If p is the third proportional to 3, 9, then what is the fourth proportional to 6, p, 4?

Question

Prepp Team · Accepted Answer

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.

Step	Calculation	Result
Find p (Third Proportional)	$\frac{3}{9} = \frac{9}{p} \implies 3p = 81 \implies p = 27$	p = 27
Find x (Fourth Proportional)	$\frac{6}{p} = \frac{4}{x} \implies \frac{6}{27} = \frac{4}{x} \implies 6x = 108 \implies x = 18$	x = 18

Revision Table: Key Concepts in Proportion

Concept	Definition	Formula	Example
Ratio	Comparison of two quantities by division.	$a:b$ or $\frac{a}{b}$	Ratio of 3 to 9 is $3:9$ or $\frac{3}{9} = \frac{1}{3}$
Proportion	Equality of two ratios.	$\frac{a}{b} = \frac{c}{d}$	$\frac{3}{9} = \frac{1}{3}$ is a proportion.
Third Proportional	For a, b, it is c such that $\frac{a}{b} = \frac{b}{c}$.	$c = \frac{b^2}{a}$	Third proportional to 3, 9 is $\frac{9^2}{3} = \frac{81}{3} = 27$.
Fourth Proportional	For a, b, c, it is d such that $\frac{a}{b} = \frac{c}{d}$.	$d = \frac{b \times c}{a}$	Fourth proportional to 6, 27, 4 is $\frac{27 \times 4}{6} = \frac{108}{6} = 18$.

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.

Prepp Team · Answer

10

Prepp Team · Answer

$\frac{3}{2}$

Prepp Team · Answer

2&radic;3