Show that 1,2,2,6 are in proportion
Answers
The numbers the said to be in proportion of the product of there extreme is equal to the product of there mean.
Here, extreme are:- 1 and 6
So, the product of extreme = 1 × 6 = 6
And,
Mean are :- 2 and 2
So, the product of mean = 2 × 2 = 4
Since , Product of extreme ≠ Product of mean
So, they are not in proportion.
In a proportion --
1st : 2nd = 3rd : 4th
-- which are the corresponding terms?
The 1st and 3rd terms, the 2nd and the 4th.
1 : 2 = 5 : 10
1 is called the first term of the proportion, 2 is the second term, 5 is the third, and 10, the fourth.
We say that 5 corresponds to 1, and 10 corresponds to 2.
2. What is the theorem of the alternate proportion?
"If four numbers are proportional, then the
corresponding terms are also proportional."
If
1st : 2nd = 3rd : 4th
then alternately,
1st : 3rd = 2nd : 4th
(Euclid, VII. 13.)
Example 1. 1 : 2 = 5 : 10.
(1 is half of 2; 5 is half of 10.)
Alternately:
1 : 5 = 2 : 10
(1 is a fifth of 5; 2 is a fifth of 10.)
Example 2.
Directly :
12 : 36 = 2 : 6. (Why?)
Alternately:
12 : 2 = 36 = 6. (Why?)
Example 3. Complete this proportion:
5 : 7 = 20 : ?
If we look directly at the ratio of 5 to 7, it is not obvious. But if we look alternately, we see that 5 is a fourth of 20:
5 : 7 = 20 : 28.
And 7 is a fourth of 28.
If we cannot solve a proportion directly, then we can solve it alternately.
Example 4. The theorem of the same multiple. Complete this proportion:
4 : 5 = 12 : ?
Solution. 4 is a third of 12 -- or we could say that 4 has been multiplied by 3. Therefore, 5 also must be multiplied by 3:
4 : 5 = 12 : 15.
proportion
As 4 is to 5, so three 4's are to three 5's.
In fact, as 4 is to 5, so any number of 4's are to an equal number of 5's.
That is called the theorem of the same multiple. It follows directly from the theorem of the alternate proportion.
3. What is the theorem of the same multiple?
"If we multiply two numbers by the same number, then the products will have the same ratio
as the numbers we multiplied."
(Euclid, VII. 17.)
We have already seen that a ratio will be preserved if we divide both terms by the same number.
Example 5. Complete this proportion:
6 : 7 = ? : 28
Solution. 7 has been multiplied by 4 to give 28. Therefore, 6 also must be multiplied by 4:
6 : 7 = 24 : 28.
To solve that proportion --
6 : 7 = ? : 28
-- we could say:
"7 goes into 28 four times. Four times 6 is 24."
All the Examples and Problems in this lesson should be simple, mental calculations.
Example 6. Solve this proportion:
2 : 3 = 12 : ?
Solution. "2 goes into 12 six times. Six times 3 is 18."
2 : 3 = 12 : 18.
In fact, consider these columns of the multiples of 2 and 3:
2 3
4 6
6 9
8 12
10 15
12 18
14 21
And so on.
Now, 2 is two thirds of 3. (Lesson 17.) And each multiple of 2 is two thirds of that same multiple of 3:
4 is two thirds of 6.
6 is two thirds of 9.
8 is two thirds of 12.
And so on. In fact, those are the only natural numbers where the first will be two thirds of the second.
Note that each pair have a common divisor. And upon dividing by that divisor, the quotients in every case are 2 and 3. That is the theorem of the common divisor. 2 and 3 are the lowest terms. They are the smallest numbers which have the ratio "two thirds."
Example 7. Name three pairs of numbers such that the first is three fifths of the second.
Solution. The elementary pair are 3 and 5. To generate others, take the same multiple of both: 6 and 10, 9 and 15, 12 and 20, and so on.
Example 8. 27 is three fourths of what number?
Solution. Proportionally:
3 : 4 = 27 : ?
"3 goes into 27 nine times. Nine times 4 is 36."
3 : 4 = 27 : 36.
27 is three fourths of 36.
Only a multiple of 3 can be three fourths of another number, which must be that same multiple of 4.
As 3 is to 4, so any number of 3's are to an equal number of 4's.