Understanding Fourth Proportional and Ratios

This question asks us to find the ratio between two quantities, where each quantity is a fourth proportional of a given set of three numbers. Let's first understand what a fourth proportional is.

If four quantities \(a, b, c, d\) are in proportion, it means that the ratio of the first two is equal to the ratio of the last two. This is written as \(a:b = c:d\). In this proportion, \(d\) is called the fourth proportional to \(a, b,\) and \(c\). This can also be written as the equation:

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

To find the fourth proportional \(d\), we can rearrange this equation:

\(a \times d = b \times c\)

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

Now, let's solve the problem by finding the two fourth proportionals.

Calculating the First Fourth Proportional

We are given the numbers 2, 5, and 6. We need to find the fourth proportional, let's call it \(x\).

So, we have the proportion \(2:5 = 6:x\).

Using the formula or setting up the fraction:

\(\frac{2}{5} = \frac{6}{x}\)

To solve for \(x\), we can cross-multiply:

\(2 \times x = 5 \times 6\)

\(2x = 30\)

Now, divide both sides by 2:

\(x = \frac{30}{2}\)

\(x = 15\)

So, the fourth proportional of 2, 5, 6 is 15.

Calculating the Second Fourth Proportional

Next, we are given the numbers 6, 8, and 9. We need to find the fourth proportional, let's call it \(y\).

So, we have the proportion \(6:8 = 9:y\).

Using the fraction form:

\(\frac{6}{8} = \frac{9}{y}\)

To solve for \(y\), cross-multiply:

\(6 \times y = 8 \times 9\)

\(6y = 72\)

Now, divide both sides by 6:

\(y = \frac{72}{6}\)

\(y = 12\)

So, the fourth proportional of 6, 8, 9 is 12.

Finding the Ratio of the Fourth Proportionals

The question asks for the ratio of the first fourth proportional (which is 15) and the second fourth proportional (which is 12).

The ratio is \(15:12\).

To simplify this ratio, we need to find the greatest common divisor (GCD) of 15 and 12. The divisors of 15 are 1, 3, 5, and 15. The divisors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common divisor is 3.

Divide both parts of the ratio by the GCD (3):

\(\frac{15}{3} = 5\)

\(\frac{12}{3} = 4\)

So, the simplified ratio is \(5:4\).

Let's summarize the calculations in a table:

The ratio of the fourth proportional of 2, 5, 6 and the fourth proportional of 6, 8, 9 is \(15:12\), which simplifies to \(5:4\).

Revision Table: Ratio and Proportion Concepts

Additional Information: Types of Proportionals

Besides the fourth proportional, there are other related terms in proportions:

• Third Proportional: If \(a, b, c\) are in continuous proportion (\(a:b = b:c\)), then \(c\) is called the third proportional to \(a\) and \(b\). This means \(\frac{a}{b} = \frac{b}{c}\), so \(b^2 = ac\), or \(c = \frac{b^2}{a}\).
• Mean Proportional: If \(a, b, c\) are in continuous proportion (\(a:b = b:c\)), then \(b\) is called the mean proportional between \(a\) and \(c\). This means \(\frac{a}{b} = \frac{b}{c}\), so \(b^2 = ac\), or \(b = \sqrt{ac}\).

Understanding these terms helps in solving various problems related to ratio and proportion.