What is the mean proportional between 3 and 27?
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Finding the Mean Proportional
The question asks us to find the mean proportional between two numbers, 3 and 27. Let's understand what a mean proportional is and how to calculate it.
Understanding Mean Proportional
The mean proportional between two positive numbers, say 'a' and 'b', is a number 'x' such that the ratio of 'a' to 'x' is equal to the ratio of 'x' to 'b'. This can be written as:
\(\frac{a}{x} = \frac{x}{b}\)
Cross-multiplying this equation gives:
\(x^2 = ab\)
To find 'x', we take the square root of the product of 'a' and 'b':
\(x = \sqrt{ab}\)
So, the mean proportional between two numbers is the square root of their product.
Calculating the Mean Proportional between 3 and 27
In this problem, the two numbers are 3 and 27. We need to find the mean proportional, let's call it 'x'.
Using the formula \(x = \sqrt{ab}\), where \(a = 3\) and \(b = 27\):
- First, multiply the two numbers: \(3 \times 27\).
- \(3 \times 27 = 81\).
- Next, find the square root of the product: \(\sqrt{81}\).
- The square root of 81 is 9, because \(9 \times 9 = 81\).
Therefore, the mean proportional between 3 and 27 is 9.
| Number 1 (a) | Number 2 (b) | Product (\(a \times b\)) | Mean Proportional (\(\sqrt{ab}\)) |
|---|---|---|---|
| 3 | 27 | \(3 \times 27 = 81\) | \(\sqrt{81} = 9\) |
The calculation confirms that the mean proportional between 3 and 27 is indeed 9.
Revision Table: Key Concepts
| Concept | Explanation | Formula |
|---|---|---|
| Ratio | A comparison of two quantities. | \(a:b\) or \(\frac{a}{b}\) |
| Proportion | An equality between two ratios. | \(\frac{a}{b} = \frac{c}{d}\) |
| Mean Proportional | A number 'x' where \(\frac{a}{x} = \frac{x}{b}\), for numbers 'a' and 'b'. | \(x = \sqrt{ab}\) |
Additional Information: Geometric Mean
The mean proportional between two numbers is also known as their geometric mean. The geometric mean is a type of average that is useful for sets of positive numbers that are interpreted according to their product, like rates of growth.
- For two numbers 'a' and 'b', the geometric mean is \(\sqrt{ab}\).
- For 'n' numbers \((x_1, x_2, ..., x_n)\), the geometric mean is the nth root of their product: \(\sqrt[n]{x_1 x_2 ... x_n}\).
- The mean proportional is the specific case of the geometric mean for two numbers.
Understanding the geometric mean provides a broader context for the concept of the mean proportional between two numbers.
Important Questions from Mean Proportional
When x is subtracted from each of 24, 30, 36 and 46, the numbers so obtained in this order, are in proportion. What is the mean proportional between (2x + 3) and (3x + 2)?
What is the mean proportional of 40 and 90?
The first, third and fourth terms of a proportion are 8, 16 and 6 respectively. What is the second term?
The mean proportional between 0.16 and 0.64 is:
The mean proportion of the two numbers is 9 and the third proportion is 243. What will be the average of those numbers?
