Given \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\). Which one of the following statements is TRUE?
9 inches = 0.25 yards
The question asks us to identify the true statement given the equation \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\). This equation involves fractional exponents, specifically the power of \(\frac{1}{2}\), which represents a square root.
Equation Analysis: Inches and Yards
The given equation is:
\[{\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}} = {\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}\]
We know that an exponent of \(\frac{1}{2}\) is equivalent to taking the square root. So, the equation can be conceptually understood as:
\[\sqrt{9{\rm{\;inches}}} = \sqrt{0.25{\rm{\;yards}}}\]
To eliminate the fractional exponents and simplify the equation to a direct comparison like "X inches = Y yards", we can square both sides of the original equation. Squaring an expression raised to the power of \(\frac{1}{2}\) will remove the exponent, as \({\left( {a^{\frac{1}{2}}} \right)^2} = a^{\frac{1}{2} \times 2} = a^1 = a\).
Solution Steps: Inches and Yards Equation
Let's square both sides of the given equation:
\[{\left( {{\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2}}}} \right)^2} = {\left( {{\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2}}}} \right)^2}\]
Applying the exponent rule \({\left( {a^m} \right)^n} = a^{m \times n}\), we multiply the exponents:
- For the left side: \({\left( {9{\rm{\;inches}}} \right)^{\frac{1}{2} \times 2}} = {\left( {9{\rm{\;inches}}} \right)^1} = 9{\rm{\;inches}}\)
- For the right side: \({\left( {0.25{\rm{\;yards}}} \right)^{\frac{1}{2} \times 2}} = {\left( {0.25{\rm{\;yards}}} \right)^1} = 0.25{\rm{\;yards}}\)
Therefore, after squaring both sides, the equation simplifies to a clear equality:
\[9{\rm{\;inches}} = 0.25{\rm{\;yards}}\]
Statement Comparison: Finding the True Statement
Now, let's compare our derived statement with the given options to find which one is TRUE.
| Option | Statement | Analysis |
|---|---|---|
| 1 | \(3{\rm{\;inches}} = 0.5{\rm{\;yards}}\) | This statement would be obtained if we took the square root of the numerical values (i.e., \(\sqrt{9} = 3\) and \(\sqrt{0.25} = 0.5\)). However, the given equation already expresses the equality of the square roots of the total expressions. To find the direct relationship without the exponent, we square both sides of the original equation. |
| 2 | \(9{\rm{\;inches}} = 1.5{\rm{\;yards}}\) | This statement does not match our derived result of \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\). |
| 3 | \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\) | This statement exactly matches our derived result from simplifying the given equation. This is the true statement. |
| 4 | \(81{\rm{\;inches}} = 0.0625{\rm{\;yards}}\) | This statement would be obtained by squaring both sides of the true statement \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\) (i.e., \(9^2 = 81\) and \(0.25^2 = 0.0625\)). The question asks for the statement directly implied by the *given* equation, not a further transformation of its simplified form. |
From the comparison, it is clear that the statement \(9{\rm{\;inches}} = 0.25{\rm{\;yards}}\) is the true statement derived directly from the given equation.
Important Questions from Numerical Estimation
The number of digits you have to type to write all the page numbers of a book starting from I (first page) is 2019. What is the number of pages in that book?
Two fair dice are thrown. The number of cases where the number appearing on the upper face of the first die is not less than that on the lower face of the second die is
The number of Hens, Ducks, and Goats in farm P are 65, 91 and 169, respectively. The total number of Hens, Ducks and Goats in a nearby farm Q is 416. The ratio of hens : ducks : goats in farm Q is 5 : 14 : 13. All the hens, ducks and goats are sent from farm Q to farm P.
The new ratio of hens : ducks : goats in farm P is ____________.
A person divided an amount of Rs. 100,000 into two parts and invested in two different schemes. In one he got 10% profit and in the other he got 12%. If the profit percentages are interchanged with these investments he would have got Rs.120 less. Find the ratio between his investments in the two schemes.
S, M, E and F are working in shifts in a team to finish a project. M works with twice the efficiency of others but for half as many days as E worked. S and M have 6 hour shifts in a day, whereas E and F have 12 hours shifts. What is the ratio of contribution of M to contribution of E in the project?
