1:x = x:64. what one number can replace x?

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29-03-2025

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This article explores how to solve the proportion 1:x = x:64, finding the single number that satisfies this equation. We'll walk through the steps, explaining the mathematical concepts involved. Understanding proportions is crucial in many areas, from scaling recipes to calculating ratios in various fields.

Understanding Proportions

A proportion is a statement that two ratios are equal. A ratio compares two quantities. In the given proportion, 1:x and x:64 are ratios. The colon (:) represents division, so 1:x can be written as 1/x.

The proportion 1:x = x:64 means:

1/x = x/64

Solving for x

To solve for 'x', we'll use cross-multiplication. This method involves multiplying the numerator of one ratio by the denominator of the other, and setting the products equal.

Step 1: Cross-Multiply

(1) * (64) = x * x

This simplifies to:

64 = x²

Step 2: Find the Square Root

To isolate 'x', we take the square root of both sides of the equation:

√64 = √x²

This gives us two possible solutions:

x = 8 or x = -8

Understanding the Two Solutions

We've found two values for x that satisfy the equation: 8 and -8. However, depending on the context, one solution might be more appropriate than the other. Since ratios often represent real-world quantities (like lengths or amounts), a negative solution might not always make sense.

Let's check both solutions:

• If x = 8: 1:8 = 8:64 simplifies to 1/8 = 8/64, which is true (both equal 1/8).

• If x = -8: 1:-8 = -8:64 simplifies to -1/8 = -1/8, which is also true.

Conclusion: The Number that Replaces x

Therefore, the number that can replace 'x' in the proportion 1:x = x:64 is 8 or -8. While both solutions are mathematically correct, the practical application will determine which is the more relevant answer. In many cases, the positive solution (8) will be the one used. This problem demonstrates the power of proportions and how solving them can lead to multiple, valid solutions.