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Q. If 1 yard of cloth cost $2, what will 6 yards cost? How many numbers are considered in this question? What are they called? How many were known or given? Name the terms. How many times does the 2nd term contain the first? How many times does the fourth contain the third ? How many times is the second term greater than the first? When are four numbers in proportion? How are they written? How are they read? What terms of a proportion express quantities of the same kind ? If 1 yard of cloth cost $3, what will 2 yards cost at the same rate ? What will 3 cost ? 4? 5? 6? 7 ? 8? 9? 10 ? If one yard of cloth cost $4 what will 2 yards cost? What will 3 yards cost? 4 ? 5? 6? 7? 8? 9? 10 ?
§ 139. The numbers
2; 4::8 : 16 are in proportion since the 2nd term is two times greater than the 1st, and the 4th term two times greater than the 3rd. And when four numbers are in proportion, the quotient of the 2nd term divided by the 1st, is equal to the quotient of the 4th term divided by the 3rd. This quotient is called the ratio of the proportion. Thus 2 is the ratio of the proportion
2:4 ::8:16; The ratio of two numbers, simply expresses how many times the second number contains the first. Hence, it is equal to the quotient of the second number divided by the first.
Thus, the ratio of 3 to 9 is 3, since 9:3=3. The ratio of 2 to 4 is 2, since 4 -2=2.
We may also compare a larger number with a less. For example, the ratio of 4 to 2 is }; for, 2 =4=. The ratio of 9 to 3 is , since 3 +9= .
In every proportion, the ratio of the 1st term to the 2nd, is equal to the ratio of the 3rd term to the 4th.
EXAMPLES. 1. What is the ratio of 9 to 18 ?
Ans. 2. 2. What is the ratio of 6 to 24 ?
Ans. 2. What is the ratio of 12 to 48 ?
Ans. 4. What is the ratio of 11 to 13?
Ans. 1,?. 5. What part of 20 is 4? or what is the ratio of 20 to 4?
6. What part of 100 is 30 ? or what is the ratio of 100 to 30? 7. What are the ratios of the proportions · 3 9 12 36.
Ans. Q: When four numbers are in proportion, what is the second term divided by the first equal to ? What is this quotient called? What does the ratio of two numbers express ? What is it equal to? What is the ratio of 1 to 5? Of 2 to 8? Of 3 to 27 ? Of 6 to 36 ? Of 12 to 144 ? Of 9 to 81 ? Of 10 to 100? Of 10 to 1? Of 12 to 2 ? Of 8 to 2? Of 8 to 1? In every proportion what is the ratio of the Ist term to the 2nd equal to ?
$ 140. Ex. 2. If 416. of tea cost $8 what will 121b. cost at the same rate ?
16. 16. As 4 : 12 : :8 : Ans.
4x8=3x8=24. $24 the cost of 1216. of tea.
It is evident that the 4th term, or cost of 121b. of tea, must be as many times greater than $8 as 121b. is greater than 416. But since the quotient of 12 divided by 4 expresses how many times 12 is greater than 4, it follows that the fourth term will be equal to $8 multiplied by this quotient: that is, equal to $8 multiplied by 3, or equal to $24. But we obtain the same result whether we multiply the 3rd term $8 by the quotient 3, or first multiply it by the 2nd term and then divide the product by the 1st term; and the same may be shown for every proportion.
Hence we conclude,
That the 4th term of every proportion may be found by multiplying the 2nd and 3rd terms together, and dividing their product by the 1st term.
Q. How do you find the fourth term of a proportion, when the first three terms are known ?
1. The first three terms of a proportion are 1, 2, and 3 : what is the 4th ?
Ans. 6. 2. The first three terms are 6, 2, and 1: what is the 4th ?
Ans. } 3. The first three terms are 10, 3, and 1: what is the 4th ?
Ans. § 141. The 1st and 4th terms of a proportion are called the two extremes, and the 2nd and 3rd terms are called the two means.
Now, since the 4th term is obtained by dividing the product of the 2nd and 3rd terms by the 1st term, and since the product of the divisor by the quotient is equal to the dividend, it follows,
That in every proportion the product of the two extremes is equal to the product of the two means.
Thus, in the first example,
1:6 :: 2 : 12 we have, 1x12=6x2=12 and in the proportion, 4 : 12 ::8: 24 4X24=8X12=96.
Q. What is the product of the extremes equal to? If the product of the extremes be divided by one of them what will the quotient be? If it be divided by one of the means, what will the quotient be?
§ 142. The Rule of Three takes its name from the circumstance that three numbers are always given to find a fourth, which shall bear the same proportion to one of the given numbers as exists between the other two. We have, for finding the 4th term, the following
GENERAL RULE. 1. Reduce the two numbers which have different names from the answer sought, to the lowest denomination named in either of them.
II. Set the number which is of the same kind with the answer sought in the third place, and then consider from the nature of the question whether the answer will be greater or less than the third term.
III. When the answer is greater than the third term, write the least of the remaining numbers in the first place, but when it is less place the greater there.
IV. Then multiply the second and third terms together and
divide the product by the first term : the quotient will be the fourth term or answer sought, and will be of the same denomination as the third term.
Ex. 3. If 48 yards of cloth cost $67,25, what will 144 yards cost at the same rate ?
In this example, as the answer is to be dol yd. yd. lars, we place the $67,25 48 : 144 :: 67,25 : Ans. in the 3rd term. Then,
144 as 144 yards of cloth
26900 will cost more than 48
26900 yards, the fourth term
6725 must be greater than the
48 9684,00($201,75 third, and therefore, we
96 write the least of the
84 two remaining numbers in the first place. The
48 product of the 2nd and
360 3rd terms is 9684,00 :
336 then dividing by the 1st
240 term we obtain $201,75
240 for the cost of 144 yards of cloth.
Q. From what does the Rule of Three take its name? What is the first thing to be done in stating a question? Which number do you make the third term ? How do you determine which to put in the first? After stating the question, how do you find the 4th term ? What will be its denomination ?
Ex. 4. If 6 men can dig a certain ditch in 40 days, how many days would 30 men be employed in digging it? As the answer must be days,
OPERATION. the 40 days are written in the men men days days. third place. Then as it is 30 : 6 :: 40 : Ans. evident that 30 men will do
6 the same work in a shorter
30)240 days. time than 6 men, it is plain
Ans. 8 days. that the fourth term must be less than the third : therefore, 30 men, the greater of the remaining numbers, is written in the first term. Besides, it is plain that the fourth term must be just so many times less than 40, as 6 is less than 30.
Ex. 5. If 25 yards of cloth cost £2 38 4d, what will 5 yards cost at the same rate ?
OPERATION. When we come to yd. yd. £
d. divide the product of 25 : 5 :: 2 3 4 : Ans. the 2nd and 3rd terms
5 by the first, it is found
25)£10 16s 8d the £10 does not con
20 tain 25. We then reduce to the next
200 lower denomination and divide as in divi
16 vision of denominate
200 Ex. 6. If 3cwt. of sugar cost £9 2s 0d, what will 4cwt. 3qr. 261b. cost at the same rate ?
3cwt. 4cwt. 3qr. 2676.
£9 2s Od
558 17472 10920
10920 We first reduce the 1st and 2nd terms to pounds,
1008 then the 3rd term to pence.
2106 The answer comes out in
2016 pence, and is afterwards
20) 3028.3d reduced to pounds shil
£15 . 23 lings and pence.
672 2352 2352
Ans. £15 2s 3d.