Find the compound ratio of 3:5, 5:7, and 7:9.

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Points to Remember:

  • Compound ratio involves multiplying corresponding terms of simple ratios.
  • The result is a single ratio representing the combined effect of the individual ratios.

Introduction:

A compound ratio is a ratio formed by multiplying two or more simple ratios. It represents the combined effect of the individual ratios. For instance, if we have the ratios A:B and B:C, the compound ratio is A:C, obtained by multiplying the antecedents (A and B) and consequents (B and C) separately. This concept is frequently used in various fields, including mathematics, finance, and even recipe scaling. This question asks us to find the compound ratio of 3:5, 5:7, and 7:9.

Body:

Calculating the Compound Ratio:

To find the compound ratio of 3:5, 5:7, and 7:9, we multiply the antecedents (the first terms of each ratio) together and the consequents (the second terms of each ratio) together.

  • Antecedents: 3 x 5 x 7 = 105
  • Consequents: 5 x 7 x 9 = 315

Therefore, the compound ratio is 105:315.

Simplifying the Compound Ratio:

The compound ratio 105:315 can be simplified by finding the greatest common divisor (GCD) of 105 and 315. The GCD of 105 and 315 is 105. Dividing both terms by 105, we get:

105 ÷ 105 = 1
315 ÷ 105 = 3

Thus, the simplified compound ratio is 1:3.

Applications of Compound Ratios:

Compound ratios are used in various real-world scenarios. For example:

  • Scaling Recipes: If a recipe calls for a ratio of ingredients (e.g., flour:sugar:butter), and you want to scale the recipe up or down, you would use compound ratios to maintain the correct proportions.
  • Financial Calculations: Compound ratios can be used in financial calculations involving interest rates, exchange rates, and other proportional relationships.
  • Map Scales: Map scales often utilize compound ratios to represent the relationship between distances on a map and actual distances on the ground.

Conclusion:

The compound ratio of 3:5, 5:7, and 7:9 is initially calculated as 105:315. However, simplifying this ratio by finding the greatest common divisor (105) yields the simplified and more useful ratio of 1:3. This demonstrates the importance of simplifying ratios to their simplest form for clarity and ease of understanding. The concept of compound ratios is fundamental in various mathematical and practical applications, highlighting its significance in proportional reasoning and problem-solving across different fields. Understanding and applying compound ratios effectively contributes to a more robust understanding of quantitative relationships.

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