Understanding Third and Fourth Proportionals in Math

This problem involves finding two types of proportionals: the third proportional and the fourth proportional. Let's break down the steps to solve it.

Step 1: Find the Third Proportional (p) to 3 and 9

The third proportional to two numbers, say 'a' and 'b', is a number 'c' such that the ratio of the first two is equal to the ratio of the second and third. This can be written as:

\(\frac{a}{b} = \frac{b}{c}\)

In this question, 'a' is 3, 'b' is 9, and the third proportional is 'p'. So, we have:

\(\frac{3}{9} = \frac{9}{p}\)

To solve for 'p', we can cross-multiply:

\(3 \times p = 9 \times 9\)

\(3p = 81\)

Now, divide both sides by 3:

\(p = \frac{81}{3}\)

\(p = 27\)

So, the value of p, the third proportional to 3 and 9, is 27.

Step 2: Find the Fourth Proportional to 6, p, and 4

The fourth proportional to three numbers, say 'a', 'b', and 'c', is a number 'd' such that the ratio of the first two is equal to the ratio of the third and fourth. This can be written as:

\(\frac{a}{b} = \frac{c}{d}\)

In this question, the three numbers are 6, p, and 4, and we need to find the fourth proportional. Let's call the fourth proportional 'x'. The given numbers are 6, p (which we found to be 27), and 4. So, we have:

• First number (a) = 6
• Second number (b) = p = 27
• Third number (c) = 4
• Fourth proportional (d) = x

Plugging these values into the formula:

\(\frac{6}{27} = \frac{4}{x}\)

To solve for 'x', we again cross-multiply:

\(6 \times x = 27 \times 4\)

\(6x = 108\)

Now, divide both sides by 6:

\(x = \frac{108}{6}\)

\(x = 18\)

The fourth proportional to 6, p (or 27), and 4 is 18.

Final Answer

The value of p (the third proportional to 3, 9) is 27. The fourth proportional to 6, p, 4 is 18.

Revision Table: Key Concepts in Proportion

Additional Information on Direct and Inverse Proportion

Understanding direct and inverse proportion can help solve many problems related to ratios and proportions.

• Direct Proportion: Two quantities are directly proportional if an increase in one quantity causes a proportional increase in the other, and a decrease causes a proportional decrease. Their ratio is constant. If y is directly proportional to x, then \(\frac{y}{x} = k\) (constant). Example: The cost of apples is directly proportional to the number of apples bought.
• Inverse Proportion: Two quantities are inversely proportional if an increase in one quantity causes a proportional decrease in the other, and vice versa. Their product is constant. If y is inversely proportional to x, then \(y \times x = k\) (constant). Example: The speed of a vehicle is inversely proportional to the time taken to cover a fixed distance.

These concepts build upon the basic understanding of ratios and proportion used in finding third and fourth proportionals.