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If 4+ 1n = 55n, what is the value of n?

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If 4+ 1n = 55n, what is the value of n?

by Max@Math Revolution » Wed Jan 02, 2019 11:15 pm

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[Math Revolution GMAT math practice question]

If $$\sqrt{4+\frac{1}{n}}=5\sqrt{\frac{5}{n}}$$ , what is the value of n?

A. 31
B. 41
C. 51
D. 61
E. 71
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by fskilnik@GMATH » Thu Jan 03, 2019 2:58 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If $$\sqrt{4+\frac{1}{n}}=5\sqrt{\frac{5}{n}}$$ , what is the value of n?

A. 31
B. 41
C. 51
D. 61
E. 71
$$? = n$$
$$\sqrt {{{4 \cdot n} \over n} + {1 \over n}} = 5\,\,\sqrt {{5 \over n}} \,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{squaring}}} \,\,\,\,{{4n + 1} \over n} = {{25 \cdot 5} \over n}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,4n = 125 - 1\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = n = {{120 + 4} \over 4} = 31\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left( {\rm{A}} \right)$$

Important: whenever we "square an equation" (put it to an EVEN power), we may "CREATE" new roots.
That´s why we have to check, at the end, whether each POTENTIAL root is in fact a root of the original equation.
(From the alternative choices, this test is not needed, of course.)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Brent@GMATPrepNow » Thu Jan 03, 2019 11:09 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

If $$\sqrt{4+\frac{1}{n}}=5\sqrt{\frac{5}{n}}$$ , what is the value of n?

A. 31
B. 41
C. 51
D. 61
E. 71
GIVEN: √(4 + 1/n) = 5√(5/n)
Square both sides to get: [√(4 + 1/n)]² = [5√(5/n)]²
Simplify left side and rewrite right side: 4 + 1/n = [5√(5/n)][5√(5/n)]
Rewrite right side again: 4 + 1/n = (5)(5)√(5/n)√(5/n)
Simplify right side again: 4 + 1/n = 25(5/n)
Simplify right side: 4 + 1/n = 125/n
Multiply both sides by n to get: 4n + 1 = 125
Subtract 1 from both sides to get: 4n = 124
Solve: n = 124/4 =31

Answer: A

Cheers,
Brent
Brent Hanneson - Creator of GMATPrepNow.com
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by Max@Math Revolution » Sun Jan 06, 2019 5:08 pm
=>

$$\sqrt{4+\frac{1}{n}}=5\sqrt{\frac{5}{n}}$$
$$\sqrt{\frac{4n+1}{n}}=5\sqrt{\frac{5}{n}}$$
$$\frac{4n+1}{n}=25\left(\frac{5}{n}\right)$$
$$\frac{4n+1}{n}=\frac{125}{n}$$
$$4n+1=125$$
$$4n=124$$
$$n=31$$

Therefore, the answer is A.
Answer: A
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