Given (9 inches) ½ = (0.25 yards) ½, which one of the following statements is TRUE?
9 inches = 0.25 yards
Inches and Yards: Understanding the Given Equation
The problem presents a mathematical equation involving two different units of measurement: inches and yards. Our task is to analyze this equation and identify which of the provided statements is mathematically true based on the initial relationship.
The given equation is:
$(9 \text{ inches})^{\frac{1}{2}} = (0.25 \text{ yards})^{\frac{1}{2}}$
In simpler terms, the expression $(\text{value})^{\frac{1}{2}}$ denotes the square root of that value. So, the equation can be read as the square root of 9 inches is equal to the square root of 0.25 yards.
Equation Simplification: Finding the True Statement
To convert the equation into a more straightforward form and find an equivalent true statement, we need to eliminate the square root from both sides. The operation that undoes a square root is squaring. Therefore, we will square both sides of the given equation.
Let's follow these steps to simplify the equation:
- Step 1: Start with the original equation.
$(9 \text{ inches})^{\frac{1}{2}} = (0.25 \text{ yards})^{\frac{1}{2}}$ - Step 2: Square both sides of the equation to remove the fractional exponent.
When you square a term with an exponent of $\frac{1}{2}$, the exponents multiply: $(\text{value}^{\frac{1}{2}})^2 = \text{value}^{(\frac{1}{2} \times 2)} = \text{value}^1 = \text{value}$.
So, applying this to both sides:
$((9 \text{ inches})^{\frac{1}{2}})^2 = ((0.25 \text{ yards})^{\frac{1}{2}})^2$ - Step 3: Perform the squaring operation.
This simplifies the equation to:
$9 \text{ inches} = 0.25 \text{ yards}$
This simplified equation, $9 \text{ inches} = 0.25 \text{ yards}$, represents a true statement derived directly from the given condition.
Options Analysis: Identifying the Correct Statement
Now, we will compare our derived true statement, $9 \text{ inches} = 0.25 \text{ yards}$, with each of the provided options to determine which one is correct.
| Option | Statement | Verification Against $9 \text{ inches} = 0.25 \text{ yards}$ |
|---|---|---|
| 1 | $3 \text{ inches} = 0.15 \text{ yards}$ | If $9 \text{ inches} = 0.25 \text{ yards}$ is true, then taking the square root of both sides gives $\sqrt{9} \text{ inches} = \sqrt{0.25} \text{ yards}$, which simplifies to $3 \text{ inches} = 0.5 \text{ yards}$. Since $0.15 \neq 0.5$, this option is FALSE. |
| 2 | $9 \text{ inches} = 1.5 \text{ yards}$ | Our derived true statement is $9 \text{ inches} = 0.25 \text{ yards}$. Since $1.5 \neq 0.25$, this option is FALSE. |
| 3 | $9 \text{ inches} = 0.25 \text{ yards}$ | This statement is identical to the equation we derived by squaring both sides of the original given equation. Therefore, this option is TRUE. |
| 4 | $81 \text{ inches} = 0.0625 \text{ yards}$ | If $9 \text{ inches} = 0.25 \text{ yards}$ is true, then squaring both sides (including units) would result in $(9 \text{ inches})^2 = (0.25 \text{ yards})^2$, which simplifies to $81 \text{ (inches)}^2 = 0.0625 \text{ (yards)}^2$. However, the option states $81 \text{ inches} = 0.0625 \text{ yards}$ (without squared units). As a direct linear equivalence, this statement is FALSE. |
True Statement: Final Conclusion
After carefully simplifying the initial equation and evaluating each of the provided options, it is clear that the statement "$9 \text{ inches} = 0.25 \text{ yards}$" is the only one that holds true. It is a direct mathematical consequence of the given relationship.
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