What is the third proportional to 9 and 36?
144
Understanding the Third Proportional Concept
The question asks for the third proportional to two numbers, 9 and 36. In a continued proportion involving three numbers, say a, b, and c, the relationship is expressed as \(a:b :: b:c\). Here, 'a' is the first proportional, 'b' is the mean proportional, and 'c' is the third proportional. This relationship can also be written as a fraction: \(\frac{a}{b} = \frac{b}{c}\).
To find the third proportional 'c' to two given numbers 'a' and 'b' (where 'b' is the mean proportional), we use the formula derived from the proportion:
\(\frac{a}{b} = \frac{b}{c}\)
By cross-multiplication, we get:
\(a \times c = b \times b\)
\(a \times c = b^2\)
And finally, to find 'c':
\(c = \frac{b^2}{a}\)
Calculating the Third Proportional to 9 and 36
In this problem, the first number is 9, which corresponds to 'a' in our formula, and the second number is 36, which corresponds to 'b' (the mean proportional). We need to find 'c', the third proportional.
- First Proportional (a) = 9
- Mean Proportional (b) = 36
- Third Proportional (c) = ?
Using the formula \(c = \frac{b^2}{a}\), we substitute the given values:
\(c = \frac{36^2}{9}\)
Now, we calculate the value:
\(c = \frac{36 \times 36}{9}\)
We can simplify the calculation by dividing 36 by 9:
\(c = \frac{4 \times 9 \times 36}{9}\)
\(c = 4 \times 36\)
Performing the multiplication:
\(c = 144\)
So, the third proportional to 9 and 36 is 144.
| Step | Description | Calculation |
|---|---|---|
| 1 | Identify a and b | a = 9, b = 36 |
| 2 | Write the formula | \(c = \frac{b^2}{a}\) |
| 3 | Substitute values | \(c = \frac{36^2}{9}\) |
| 4 | Simplify | \(c = \frac{1296}{9}\) or \(c = \frac{36 \times 36}{9}\) |
| 5 | Calculate result | \(c = 144\) |
Therefore, the third proportional to 9 and 36 is 144.
Revision Table: Key Ratio and Proportion Terms
| Term | Definition | Example (using a:b::c:d) |
|---|---|---|
| Ratio | Comparison of two quantities by division. | a:b (read as 'a is to b') |
| Proportion | Equality of two ratios. | a:b :: c:d (read as 'a is to b as c is to d') |
| Extremes | The first and fourth terms in a proportion (a and d). | In 2:3 :: 4:6, 2 and 6 are extremes. |
| Means | The second and third terms in a proportion (b and c). | In 2:3 :: 4:6, 3 and 4 are means. |
| Product of Extremes | The product of the first and fourth terms. | \(a \times d\) |
| Product of Means | The product of the second and third terms. | \(b \times c\) |
| Property of Proportion | Product of extremes equals product of means. | \(a \times d = b \times c\) |
| Continued Proportion | When three or more quantities are such that the ratio of the first to the second is equal to the ratio of the second to the third, and so on (a:b::b:c::c:d...). | 3, 6, 12 are in continued proportion because 3:6 :: 6:12 (\(\frac{3}{6} = \frac{6}{12} = \frac{1}{2}\)). |
| Third Proportional | In a continued proportion a:b::b:c, 'c' is the third proportional to 'a' and 'b'. | If 3, 6, c are in continued proportion, c is the third proportional. \(\frac{3}{6} = \frac{6}{c} \implies 3c = 36 \implies c = 12\). |
| Mean Proportional | In a continued proportion a:b::b:c, 'b' is the mean proportional between 'a' and 'c'. | If 3, b, 12 are in continued proportion, b is the mean proportional. \(\frac{3}{b} = \frac{b}{12} \implies b^2 = 36 \implies b = \sqrt{36} = 6\). |
Additional Information on Proportionality
Understanding proportions is fundamental in various areas of mathematics and real life, including scaling recipes, converting currencies, interpreting maps, and solving problems in physics and engineering. The concept of the third proportional is a specific case of continued proportion, which highlights how quantities can be related sequentially.
For three numbers a, b, and c to be in continued proportion, the ratio of the first to the second must be equal to the ratio of the second to the third. This condition, \(\frac{a}{b} = \frac{b}{c}\), is key. It implies that the middle term (b) is the geometric mean of the first (a) and the third (c), because \(b^2 = ac\), so \(b = \sqrt{ac}\). This is why 'b' is often called the mean proportional or geometric mean.
In the context of the question about the third proportional to 9 and 36, 9 is the first term, 36 is the mean proportional, and we are finding the third term. The calculation \(c = \frac{b^2}{a}\) is a direct application of the continued proportion definition.
Important Questions from Third Proportional
What is the third proportional to 16 and 40?
When the same number is subtracted from each of 7, 9, 11 and 15, the resulting numbers are in proportion. The number subtracted is:
If L : M = 3 : 5 and M : N = 2 : 3, then N : L = ?
A. 2 : 1
B. 5 : 2
C. 3 : 2
D. 1 : 2
What is the third proportional to 16 and 24 ?
The third proportional to 4 and 6 is:
