If the numbers 2, x, x, 8 are in proportion, find x.

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Points to Remember:

  • The concept of proportion.
  • Cross-multiplication in proportions.
  • Solving for an unknown variable in an equation.

Introduction:

Proportionality is a fundamental concept in mathematics expressing the equality of two ratios. If four numbers a, b, c, and d are in proportion, it means that the ratio a/b is equal to the ratio c/d. This is often written as a:b::c:d or a/b = c/d. This relationship implies a consistent relationship between the numbers; a change in one number proportionally affects the others. In this question, we are given three numbers (2, x, x, 8) that are in proportion, and we need to find the value of x. This requires understanding and applying the principles of proportionality to solve for the unknown variable.

Body:

Setting up the Proportion:

Since the numbers 2, x, x, and 8 are in proportion, we can set up the equation as follows:

2/x = x/8

Solving for x:

To solve for x, we use cross-multiplication:

2 * 8 = x * x

16 = x²

Taking the square root of both sides:

x = ±√16

x = ±4

Therefore, x can be either 4 or -4.

Verification:

Let’s verify both solutions:

  • If x = 4: The proportion becomes 2/4 = 4/8, which simplifies to 1/2 = 1/2. This is true.
  • If x = -4: The proportion becomes 2/(-4) = (-4)/8, which simplifies to -1/2 = -1/2. This is also true.

Discussion of Solutions:

Both positive and negative values of x satisfy the proportional relationship. The context of the problem might dictate which solution is more appropriate. For instance, if x represents a physical quantity (like length or weight), a negative value might be unrealistic. However, purely mathematically, both solutions are valid.

Conclusion:

In conclusion, if the numbers 2, x, x, and 8 are in proportion, then x can be either 4 or -4. Both values satisfy the equation derived from the proportional relationship. The choice of which value to use depends on the specific context of the problem. Understanding the concept of proportion and the ability to solve for unknown variables using cross-multiplication are crucial skills in mathematics and have wide applications in various fields, from physics and engineering to economics and finance. The ability to solve such problems demonstrates a fundamental understanding of mathematical relationships and problem-solving techniques.

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