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What is the value of \(\sqrt{\frac{20\%}{40\%}}\)?
A. \((\frac{1}{2})\%\)
B. \((\frac{1}{\sqrt{2}})\%\)
C. \((\frac{50}{\sqrt{2}})\%\)
D. \((50\sqrt{2})\%\)
E. \((100\sqrt{2})\%\)
A. \((\frac{1}{2})\%\)
B. \((\frac{1}{\sqrt{2}})\%\)
C. \((\frac{50}{\sqrt{2}})\%\)
D. \((50\sqrt{2})\%\)
E. \((100\sqrt{2})\%\)
17 May 2022, 08:20
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Official Solution:
What is the value of \(\sqrt{\frac{20\%}{40\%}}\)?
A. \((\frac{1}{2})\%\)
B. \((\frac{1}{\sqrt{2}})\%\)
C. \((\frac{50}{\sqrt{2}})\%\)
D. \((50\sqrt{2})\%\)
E. \((100\sqrt{2})\%\)
Worth knowing that "per cent" from Latin literally means "per one hundred" or "out of one hundred", so for example, \(x\%\) is \(\frac{x}{100}\) (\(x\) per one hundred) and say \(5\%\) is \(\frac{5}{100} = 0.05\) (5 out of one hundred). On the other hand, we can write 0.05 as \(0.05*100\% = 5\%\) and say 20 as \(20*100\% = 2000\%\).
Basically, "%" symbol just means "per 100", or algebraically, "/100". Thus:
To drop "%" symbol, so to convert the percentage into a ratio, just divide by 100: \(m\% = \frac{m}{100}\). For example, \(10\%=\frac{10}{100}=\frac{1}{10}=0.1\) and \(400\%=\frac{400}{100}=4\).
To get "%" symbol, so to convert the ratio into a percentage, just multiply by 100%: \(n=n*100\%\) (100% is just 100/100 = 1, so we are essentially multiplying by 1). For example, \(0.4=0.4*100\%=40\%\) and \(15=15*100\%=1500\%\).
To sum up: \(x\%\) and \(\frac{x}{100}\) are just two different ways of writing the same thing: as a percentage and as a ratio.
Back to the question..
Now, there are certain algebraic rules you can apply when there are "%" symbols in an equation, and if you know them (though they are no necessary for the GMAT), then you could directly write \(\sqrt{\frac{20\%}{40\%}}=\sqrt{\frac{20}{40}}\) but even if you are not aware of these rules, then you can apply the above method of converting the percentage into a ratio:
\(\sqrt{\frac{20\%}{40\%}}=\sqrt{\frac{(\frac{20}{100})}{(\frac{40}{100})}}=\sqrt{\frac{20}{100}*\frac{100}{40}}=\sqrt{\frac{20}{40}}\).
Next:
\(\sqrt{\frac{20}{40}}=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}\).
Notice that the answer choices are as a percentage, so to get "%" symbol, to convert the ratio into a percentage, just multiply by 100%:
\(\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}*100\%=\frac{100}{\sqrt{2}}\%\).
Still no match with any of the options, so let's rationalize the fraction:
\(\frac{100}{\sqrt{2}}\%=(\frac{100}{\sqrt{2}})*(\frac{\sqrt{2}}{\sqrt{2}})\%=(50\sqrt{2})\%\).
Answer: D
What is the value of \(\sqrt{\frac{20\%}{40\%}}\)?
A. \((\frac{1}{2})\%\)
B. \((\frac{1}{\sqrt{2}})\%\)
C. \((\frac{50}{\sqrt{2}})\%\)
D. \((50\sqrt{2})\%\)
E. \((100\sqrt{2})\%\)
Worth knowing that "per cent" from Latin literally means "per one hundred" or "out of one hundred", so for example, \(x\%\) is \(\frac{x}{100}\) (\(x\) per one hundred) and say \(5\%\) is \(\frac{5}{100} = 0.05\) (5 out of one hundred). On the other hand, we can write 0.05 as \(0.05*100\% = 5\%\) and say 20 as \(20*100\% = 2000\%\).
Basically, "%" symbol just means "per 100", or algebraically, "/100". Thus:
To drop "%" symbol, so to convert the percentage into a ratio, just divide by 100: \(m\% = \frac{m}{100}\). For example, \(10\%=\frac{10}{100}=\frac{1}{10}=0.1\) and \(400\%=\frac{400}{100}=4\).
To get "%" symbol, so to convert the ratio into a percentage, just multiply by 100%: \(n=n*100\%\) (100% is just 100/100 = 1, so we are essentially multiplying by 1). For example, \(0.4=0.4*100\%=40\%\) and \(15=15*100\%=1500\%\).
To sum up: \(x\%\) and \(\frac{x}{100}\) are just two different ways of writing the same thing: as a percentage and as a ratio.
Back to the question..
Now, there are certain algebraic rules you can apply when there are "%" symbols in an equation, and if you know them (though they are no necessary for the GMAT), then you could directly write \(\sqrt{\frac{20\%}{40\%}}=\sqrt{\frac{20}{40}}\) but even if you are not aware of these rules, then you can apply the above method of converting the percentage into a ratio:
\(\sqrt{\frac{20\%}{40\%}}=\sqrt{\frac{(\frac{20}{100})}{(\frac{40}{100})}}=\sqrt{\frac{20}{100}*\frac{100}{40}}=\sqrt{\frac{20}{40}}\).
Next:
\(\sqrt{\frac{20}{40}}=\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}\).
Notice that the answer choices are as a percentage, so to get "%" symbol, to convert the ratio into a percentage, just multiply by 100%:
\(\frac{1}{\sqrt{2}}=\frac{1}{\sqrt{2}}*100\%=\frac{100}{\sqrt{2}}\%\).
Still no match with any of the options, so let's rationalize the fraction:
\(\frac{100}{\sqrt{2}}\%=(\frac{100}{\sqrt{2}})*(\frac{\sqrt{2}}{\sqrt{2}})\%=(50\sqrt{2})\%\).
Answer: D
