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In a decimal, a bar over a digit indicates that the digit repeats indefinitely. If a and b are integers, \(a/3 - b/4 = 0.41\overline{6}\) and \(a + b = 17\), what is the value of \(a\)?
(A) 5
(B) 7
(C) 8
(D) 9
(E) 10
(A) 5
(B) 7
(C) 8
(D) 9
(E) 10
27 Oct 2021, 12:48
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Bunuel
\((\frac{a}{3 }- \frac{b}{4} = 0.41\overline{6})\) and a+b=17, and we have to find the value of a.
a=17-b
Substitute in equation \((\frac{a}{3 }- \frac{b}{4} = 0.41\overline{6})\)
\((\frac{17-b}{3 }- \frac{b}{4} = 0.41\overline{6})\)
\((\frac{17}{3 }-\frac{b}{3}- \frac{b}{4} = 0.41\overline{6})\)
\(5.\overline{6}-\frac{7b}{12 } = 0.41\overline{6}\)
\(5.66\overline{6}-0.41\overline{6}=\frac{7b}{12 } \)
\(7b=12*5.25.............7b=63.....b=9\)
Thus, \(a=17-9=8\)
C
23 Feb 2022, 05:52
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Converting a mixed decimal into a fraction (Please refer to the GMATCLUB MATH BOOK for step by step explanantion):
\(0.41\overline{6}\) = \(416-41\) = \(\frac{375}{900}\) = \(\frac{5}{12}\) (when reduced to lowest fraction)
∴ \(\frac{a}{3}-\frac{b}{4}\) = \(\frac{5}{12}\)
\(\frac{4a-3b}{12}\) = \(\frac{5}{12}\)
\(4a-3b\) = \(5\)
\(a+b = 17\) (multiplying it by 3)
\(3a + 3b = 51\) --------- (1)
\(4a-3b = 5\) --------- (2)
Upon adding equations 1 and 2 we get:
\(7a = 56\)
\(a = 8\) (C)
\(0.41\overline{6}\) = \(416-41\) = \(\frac{375}{900}\) = \(\frac{5}{12}\) (when reduced to lowest fraction)
∴ \(\frac{a}{3}-\frac{b}{4}\) = \(\frac{5}{12}\)
\(\frac{4a-3b}{12}\) = \(\frac{5}{12}\)
\(4a-3b\) = \(5\)
\(a+b = 17\) (multiplying it by 3)
\(3a + 3b = 51\) --------- (1)
\(4a-3b = 5\) --------- (2)
Upon adding equations 1 and 2 we get:
\(7a = 56\)
\(a = 8\) (C)
