The geometric mean of x and y is 6 and the geometric mean of x, y and z is also 6. Then the value of z is
6
Understanding Geometric Mean
The geometric mean is a type of average that uses the product of the values rather than the sum (as used in the arithmetic mean). For a set of n positive numbers \((a_1, a_2, \dots, a_n)\), the geometric mean (GM) is calculated as the nth root of their product.
Mathematically, the geometric mean is given by:
\( \text{GM} = \sqrt[n]{a_1 \times a_2 \times \dots \times a_n} = (a_1 \times a_2 \times \dots \times a_n)^{1/n} \)
Solving the Geometric Mean Problem
We are given two pieces of information about the geometric mean:
- The geometric mean of x and y is 6.
- The geometric mean of x, y, and z is also 6.
Step 1: Use the first piece of information
The geometric mean of two numbers, x and y, is \((xy)^{1/2}\) or \( \sqrt{xy} \). We are told this is equal to 6.
\( (xy)^{1/2} = 6 \)
To find the value of \(xy\), we can square both sides of the equation:
\( ((xy)^{1/2})^2 = 6^2 \)
\( xy = 36 \)
We now know that the product of x and y is 36.
Step 2: Use the second piece of information
The geometric mean of three numbers, x, y, and z, is \((xyz)^{1/3}\) or \( \sqrt[3]{xyz} \). We are told this is also equal to 6.
\( (xyz)^{1/3} = 6 \)
To find the value of \(xyz\), we can cube both sides of the equation:
\( ((xyz)^{1/3})^3 = 6^3 \)
\( xyz = 216 \)
We now know that the product of x, y, and z is 216.
Step 3: Find the value of z
We have two equations:
- Equation 1: \( xy = 36 \)
- Equation 2: \( xyz = 216 \)
We can substitute the value of \(xy\) from Equation 1 into Equation 2:
\( (36) \times z = 216 \)
Now, we can solve for z by dividing both sides by 36:
\( z = \frac{216}{36} \)
Performing the division:
\( z = 6 \)
Conclusion
The value of z is 6.
| Information Given | Mathematical Expression |
|---|---|
| Geometric mean of x and y is 6 | \( (xy)^{1/2} = 6 \implies xy = 36 \) |
| Geometric mean of x, y, and z is 6 | \( (xyz)^{1/3} = 6 \implies xyz = 216 \) |
Comparing our result with the given options, we find that the value of z is 6, which corresponds to Option 3.
Revision Table: Key Concepts in Geometric Mean
| Concept | Description | Formula |
|---|---|---|
| Geometric Mean (GM) | A type of average for a set of positive numbers, calculated as the nth root of their product. | For \(n\) numbers \(a_1, \dots, a_n\): \( \text{GM} = (a_1 a_2 \dots a_n)^{1/n} \) |
| GM of two numbers (x, y) | The square root of their product. | \( \text{GM}(x, y) = \sqrt{xy} \) |
| GM of three numbers (x, y, z) | The cube root of their product. | \( \text{GM}(x, y, z) = \sqrt[3]{xyz} \) |
Additional Information: Geometric Mean vs. Arithmetic Mean
It's useful to understand the difference between geometric mean and arithmetic mean.
- Arithmetic Mean (AM): Calculated by summing the values and dividing by the count. Used for simple averages. For \(n\) numbers \(a_1, \dots, a_n\), \( \text{AM} = \frac{a_1 + \dots + a_n}{n} \).
- Geometric Mean (GM): Calculated using the product of the values. Used for averages involving rates, percentages, or growth, or when dealing with numbers that multiply together to produce a result (like in this problem where \(xy\) and \(xyz\) are products).
For any set of positive numbers (unless all numbers are identical), the Arithmetic Mean is always greater than or equal to the Geometric Mean (AM ≥ GM).
Similar Questions
The mean of 5 numbers is 15. If one more number is included, the mean of 6 numbers becomes 17. What is the included number?
In an asymmetrical distribution, if the mean and median of the distribution are 270 and 220 respectively, then the mode of the data is
Average marks in Mathematics of Section A comprising 30 students is 65 and that of Section B comprising 35 students is 70. What are the average marks (approximately) of both the sections if it was detected later that an entry of 47 marks was wrongly made as 74 ?
The average height of 22 students of a class is 140 cm and the average height of 28 students of another class is 152 cm. What is the average height of students of both classes?
If the average of 9 consecutive positive integers is 55, then what is the largest integer?
A cricketer has certain average of 10 innings. In the 11 th inning, he scored 108 runs, thereby increasing his average by 6 runs. What is his new average?
In a class of 100 students, there are 70 boys whose average marks in a subject are 75. If average marks of the complete class is 72, then what is the average marks of the girls?
The average of 50 consecutive natural numbers is x. What will be the new average when the next four numbers are also included?
Consider two-digit numbers which remain the same when the digits interchange their positions. What is the average of such two-digit numbers?
The arithmetic mean of two numbers is 10 and their geometric mean is 8. What are the two numbers?
Important Questions from Average
Average of 40 numbers is 71, if the number 100 replaced by 140, then average is increased by
The captain of a football team of 11 members is 28 years old and the goalkeeper is 4 years older than him. If the ages of these two are removed, then the average age of the remaining players is two years less than the average age of the whole team. What is the average age of the team?
There are two Classes A and B having 25 and 30 students respectively. In Class-A the highest score is 21 and lowest score is 17. In Class-B the highest score is 30 and lowest score is 22. Four students are shifted from Class-A to Class-B.
Consider the following statements:
1. The average score of Class-B will definitely decrease.
2. The average score of Class-A will definitely increase.
Which of the above statements is/are correct?
The average weight of A, B, Cis 40 kg, the average weight of B, D, Eis 42 kg and the weight of Fis equal to that of B. What is the average weight of A, B, C, D, Eand F?
A cow costs more than 4 goats but less than 5 goats. If a goat costs between Rs. 600 and Rs. 800, which of the following is a most valid conclusion?
