The fourth proportional to 3, 12, 14 is:

Question

Prepp Team · Accepted Answer

Finding the Fourth Proportional: A Step-by-Step Guide

The question asks us to find the fourth proportional to the numbers 3, 12, and 14. Understanding the concept of proportion is key to solving this problem.

Understanding Proportionality

When four numbers are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. If we have four numbers $a, b, c,$ and $d$ in proportion, we write it as:

$a : b :: c : d$

This can also be written in fraction form:

$\frac{a}{b} = \frac{c}{d}$

In this case, $d$ is called the fourth proportional to $a, b,$ and $c$.

Calculating the Fourth Proportional

Given the numbers 3, 12, and 14, we need to find the fourth proportional. Let's call the fourth proportional $x$. We can set up the proportion using the given numbers:

First number ($a$) = 3

Second number ($b$) = 12

Third number ($c$) = 14

Fourth proportional ($d$) = $x$

So the proportion is:

$3 : 12 :: 14 : x$

Writing this as fractions:

$\frac{3}{12} = \frac{14}{x}$

To solve for $x$, we can cross-multiply the terms:

$3 \times x = 12 \times 14$

$3x = 168$

Now, isolate $x$ by dividing both sides by 3:

$x = \frac{168}{3}$

Performing the division:

$x = 56$

So, the fourth proportional to 3, 12, and 14 is 56.

Verification of the Proportion

Let's check if the numbers 3, 12, 14, and 56 are indeed in proportion:

Ratio of the first two numbers: $\frac{3}{12} = \frac{1}{4}$

Ratio of the last two numbers: $\frac{14}{56} = \frac{1 \times 14}{4 \times 14} = \frac{1}{4}$

Since the ratios are equal ($\frac{1}{4} = \frac{1}{4}$), the numbers 3, 12, 14, and 56 are in proportion, and 56 is the correct fourth proportional.

Concept	Description	Formula
Ratio	Comparison of two quantities by division	a : b or $\frac{a}{b}$
Proportion	Equality of two ratios	$\frac{a}{b} = \frac{c}{d}$
Fourth Proportional	The fourth term ($d$) in a proportion $a : b :: c : d$	$d = \frac{b \times c}{a}$

Revision Table: Key Proportion Concepts

Understanding different types of proportion problems is helpful for exam preparation.

Term	Definition	Example (using 3, 12, 14)
Third Proportional	If $a, b, c$ are in continuous proportion ($a:b::b:c$), $c$ is the third proportional to $a$ and $b$.	Find third proportional to 3 and 12. $3:12::12:x \implies \frac{3}{12}=\frac{12}{x} \implies 3x = 144 \implies x=48$.
Mean Proportional	If $a, b, c$ are in continuous proportion ($a:b::b:c$), $b$ is the mean proportional between $a$ and $c$.	Find mean proportional between 3 and 48. $3:x::x:48 \implies \frac{3}{x}=\frac{x}{48} \implies x^2 = 3 \times 48 = 144 \implies x=12$ (positive root).
Fourth Proportional	If $a, b, c, d$ are in proportion ($a:b::c:d$), $d$ is the fourth proportional to $a, b,$ and $c$.	Find fourth proportional to 3, 12, 14. $3:12::14:x \implies \frac{3}{12}=\frac{14}{x} \implies 3x = 168 \implies x=56$.

Additional Information on Ratios and Proportions

Ratios can be simplified by dividing both parts by their greatest common divisor. For example, the ratio 3:12 simplifies to 1:4.
Proportions are used in many areas, including scaling recipes, map reading, and understanding relationships between quantities in science and engineering.
The product of the means equals the product of the extremes in a proportion. In $a:b::c:d$, $b$ and $c$ are the means, and $a$ and $d$ are the extremes. So, $b \times c = a \times d$. This is the principle used in cross-multiplication.