Understanding Compound Ratio

The question asks us to find the compound ratio of three given ratios: 2 : 3, 4 : 7, and 5 : 6.

A compound ratio is obtained by multiplying the corresponding terms of two or more simple ratios. Specifically, you multiply the antecedents (the first terms) together and the consequents (the second terms) together.

Calculating the Compound Ratio

We are given the following ratios:

• Ratio 1: 2 : 3
• Ratio 2: 4 : 7
• Ratio 3: 5 : 6

The antecedents of these ratios are the first terms: 2, 4, and 5.

The consequents of these ratios are the second terms: 3, 7, and 6.

To find the compound ratio, we perform the following calculations:

Product of antecedents = \(2 \times 4 \times 5\)

Product of consequents = \(3 \times 7 \times 6\)

Let's calculate the products:

Product of antecedents = \(2 \times 4 = 8\), and \(8 \times 5 = 40\)

So, the product of the antecedents is 40.

Product of consequents = \(3 \times 7 = 21\), and \(21 \times 6\). To calculate \(21 \times 6\): \(20 \times 6 = 120\), and \(1 \times 6 = 6\). So, \(120 + 6 = 126\).

So, the product of the consequents is 126.

The compound ratio is the ratio of the product of the antecedents to the product of the consequents.

Compound Ratio = (Product of antecedents) : (Product of consequents)

Compound Ratio = 40 : 126

Now, we should simplify this ratio to its lowest terms by finding the greatest common divisor (GCD) of 40 and 126.

Let's find the factors of 40 and 126.

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Factors of 126: 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126

The common factors are 1 and 2. The greatest common divisor (GCD) is 2.

Divide both terms of the ratio 40 : 126 by the GCD, which is 2.

\(\frac{40}{2} : \frac{126}{2}\)

\(20 : 63\)

The simplified compound ratio is 20 : 63.

Let's check if the calculation was correct. Product of antecedents = \(2 \times 4 \times 5 = 40\). Product of consequents = \(3 \times 7 \times 6 = 21 \times 6 = 126\). Ratio = 40 : 126. Simplifying 40/126 by dividing by 2 gives 20/63. The ratio is 20 : 63.

Alternatively, you can sometimes simplify terms before multiplying. For example, in the ratios 2:3, 4:7, 5:6, we could write them as fractions \(\frac{2}{3}\), \(\frac{4}{7}\), \(\frac{5}{6}\). The compound ratio is the product of these fractions:

\(\frac{2}{3} \times \frac{4}{7} \times \frac{5}{6}\)

We can cancel out common factors between numerators and denominators.

Notice that the 2 in the first numerator and the 6 in the third denominator share a factor of 2. Divide both by 2:

\(\frac{1}{3} \times \frac{4}{7} \times \frac{5}{3}\) (The 2 becomes 1, the 6 becomes 3)

Now, multiply the remaining numerators and denominators:

Product of numerators = \(1 \times 4 \times 5 = 20\)

Product of denominators = \(3 \times 7 \times 3 = 21 \times 3 = 63\)

The resulting fraction is \(\frac{20}{63}\), which corresponds to the ratio 20 : 63.

Both methods yield the same result.

Revision Table: Key Ratio Concepts

Additional Information: Ratios in Real Life

Ratios are used widely in everyday life and various fields:

• Cooking: Recipes often use ratios to specify ingredient quantities (e.g., 2 cups flour to 1 cup sugar).
• Maps and Models: Scales on maps and models are ratios comparing a distance on the map/model to the actual distance (e.g., 1:1000).
• Finance: Financial ratios are used to analyze company performance (e.g., debt-to-equity ratio).
• Chemistry: Ratios represent the proportions of different elements in a compound.
• Art and Design: Ratios like the Golden Ratio are used for aesthetic proportions.

Understanding how to manipulate ratios, including finding compound ratios, is fundamental in many quantitative applications.