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KEY TO CARD No. 31. ly to inspire a learner with the knowledge of new rules, whereby he can solve problems of a more difficult nature.
These numbers are called terms; the first term, the second term, the third term, and the fourth term. See Lesson A. Card 31.
In stating a question, we place a colon between the first and second term, two colons between the second and third, and if the fourth number be known, we place one colon between the third and fourth, thus, 2 : 6 :: 4 : 12 and we read them as follows ; as 2 are to 6; so are 4 to 12, that is, 2 bear the same proportion to 6, as 4 do to 12. See Lesson B. Card 31.
Let us scan another example: If 2 bushels of oats cost 4 shillings, what will 7 bushels cost?
Here 2 bushels cost 4 shillings, which is at the rate of 2 shillings a bushel ; consequently, 7 bushels will cost 14 shillings, because 7 times 2 are 14; thus the first term 2, bears the same proportion to the second, as the third term 7, does to the fourth. How to know which is the first, second, and third term, before
a question is stated. The first term is a supposition, as, “If 2 bushels cost,” or, Suppose 2 bushels cost.”
The second term is that which bears the same name of the answer, as, “ 4 shillings."
The third term is of the same name of the first, and expresses a demand, by asking a question, as, “ What will 7 bushels cost." Here our ideas are naturally led to shillings for the
th term or answer, because the second term was in
if the second term had been in pence, or gallons, rts, or feet, or inches, &c. the fourth term would be
name as the second.
KEY TO CARD No. 31.
EXAMPLE If 2 shillings will buy 4 quarts of vinegar, what quantity will 8 shillings buy?
In this case the second term is in quarts, and the answer must be likewise in quarts.
Many more particulars which belong to the rule of three, must be taken into consideration; but we will pass on with a few lessons before the mind is confused with too much criticism.
LESSON 1. If 5 pounds of butter cost 80 pence, what will 9 pounds cost?
Answer, 12 shillings. OPERATION, lb. d. Ib.
5 80 :: 9 to a fourth number. That is to say, as 5 lb. are to 80d. ; so are 9 lb to a fourth number.
Multiply the second and third terms together and divide by the first. The quotient or answer will be in pence. Divide those pence by 12 to reduce them into shillings.
80 the second term.
First term, 5)720
12) 144 pence or fourth term.
LESSON 2. If 5 pounds of butter cost 80 pence, how much will i pound cost ?
Answer, 16 pence. lb. d. lb.
5 : 80 :: 1 to a fourth number. That is to say, as 5 lb. are to 80 pence; so is 1 lb. to a fourth number or answer.
Here, the third term being 1, it makes multiplication useless ; therefore only divide the second term by the first, the quotient will be the answer.
KEY TO CARD No. 31.
16 d. As the second term was in pence, so the answer is in pence.
Answer, 80 pence. In this case division is unnecessary, because the first term is 1; therefore a multiplication of the second and third term, will give the answer.
Ib. d. Ib.
1: 16 :: 5 to a fourth number. That is, as 1 lb is to 16d. so are 5 lb. to a fourth numþer; and, as the second term is in pence, so the answer will be in pence likewise.
16 second term,
80 fourth term, or answer.
pay for 12 pieces at that rate ?
Answer, £ 86 8s. Reduce the sum £57 12 into shillings, viz. 1152s.
pieces. S. pieces.
8 : 1152 :: 12 to a fourth number. Read the statement thus; as 8 pieces are to 1152 pence, so are 12 pieces to a fourth number.
OPERATION. 1152 second term.
12 third term.
1728 shillings, or fourth term.
KEY TO CARD No. 31.
Here remember, that, when the fourth term is in a low denomination, it must be reduced to its proper quantity; as, pints to gallons ; inches to feet; pounds to hundreds; and, as in this case, shillings to pounds. Therefore divide the sum 1728 shillings, by 20, the answer will be £ 86 8s.
LESSON 5. Suppose the aforesaid pieces of chints contain 432 yards in the whole, and cost £ 86 8; how much is that a yard ?
Answer, 4 shillings. Here the third term being 1, division only is necessary. £ 86 8s.=)728s.
432 : 1728 : : 1 Read thus, “ as 432 yards are to 1728 shillings; so is 1 yard to a fourth number.”
LESSON 6. If 6 horses eat 9 bushels of oats, how much will 24 horses eat in the same time? Answer, 36 bushels.
OPERATION h. b. h.
6:9: : 24 third term. Multiply by
9 the second term. Divide by 6,
36 bushels for answer. But, some of those particulars before mentioned, must come under consideration: In the Rule of Three, there are two sorts of proportion ; direct, and inverse.
That part which we have already scrutinized, is direct proportion; yet we know not why it is direct. In every KEY TO CARD No. 31.
problem* there are certain signs by which we can know direct from ikrerse proportion.
Ist. When more requires more, the proportion is direct.
If the third term be greater than the first, and the nature of the question shows that the answer must be greater than the second term, then more requires more.
EXAMPLE If one bushel of salt cost 8 shillings, what will 2 bushels cost
Here the third term is greater than the first, and we readily perceive that the answer must be greater than the second term.
AGAIN, 2nd. When less requires less, the proportion is direct. • If the third term be less than the first, and the nature of the question shows that the answer must be less than the second, then less requires less.
EXAMPLE If 4 pounds of sugar cost 6 shillings, what will 1 pound cost?
Here the third term is less than the first, and the nature of the question shows that the answer must be less than the second.
Let us become masters of direct proportion, and then contrive some rules whereby we can distinguish that which is inverse.
LESSON 7. If buttons cost 50 shillings a gross, what will one dozen cost?
Answer, 50 pence. Note.—144 make one common gross; and whatsoever number of shillings any articles cost by the gross, so many pence they will cost by the dozen; and whatever number of shillings any articles cost by the dozen, so many pence will be the price of one.
* Problem, a question, a mathematical question,