Mean refers to the average of a number. There are two types of mean: Arithmetic and Geometric mean.
What is Arithmetic Mean?
In Mathematics, the Arithmetic mean is defined as the ratio of the sum of all the given numbers to the total number of observations.
In Arithmetic mean, we find the average of a given set of numbers by adding the numbers and dividing them by the total numbers. For example, if there are 5 students whose marks are 50, 60, 70, 80 and 90. Then we can find the average marks (Arithmetic mean) by first adding them (50 +60 +70 +80 +90 = 350). Now divide the total by 5. So 350/5 = 70.
What is Geometric Mean?
In Mathematics, Geometric Mean is the average of two positive integers a and p such that a/x = x/p.
In Geometric mean, we find the average of a given set of numbers by multiplying them and then finding their root. It is also known as the Mean Proportional. It is related to ratio and proportion. In ration if a/x = p/y, then, ay = px. In the case of Mean proportional x = y which means that a/x = x/p. By cross multiplication, we get ap = x^2.
Geometric Mean is also known as mean proportional.
We can have a ratio of 2:3. It can be written as 4:6 as we multiplied both the numbers by 2. So we can say that 2:3 is proportional to 4:6. Mathematically, we can write this as,
2:3 :: 4:6
Here, 2 and 6 are extreme numbers, and 3 and 4 are the mean numbers. But, the mean numbers are different, which means that they are not the average mean proportional number. In the case of mean proportionality, the mean number is the average of the extremes.
If we want to find the mean proportional of the extremes 2 and 6, we should multiply them.
2 x 6 = 12
Now, we should find the root of 12. Root 12 can be written as 2 roots, 3 or we can use the division method to calculate root 12, which is equal to 3.464.
Steps to Find Mean Proportional Between Two Numbers
Determination of mean proportional involves the following steps:
If a and p are given, we can easily find x. Let us see how.
For example, a = 4 and p = 9. Find their mean proportional x.
Let us take mean proportional as b. So,
a/x = x/p
or, 4/x = x/9
By cross multiplication,
4 x 9 = x * x
or, 36 = x^2
Taking square root on both sides,
Sq. root of 36 = Sq. root of x^2or, 6 = x
or, x = 6
Therefore, the mean proportional of 4 and 9 is 6.
Let us take another example.
To find the mean proportion between two numbers 25 and 81.
We should know that we are supposed to find the average number, which means the answer should be between 25 and 81. But unlike the Arithmetic mean, where we add the number and divide it, in Geometric mean, we should multiply the numbers are find their root.
So here it can be represented mathematically as,
25:p :: p:81
Here 25 and 81 are the extremes and p and p are the means.
To calculate p, multiply 25 and 81
25 x 81 = 2025
Now find the square root of 2025
2025 = 5 * 5 * 9 * 9
So, root of 2025 = 45 (5*9)
How to construct Mean Proportional of 2 line segments
Let us take, for example, two lines of length 5 cm and 2.5 cm. We are supposed to draw the mean proportional of the line segments and measure the length of the mean proportional.
Construction:
Say, AB = 5 cm and BC = 2.5 cm.
Step 1: Draw a line AX. The length of AX should be greater than the total length of 2 line segments together.
Step 2: From A, draw AB = 5 cm
Step 3: From B, draw BC = 2.5 cm
Step 4: Draw a perpendicular bisector of AC bisecting at point O.
Step 5: Taking OA as radius, draw a semi-circle.
Step 6: Draw BD perpendicular to AC such that point D is touching the circumference of the semi-circle.
Step 7:
Now measure BD. It will be approx 3.4 cm. Hence the mean proportional of line segments 5 cm and 2.5 cm will be around 3.4 cm.
Conclusion
Here we discussed how to determine the mean proportional. We also learned how to calculate it and also to construct it. I hope you understood this well.
