotient of the antecedent divided by the consequent, or a fraction, what changes may be made on it without changing its value? 15. Write two equal ratios. Multiply the first term by the last term and compare their product with the product of the intermediate terms. II. An equality of ratios is a proportion. Thus 4 8 as 8 = 16 is a proportion. III. A double () is a sign of proportion. It is written between the ratios. It has sometimes been regarded as the extremities of the sign of equality (=). The sign of equality is often used in proportion instead of the double colon. 83143-13-6 E IV. A proportion must have four terms, viz, two antecedents and two consequents. When any three are given, the other may be found. V. The first and third terms of a proportion are the antecedents of the proportion and the second and fourth terms are the consequents. Thus, in the proportion 5: 8:10:16, 5 and 10 are the antecedents and 8 and 16 are the consequents. VI. The first and last terms are the extremes and the second and third are the means of a proportion. Thus, in the proportion 10:12:5:6, 10 and 6 are the extremes and 12 and 5 are the means. VII. Principles. 1. The product of the extremes is equal to the product of the means. 2. The product of the extremes divided by either of the means gives the other mean. 3. The product of the means divided by either of the extremes gives the other extreme. EXERCISES. Find the term that is wanting in the following: VIII. A ratio between any two numbers is a simple ratio. WRITTEN EXERCISES. 1. If 9 yards of silk cost $27, what will 18 yards cost? cost of It is evident that the 18 yards is greater than the cost of 9 yards; consequently the answer ought to be greater than $27. (1) By Article VII Arranging the answer sought and $27 as a couplet of a proportion, the other couplet of the proportion must be expressed to correspond with the couplet first arranged. That is in the proportion (1) the consequent of the second couplet is greater than the antecedent; therefore the consequent of the first couplet must be greater than the antecedent, and the couplet is written 9 yd. : 18 yd. Solving according to Article VII, the value of 18 yards is $54. 2. If 6 men can dig a ditch in 48 days, in what time can 8 men dig it? 3. Example: If 3 men can dig a cellar in 10 days, in how many days can 5 men dig it? Solution: Since the number required, or fourth term of the proportion, is days, the third term is 10 days. Since 5 men will dig the cellar in a less number of days than 3 men, 3 men is the second term of the proportion and 5 men the first term. Dividing the product by 3 and 10 by 5 the required term is 6 days. 5 3 : 10: what? 2 3 X 10 6. 5 In this example the number of days is an inverse ratio to the number of men; that is, the greater the number of men, the less the number of days in which they will dig the cellar. RULE I. For the third term, write that number which is of the same denomination as the number required. II. For the second term, write the GREATER of the two remaining numbers, when the fourth term is to be greater than the third; and the LESS, when the fourth term is to be less than the third. III. Divide the product of the second and third terms by the first; the quotient will be the fourth term, or number required. RULE: Select the number which is the same kind as the answer, and from the conditions of the problem discover whether the answer will be greater or less than that number. Arrange these two terms as a couplet and then arrange the terms of the other couplet to conform to the conditions of the first couplet. Divide the product the product of the extremes or the means by the single extreme or mean. The result will be the term sought. Use cancellation whenever possible to do so. 3. Four men can shovel 40 tons of coal in 2 hours. How many tons can 6 men shovel in the same time? Answer, 60 tons. 4. A can do a piece of work in 18 days and 2 hours of 10 working hours. In how many days can he do the work working 14 hours per day? Answer, 13 days. |