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16. Supposing there is a note of $100, dated March 20, 1820, upon which were the following payments, viz.
Sept. 2, 1820,
20, Dec. 7, 1821,
Ans. $13,021. 17. Supposing there is a note of $317,19, dated July 12, 1812, on which were the following payments, viz.
Oct. 17, 1812, $61,10
Ans. $144,363. 18. What is the compound interest of $ 246 for 6
PROPORTION, OR SINGLE AND DOUBLE RULE OF THREE.
1. Tut. I have taught you in the preceding Conversations, the different methods necessary for performing on all numbers, whether whole or fractional, simple or compound, the four fundamental operations of arithme. tic, viz. Addition, Subtraction, Multiplication, and Division. But in order to facilitate the application of these rules, it will be necessary to instruct you in other rules, founded on the preceding ; and that which I propose now to consider, is the single rule of three. This is sometimes called the golden rule, for on a proper application of it a great part of the business of arithmetic depends.
If 4 yards of cloth cost $8, what will 6 yards of the same cloth cost ?
2. It is plain in this example that if we knew the price of 1 yard, we could find the price of 6 yards by repeating this price 6 times. But as the price of 4 yar ds is
given, if we multiply 6 by it we shall have 4 times the price of 6 yards, consequently, if we divide this product by 4, the quotient will be the price of 6 yards, viz. $12. Here you see that the price of 4 yards is double the quantity of cloth, and the price of 6 yards must be double its quantity, or $12. If the price of 4 yards had been 3 times the quantiiy, the price of 6 yards must have been three times that quantity ; and whatever the price of 4 yards may be, the price of 6 yards must be such, that it will contain 6 as often as the price of 4 yards contains 4. The proportion here is, as 4 is to 6 so is 8 to the answer, which is 12. To denote that there is a proportion between the numbers, they are written thus,
4 : 6 :: 8 : 12
that is, 4 is the same part of 6, that 8 is of 12, or 4 is contained in 6 as often as 8 is contained in 12.
3. Again, if it were required to find how many days it would take 12 men to do a piece of work which 8 men, working at the same rate, could do in 16 days, you will see that the days should be less in proportion as the number of men is greater. Hence the proportion is not as 8 is to 12, so is 16 to the answer, but, as 12 is to 8 so is 16 to the answer, thus,
12 : 8 :: 16 : 24.
This last example is called inverse proportion, in distinction from the former, which is called direct proportion. From what has been said, you will readily understand the following rule for the
SINGLE RULE OF THREE.
Write that number, which is of the same kind with the answer, for the third term ; then consider whether the answer ought to be larger or smaller than this number; if it ought to be larger, write the largest of the other two numbers for the second term, and the least for the first; but if it ought to be smaller, write the least of the other two numbers for the second term, and the largest for the first; then multiply the two last terms together, and divide the product by the first term, and the quotient will be the answer.
If 16 bushels of corn cost $ 12, what will 32 bushels cost ?
16 :: 32 : 12
Ans. $ 24.
Here we have the price of 16 bushels given, and are required to find the price of 32 bushels at the same price
If we knew the price of one bushel, we could multiply 32 by it, and it would give the price of 32 bushels. But here we have the price of 16 bushels, and multiply 32 by it, which gives 16 times the price of 32 bushels, consequently, the quotient arising from the division of this product by 16, is the price of 16 bushels.
If 12 men perform a piece of work in 16 days, in what time will 18 men, working at the same rate, perform the same work?
18 :: 12 : 16
This is an example of inverse proportion ; the number of days decreasing as the number of men increases ; consequently the answer ought to be less than the third term, and the smallest of the other two terms, must be the second term.
If, in this example, the time in which one man could perform the work, was given, we could, by dividing that time by the number of men, find how long it would take those men to perform the work. But here we have the time, in which it took 12 men to perform the work, given to find how long it would take 18 men to do the same
work. Now it is evident, that it would take one man 12 times as long as it would take 12 men to do the work ; therefore, if we multiply 16, the number of days in which 12 men performed the work, by 12, we obtain the time in which it would take one man to perform the work, and this time, divided by 18, gives the time in which it would take 18 men to perform the work, viz. 10 days.
Pup. This appears very plain, and from what you have taught me concerning proportion, I should suppose, that almost any question might be solved by it.
Tut. This rule is used in almost all cases for transacting business, and you must not leave it till it is perfectly familiar. All the difficulty in these questions consists in stating the proportion. In every question in this rule, there are three numbers given for moving the operation. Two of these numbers are of the same kind, and must be the first and second terms of the proportion. The third number is different from the two first, and of the same kind with the answer. Should the first and second terms consist of different denominations, they must be reduced to the lowest denomination mentioned in either. If the third term consists of more than one denomination, it must be reduced to the lowest denomination mentioned in it, and the answer will ever be in the same denomination in which the third term is.
If 5 cwt. of beef cost $6, what will 9 cwt. 2 qr. 10 lb. cost ?
5 9 2 10 :: 6
If $12 will buy 10 cwt. I qr. of beef, how much will $16 buy?
12 : 16 :: 10 1
4. “ Proportion is also applied to questions, in which the relation of the quantity required, to the given quantity of the same kind, depends upon several circumstances combined together; it is then called Compound Proportion, or Double Rule of Three."
If a man travel 240 miles in 10 days, when the days are 12 hours long, how far will he travel in 12 days when they are 16 hours long?
In this example, there are two proportions to be considered, viz. of the days, and the hours in each day, or the number of miles the man travels depends on the number of days he travels, and the number of hours be travels,