Problem Solving

e-GMAT

If the ratio of the present age of Anna and Paula is 1:2, what could be the ratio of their respective ages 8 years ago?

A. 3:8
B. 4:7
C. 3:5
D. 2:3
E. 4:5

OA A.

AAPL wrote:e-GMAT

If the ratio of the present age of Anna and Paula is 1:2, what could be the ratio of their respective ages 8 years ago?

A. 3:8
B. 4:7
C. 3:5
D. 2:3
E. 4:5

OA A.

With a little thinking, no calculations are necessary with this question.

Anna is currently half of Paula's age.

In subtracting a fixed number, 8, from each, it should be clear that the percentage reduction in Anna's age is greater than that for Paula.

So the ratio 8 years ago should be less than the ratio now. Conveniently, only one of the choices [spoiler]A 3/8[/spoiler] is less than 1/2

AAPL wrote:e-GMAT
If the ratio of the present age of Anna and Paula is 1:2, what could be the ratio of their respective ages 8 years ago?
A. 3:8
B. 4:7
C. 3:5
D. 2:3
E. 4:5

I do believe regor60 ´s solution is as good as it gets!

The alternate approach I offer (below) is for a "fictitious scenario" in which at least two alternative choices were (a priori) possible!
(To be honest: I would not have regor60´s idea... therefore my solution below - less smart, less efficient - would be the way I would proceed...that´s life!)

In "Which of the following..." questions we (usually) "test" each choice offered, but before that, let´s follow our method: explicit the FOCUS and relate it to DATA:
\[\begin{array}{*{20}{c}}
{A = k} \\
{P = 2k}
\end{array}\,\,\,\,\,\,\,\left( {k > 0} \right)\]
\[?\,\,\,\,\,:\,\,\,\,\,\,\frac{{k - 8}}{{2k - 8}}\,\,\, = \,\,\,\underline {{\text{altern}}{\text{.}}\,\,{\text{choice}}} \,\,\,,\,\,\,{\text{which?}}\]
\[\left( A \right)\,\,\,\,\frac{3}{8}\,\,\,\mathop = \limits^? \,\,\,\frac{{k - 8}}{{2k - 8}}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,3\left( {2k - 8} \right)\,\,\mathop = \limits^? \,\,8\left( {k - 8} \right)\,\,\,\, \Leftrightarrow \,\,\, \ldots \,\,\, \Leftrightarrow \,\,k\,\mathop = \limits^? 20\]
\[k = 20 > 0\,\,{\text{viable}}\,\,\,\,\,\,\left( {A = 20{\text{y}}\,\,\,,\,\,\,\,P = 40{\text{y}}} \right)\,\,\,\,\, \Rightarrow \,\,\,{\text{we}}\,\,{\text{are}}\,\,{\text{done!}}\]

(Other alternative choices must get nonpositive values for k, but you don´t need to check that: it is the examiner´s burden to offer only one correct/viable alternative choice among all presented!)

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

Regards,
fskilnik.