Finding the Fourth Proportional Explained

The question asks us to find the fourth proportional to the numbers 17, 272, and 19. Let's first understand what a proportional is in this context.

When four numbers, say \(a\), \(b\), \(c\), and \(d\), are in proportion, it means that the ratio of the first two numbers is equal to the ratio of the last two numbers. This is written as \(a : b :: c : d\). The double colon (::) means "is to" or "as". In this proportion, \(d\) is called the fourth proportional to \(a\), \(b\), and \(c\).

Mathematically, the proportion \(a : b :: c : d\) can be written as an equation of ratios:

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

To find the fourth proportional \(d\), we can rearrange this equation. By cross-multiplication, we get \(a \times d = b \times c\). Then, solving for \(d\), we find \(d = \frac{b \times c}{a}\).

Applying the Concept to Find the Fourth Proportional

In our problem, the given numbers are 17, 272, and 19. We need to find the fourth proportional, which we can call \(x\). So, the numbers are in proportion as 17, 272, 19, and \(x\).

Setting up the proportion based on the given numbers:

\(17 : 272 :: 19 : x\)

Now, let's write this proportion as an equation of fractions:

\(\frac{17}{272} = \frac{19}{x}\)

Step-by-Step Calculation of the Fourth Proportional

To find the value of \(x\), the fourth proportional, we can cross-multiply the equation \(\frac{17}{272} = \frac{19}{x}\).

1. Cross-multiplying gives:

   \(17 \times x = 272 \times 19\)

2. Now, to isolate \(x\), we divide both sides of the equation by 17:

   \(x = \frac{272 \times 19}{17}\)

3. We can simplify this by dividing 272 by 17 first:

   \(272 \div 17 = 16\)

4. Substitute the result back into the equation for \(x\):

   \(x = 16 \times 19\)

5. Finally, perform the multiplication:

   \(x = 304\)

So, the fourth proportional to 17, 272, and 19 is 304.

Revision Table: Key Concepts in Proportion

Understanding proportions is fundamental. Here's a quick review:

Additional Information on Ratios and Proportions

Ratios and proportions are widely used in mathematics and various real-world applications.

• Direct Proportion: Two quantities are in direct proportion if an increase in one quantity causes a proportional increase in the other, and vice versa. For example, the cost of apples is directly proportional to the number of apples bought.
• Inverse Proportion: Two quantities are in inverse proportion if an increase in one quantity causes a proportional decrease in the other, and vice versa. For example, the speed of a car and the time taken to cover a fixed distance are inversely proportional.
• Continued Proportion: When three numbers \(a\), \(b\), and \(c\) are such that \(a:b :: b:c\), they are said to be in continued proportion. Here, \(b\) is called the mean proportional between \(a\) and \(c\). The relationship is \(b^2 = ac\).

Understanding these concepts helps solve various problems related to ratios, proportions, speed, time, distance, work, etc.