Thus, if the cause be doubled, the effect will be doubled, if trebled, the effect will be trebled. Hence this PRINCIPLE. Any CAUSE is to another similar CAUSE, as the EFFECT of the first CAUSE is to the EFFECT of the second CAUSE. ILLUSTRATION.-If 3 men make 6 rods of road in a day, how many rods can 4 men make? 3 men. : 4 men. :: 6 rods. : 8 rods. Ans. 1st cause. 2d cause. 1st effect. 2d effect. The effect of the 2d cause, being the 4th term of a proportion, is found by multiplying the 3d term by the 2d, and dividing the product by the 1st. Art. 202. NOTE. In reviewing, the questions may be stated by this principle; or it may be used at first instead of the Rule, Art. 203. ART. 205. COMPOUND PROPORTION. A Compound Proportion contains one or more compound ratios. Thus, Art. 195. 2: 61 7 42, is a compound proportion. Or, 2X4 6X8 :: 7 : 42. This, sometimes called the Double Rule of ThREE, is applied to the solution of questions requiring more than one statement in Simple Proportion. 1. If 2 men earn $20 in 5 days, what sum can 6 men earn in 10 days? Here, the sum earned depends on two things: the number of men, and the number of days; that is, on the ratio of 2 men to 6 men, and on the ratio of 5 days to 10 days. First find the sum earned by 6 men in the same time with 2 men. men. men. 26 $ $ 20: 60=sum earned by 6 men in 5 days. Knowing the sum earned in 5 days, the sum earned in 10 days can be found thus: days. days. 5 10 $ 60: 120 sum earned by 6 men in 10 days. REVIEW.-204. On what is Cause and Effect based? What is the Principle? How find the effect of the 2d cause? By examining these proportions, it is seen that the ratio of the third term to the fourth, depends on two ratios 1st, the ratio of 2 men to 6 men; 2d, the ratio of 5 da. to 10 da. The first ratio is 3, and the second 2; their product, 3×2=6, is the ratio of the third term to the required term. Hence, The ratio, which the 3d term has to the 4th, is compounded of the ratios of 2 to 6 and of 5 to 10: write the two simple proportions Or, 2X5 6 X 10 :: 20 fourth term. 20X6X10=1200; 2X5=10; and 1200÷10=$120. Ans. *2. If a man travel 24 mi. in 2 da., by walking 4 hr. a day; at the same rate, how far will he travel in 10 da., walking 8 hr. a day? Ans. 240 mi. Rule for Compound Proportion.-1. Write for the 3d term, that number which is of the denomination required in the answer. 2. Take any two numbers of the same kind, and arrange them as in Simple Proportion. Art. 203, Rule 2, page 201. 3. Arrange any other two numbers of the same kind, in like manner, till all are used. 4. Multiply the third term by the continued product of the second terms; divide the result by the continued product of the first terms: the quotient will be the fourth term, or answer. NOTES.-1. If the terms in any couplet are of different denominations, reduce them to the same. If the third term consists of more than one denomination, reduce it to the lowest named. 2. The examples may be solved by two or more statements in Simple Proportion, or by Analysis, Art. 263. Also, by Cause and Effect, Art. 204, see statement, page 208. REVIEW.-205. What does Compound Proportion contain To what is it applied? In stating a question, what number is put for the third term, Rule? How are the other numbers arranged? 205. How is the fourth term found? NOTE 1. If the terms in any couplet are of different denominations, what is required? What if the third term contains more than one denomination? 3. If 6 men, in 10 days, build a wall 20 ft. long, 3ft. high, and 2 ft. thick in how many days could 15 men build a wall 80 ft. long, 4ft. high, and 3 ft. thick? The terms of the 2d cause and 1st effect form the divisors; those of the 1st cause and 2d effect, the dividends. The x shows the place of the required term. 4. If 16 men build 18 rods of fence in 12 days, how many men can build 72rd. in 8 da. ? Ans. 96 men. 5. If 6 men spend $150 in 8mon., how much will 15 men spend in 20 mon.? Ans. $937.50 6. I travel 217 mi. in 7 days of 6 hr. each, how far can I travel in 9 da. of 11 hr. each? Ans. 511 mi. 7. If $100 gain $6 in 12 months, what sum will $75 gain in 9 mon.? Ans. $3.375 -8. If 100 lb. be carried 20 mi. for 20 cts., how far may 10100 lb. be carried for $60.60 ? Ans. 60 mi. 9. To carry 12 cwt. 3 qr. 400 mi., costs $57.12: what will it cost to carry 10 tuns 75 mi.? Ans. $168. 40 rd. long, 5 ft. 10. If 18 men, in 15 da., build a wall high, 4ft. thick; in what time could 20 men build a wall 87 rd. long, 8ft. high, and 5 ft. thick? Ans. 58 da. 11. If 180 men, in 6 da. of 10 hr. each, dig a trench 200 yd. long, 3 yd. wide, 2 yd. deep; in how many days can 100 men, working 8 hr. a day, dig a trench 180 yd. long, 4 yd. wide, and 3 yd. deep? Ans. 24.3 da. For additional problems, see Ray's Test Examples. XV. ALIQUOTS, OR PRACTICE. ART. 206. One number is an aliquot part of another, when it will exactly divide it (Art. 110). Thus, 5 cents, 10 cts., 20 cts., &c., are aliquot parts of $1. TABLE OF ALIQUOT PARTS OF A DOLLAR. ART. 207. To find the cost of articles, when the price is an aliquot part, or aliquot parts of a dollar. 1. What cost 24 yards of muslin, at 25 cts. a yd.? SOLUTION.—If the price was $1 a yard, the cost would be $24; and, since 25 cents is of a dollar, the cost at 25 cts. a yard will and 1 of $24=$6. Ans. be the cost at $1; 2. What cost 16 yards of calico, at 37 cts. a yard? SOLUTION. If the price was $1 a yard, the cost would be $16. At 25 cts. a yard, the cost would be the cost at $1; 1 of $16=$4., Again; since 12 cts. is of 25 cts., the cost at 121 cts. a yard, will be the cost at 25 cts.; of $4=$2. But, the cost at 371cts. a yard, is equal to the sum of the costs at 25cts. and at 121⁄2 cts.; $4+$2=$6. Ans. REVIEW. 206. When is one number an aliquot part? Give examples. What cost 24 yd. silk at 621 cts. a yd.? Ans. $15. Bule Dr Case L-Tike such aliquot parts of the cost à do đơ mây đe necessary to find the cost at the given price. & What cost 48 yd. of linen, at 683 cts. per yd.? 12 ori, ac Sets, a bu.? bes, at 311 ets. a lb.? Ans. $14.061 *, * 8'48. a yd.? Ans. $23.62ļ veliers, a bu.? Ans. $30.00 21. Joch, at $3.12 per yd.? YARWA Z2) = 29st of 20 yd. at $1. 3 |
