Problem Solving

Anna wants to distribute chocolates among her four children in the ratio 1/2 : 1/5 : 1/6 : 1/12. How many minimum chocolates should she buy, so that she can distribute the chocolates in the given ratio?

A. 30
B. 45
C. 57
D. 90
E. 120

The OA is C.

Source: e-GMAT

swerve wrote:Anna wants to distribute chocolates among her four children in the ratio 1/2 : 1/5 : 1/6 : 1/12. How many minimum chocolates should she buy, so that she can distribute the chocolates in the given ratio?

A. 30
B. 45
C. 57
D. 90
E. 120

Let the total number of chocolates = the LCM of the four denominators 2, 5, 6 and 12 = 60.
Number of chocolates given to the four children:
(1/2)(60) = 30
(1/5)(60) = 12
(1/6)(60) = 10
(1/12)(60) = 5
Total number of chocolates = 30+12+10+5 = 57.

The correct answer is C.

swerve wrote:Anna wants to distribute chocolates among her four children in the ratio 1/2 : 1/5 : 1/6 : 1/12. How many minimum chocolates should she buy, so that she can distribute the chocolates in the given ratio?

A. 30
B. 45
C. 57
D. 90
E. 120
Source: e-GMAT

\[? = \min \left( {{\text{Total}}} \right)\]
\[\frac{1}{2}\,\,:\,\,\frac{1}{5}\,\,:\,\,\frac{1}{6}\,\,:\,\,\frac{1}{{12}}\,\,\,\,\mathop \Leftrightarrow \limits_{:\,\,60}^{ \cdot \,\,60} \,\,\,\,30:12:10:5\]
\[\left\{ \begin{gathered}
{\text{Child}}\,1 = 30k \hfill \\
{\text{Child}}\,2 = 12k \hfill \\
{\text{Child}}\,3 = 10k \hfill \\
{\text{Child}}\,4 = 5k \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\left( {k > 0} \right)\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,Total = 57k\,\,\,\,\,\,\mathop \Rightarrow \limits^{k\,\,\operatorname{int} \,\,\left( * \right)} \,\,\,\,\,\,? = \min \,\,\left( {{\text{Total}}} \right)\,\,\, = \,\,57 \cdot 1 = 57\]
$$\left( * \right)\,\,\,\,\left\{ \matrix{
5k\,\, = {\mathop{\rm int}} \hfill \cr
2k = 12k - 10k\,\, = \,\,{\mathop{\rm int}} - {\mathop{\rm int}} = {\mathop{\rm int}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,\,\,k = 5k - 2 \cdot \left( {2k} \right) = {\mathop{\rm int}} - 2 \cdot {\mathop{\rm int}} = {\mathop{\rm int}} $$


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

Regards,
Fabio.