Finding the Compound Ratio of 3:5, 5:7, and 7:9

**1. Understanding Compound Ratio:

• Compound Ratio: The compound ratio of a set of ratios is found by multiplying the numerators together and the denominators together. For example, to find the compound ratio of ratios
  $A:BA:B$ 

  A:B, $C:DC:D$ 

  C:D, and $E:FE:F$ 

  E:F, the compound ratio is calculated as $(A\times C\times E):(B\times D\times F)(A\; \backslash times\; C\; \backslash times\; E)\; :\; (B\; \backslash times\; D\; \backslash times\; F)$ 

  (A×C×E):(B×D×F).

**2. Given Ratios:

• Ratios to be Combined: 3:5, 5:7, and 7:9.

**3. Calculating the Compound Ratio:

• Numerators Multiplication: Multiply the numerators of the given ratios:

  $Numerators=3\times 5\times 7=105\backslash text\{Numerators\}\; =\; 3\; \backslash times\; 5\; \backslash times\; 7\; =\; 105$Numerators=3×5×7=105

• Denominators Multiplication: Multiply the denominators of the given ratios:

  $Denominators=5\times 7\times 9=315\backslash text\{Denominators\}\; =\; 5\; \backslash times\; 7\; \backslash times\; 9\; =\; 315$Denominators=5×7×9=315

• Compound Ratio: Therefore, the compound ratio is:

  $Compound Ratio=\frac{105}{315}\backslash text\{Compound\; Ratio\}\; =\; \backslash frac\{105\}\{315\}$Compound Ratio=315105​

**4. Simplifying the Compound Ratio:

• Simplification: To simplify the ratio
  $\frac{105}{315}\backslash frac\{105\}\{315\}$ 

  315105​, divide both the numerator and the denominator by their greatest common divisor, which is 105:

  $\frac{105÷105}{315÷105}=\frac{1}{3}\backslash frac\{105\; \backslash div\; 105\}\{315\; \backslash div\; 105\}\; =\; \backslash frac\{1\}\{3\}$315÷105105÷105​=31​

• Simplified Ratio: Thus, the simplified compound ratio is:

  $1:31:3$1:3

**5. Recent Examples and Applications:

• Finance and Investment: Understanding compound ratios is useful in finance for analyzing compounded growth rates. For example, investment portfolios with multiple assets often require the calculation of combined ratios to assess overall returns.
• Engineering and Architecture: In fields like engineering and architecture, compound ratios are used to determine proportions and scales when combining different ratios for design and construction purposes. For instance, in designing a multi-component system, engineers may need to combine different ratios of materials.

**6. Conclusion:

• Result: The compound ratio of 3:5, 5:7, and 7:9 is 1:3 after simplification. Calculating compound ratios helps in various practical applications where multiple ratios need to be combined to understand overall proportions and relationships.