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.
Important Questions from Numerical Reasoning
Among 150 faculty members in an institute, 55 are connected with each other through Facebook and 85 are connected through WhatsApp. 30 faculty members do not have Facebook or WhatsApp accounts. The number of faculty members connected only through Facebook accounts is ______________.
X is 1 km northeast of Y. Y is 1 km southeast of Z. W is 1 km west of Z. P is 1 km south of W. Q is 1 km east of P. What is the distance between X and Q in km?
- Two numbers are, respectively, 28% and 25% less than a third number. What percent is the first number of the second number?
A dealer sold three-forth (3/4th) of his articles at a gain of 20% and the remaining articles at the cost price. Find the gain earned by him in the whole transaction.
78, 65, 82, 69, 86, ?
