Understanding Ratio Problems with Fractions

This question involves working with ratios, specifically converting a ratio with fractions into a simpler integer ratio and then using that ratio to find another related ratio. Let's break down the steps involved to solve this ratio problem effectively.

Step 1: Convert the Fractional Ratio a:b:c to an Integer Ratio

The given ratio is \(a : b : c = \frac{1}{4} : \frac{1}{3} : \frac{1}{2}\). To convert this fractional ratio into a simple integer ratio, we need to find the least common multiple (LCM) of the denominators of the fractions (4, 3, and 2).

• The denominators are 4, 3, and 2.
• The LCM of 4, 3, and 2 is 12.

Now, multiply each term in the ratio by the LCM (12):

• \(a = \frac{1}{4} \times 12 = 3\)
• \(b = \frac{1}{3} \times 12 = 4\)
• \(c = \frac{1}{2} \times 12 = 6\)

So, the integer ratio \(a : b : c\) is \(3 : 4 : 6\).

Step 2: Calculate the Individual Ratios \( \frac{a}{b}, \frac{b}{c}, \frac{c}{a} \)

Using the integer values we found for a, b, and c (which are proportional to the original fractional values), we can calculate the required ratios:

• \( \frac{a}{b} = \frac{3}{4} \)
• \( \frac{b}{c} = \frac{4}{6} = \frac{2}{3} \) (simplifying the fraction)
• \( \frac{c}{a} = \frac{6}{3} = 2 = \frac{2}{1} \) (simplifying and writing as a fraction)

Step 3: Form the Ratio \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \) and Convert to an Integer Ratio

The ratio we need to find is \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \), which is \( \frac{3}{4} : \frac{2}{3} : \frac{2}{1} \). Again, we have a ratio with fractions. To convert this to an integer ratio, we find the LCM of the denominators (4, 3, and 1).

• The denominators are 4, 3, and 1.
• The LCM of 4, 3, and 1 is 12.

Multiply each term in the ratio \( \frac{3}{4} : \frac{2}{3} : \frac{2}{1} \) by the LCM (12):

• First term: \( \frac{3}{4} \times 12 = 3 \times 3 = 9 \)
• Second term: \( \frac{2}{3} \times 12 = 2 \times 4 = 8 \)
• Third term: \( \frac{2}{1} \times 12 = 2 \times 12 = 24 \)

So, the integer ratio \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \) is \(9 : 8 : 24\).

Summary of Calculations

The final calculated ratio \( \frac{a}{b} : \frac{b}{c} : \frac{c}{a} \) is \(9 : 8 : 24\). This corresponds to one of the given options.

Revision Table: Key Ratio Concepts

Additional Information: Properties of Ratios

• Ratios can be simplified by dividing all parts by the same non-zero number.
• Ratios can be converted into fractions or percentages.
• If \(a : b = x : y\), then \( \frac{a}{b} = \frac{x}{y} \).
• Compound ratio of \(a:b\) and \(c:d\) is \((ac):(bd)\).
• Duplicate ratio of \(a:b\) is \(a^2:b^2\).
• Sub-duplicate ratio of \(a:b\) is \( \sqrt{a}:\sqrt{b} \).
• Triplicate ratio of \(a:b\) is \(a^3:b^3\).
• Sub-triplicate ratio of \(a:b\) is \( \sqrt[3]{a}:\sqrt[3]{b} \).