Understanding Proportions: Solving 5:9 16:27

In mathematical terms, a proportion is an equation stating that two ratios are equal. For instance, the proportion 5:9 16:27 illustrates the relationship between two pairs of numbers. This article will delve into the logical steps to solve such a proportion problem and provide a detailed explanation of the process.

Solving the Proportion 5:9 16:27

To solve the proportion 5:9 16:27, we start by expressing it as a mathematical equation:

5/9 16/27

The next step involves cross-multiplication to determine if the two ratios are indeed equivalent. Cross-multiplication means multiplying the numerator of one fraction by the denominator of the other, and setting the products equal to one another. In this case, we have:

5 * 27 16 * 9

Solving this equation, we get:

135 144

This result shows that the two ratios are not equivalent, which implies there might be an error in the given problem or the interpretation of the equation.

Correcting and Verifying the Proportion

However, to understand the problem better, let's consider the possibility of performing further algebraic manipulation. We can start by solving for a missing number to make the given proportions equal. This can be done using the concept of solving equations.

First, let's isolate the fraction with the unknown:

(16/27) – (5/9) (16 – 15) / 27 1/27

Then, to fill the blanks, we need to find the correct proportion that maintains the equivalence:

5:9 : 1:27 16:27 Ans.

This means that the original problem might need to be adjusted for it to be a correct proportion. Let's explore this with a corrected example.

Correcting the Proportion 5:9 16:2x

To ensure that 5:9 is equal to 16:2x, we need to find the value of x. This involves setting up the proportion and solving for x:

5/9 16/2x

Cross-multiplying gives us:

5 * 2x 9 * 16

Simplifying further:

1 144

Solving for x:

x 144 / 10 14.4

Therefore, the corrected proportion is:

5:9 16:14.4

This proves that the original problem can be adjusted to maintain a correct proportion.

Conclusion

In conclusion, solving proportions like 5:9 16:27 requires a clear understanding of ratios and algebraic manipulation. By cross-multiplying, we can verify the equivalence of the given ratios or solve for unknown values to maintain the proportion. This knowledge is essential in various fields, including mathematics, engineering, and science, where proportional relationships are frequently employed.

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By understanding these concepts, you can effectively solve problems involving proportions and ratios, ensuring accuracy in your mathematical calculations.