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What is the quotient arising from dividing one number by another of the same kind called ? What is the ratio of 12 to 2? Of 12 to 3? Of 12 to 4? What does the ratio of two quantities show? Can a ratio exist between two quantities of different denominations ? Is there a ratio between 12 feet and 3 pounds ? Can the ratio be a denominate number? How are four quantities related when the ratio of the first to the second is the same as the ratio of the third to the fourth? Which are called extremes ? Which are called means? To what is the product of the extremes equal ! is the product of the extremes be divided by one of the means, what will the quotieng bo? How many terms of a proportion must be known in order to find the others ?
1. Let us endeavor to find the value of 24 yards of cloth, on the supposition that 8 yards are worth $12.
It is obvious that the value sought must be as many times greater than $12 as 24 yards is greater than 8 yards. Hence, there is the same ratio between $12 and the value sought, as there is between 8 yards and 24 yards. Consequently, we have this proportion:
8 yards : 24 yards : : $12 : value sought. Taking the product of the means, we have 24 x 12= 288. This, divided by the first term, gives 298=36 for the fourth term sought, which must be of the same kind as the third term; therefore, $36 is the value of 24 yards.
Nore.-When we take the product of the means we do not multipiy the 24 yards by 12 dollars, but simply multiply 24, the number denoting he yards, by 12, the number denoting the dollars. The product, 288, is neither yards nor dollars, but 288 units. When we divide this product by the first term of the proportion, we do not divide by 8 yards, but simply divide by 8, the number denoting the yards. The quotient, 36, gives the fourth term of the proportion; and since the fourth term is of the same denominate value as the third term, our fourth term, or answer, must be 36 dollars.
2. What will 312 pounds of coffee cost, if 25 pounds cost $3.25 ?
In this example, the ratio of 25 pounds to 312 pounds, is the same as the ratio of $3.25 to the number of dollars sought. Hence,
25 pounds : 312 pounds : : $3.25': the answer.
Here we first multiply the means together; we then divide the product by the first term.
Since there is a ratio between the third and fourth terms it follows that they must be of the same denominate value. Hence, of the three quantities given, we may always take for the third term of our proportion the quantity which is of the same kind as the answer required; then, if the answer sought is to be greater than this third term, the second term must exceed the first; but if the answer sought is to be less than this third term, then the second term must be less than the first.
109. From what has been said and done, we deduce this first form for the
.: RULE OF THREE. 1. Form a proportion by placing for the third term, the quantity which is of the same kind as the answer sought ; the two remaining quantities must be taken for the first and second terms, observing to take the larger of the two quantities for the second term, when the answer sought is to exceed the third term ; but to take the smaller of the two quantities for
the second term, when the answer is to be less than the third
Nore.-Since there is a ratio between the first and second terms, they must be reduced to the same denominate value. Also, the third term must be reduced to its lowest denomination; then the quotient found by dividing the product of the means by the first term, will be of the same denomination as the third term.
In stating questions in the Rule of Three, which quantity must be taken for the third term? Of the two remaining quantities, which is to be taken for the second term ? After the question is stated, how do you proceed to find the answer? Is it ever necessary to make any reduction in the terms before multiplying and dividing? What are these reductions ? The answer when found, will be of the same name sus which term ?
1. What is the cost of 6 cords of wood, at $7 for 2 cords?
2 cords : 6 cords : : $7 : Ans.
3. If there are 9 weeks in 63 days, how many weeks are there in 365 days? 63 days : 365 : : 9 weeks.
. 4. If a railroad car goes 17 miles in 45 minutes, how far will it go in 5 hours ?,
45 minutes : 5 hours : : 17 miles. or 45 6 : 300 minutes : : 17 66
5. If $100 will gain $7 in one year, how long will it require to gain $100 ?
Ans. 144 years. 6. If 3 paces or common steps of a person is equal to 2 yards, how many yards will 480 paces make?
Ans. 320 yards. 7. If 15 men can raise a wall of masonry 12 feet in one week, how many will be necessary to raise it 20 feet in the same time?
Ans. 25 men, 8. If 5 tons of coal, of 2000 pounds each, will last 3
months of 30 days each, how much will be consumed in 3 weeks, or 21 days ? Ans. 1 ton, or 2000 pounds.
9. If 9 bushels of wheat make 2 barrels of flour, how many bushels will be required to make 13 barrels ?
Ans. 614 bushels. 10. If a steamboat of 242 feet in length move 15 miles in one hour, how many seconds will it require to move its own length ?
Ans. 11 seconds. 11. If a steamboat of 242 feet in length move 15 miles an hour, how many times its own length will it move in 11 hours ?
Ans. 3600 times. 12. A reservoir has a pipe capable of discharging 30 gallons in one minute, what time will be necessary to discharge 15 hogsheads ?
Ans. 31 minutes. 13. If a man can mow 9 acres of grass in 3. days of 10 hours each, how long will it require for him to mow 21 acres ?
Ans. 81 days. 14. If 100 pounds of galena, or lead ore, yield 83 pounds of pure metal, how much pure metal will 7 tons of galena produce, if we reckon 2240 pounds to the ton?
Ans. 130144 pounds. 15. If 12 barrels of flour are worth $54, what is the value of 42 barrels at the same rate ?
12 barrels : 42 barrels : : $54
12)2268(189 dollars. Ans.