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Albert spent $100 in total on four different types of

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Albert spent $100 in total on four different types of

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Source: Veritas Prep

Albert spent $100 in total on four different types of scratch tickets, and he did not spend more than $40 on any one type of scratch ticket. How much did he spend on the scratch ticket on which he spent the least?

1) Albert spent twice as much on C-type tickets as he did on G-type tickets.
2) Albert spent at least $20 on each type of ticket.

The OA is C.

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BTGmoderatorLU wrote:
Source: Veritas Prep

Albert spent $100 in total on four different types of scratch tickets, and he did not spend more than $40 on any one type of scratch ticket. How much did he spend on the scratch ticket on which he spent the least?

1) Albert spent twice as much on C-type tickets as he did on G-type tickets.
2) Albert spent at least $20 on each type of ticket.
All numbers presented below are in dollars.
$$B,C,D,G\,\, \ge 0\,\,\,(\$ \,\,{\rm{spent}}\,\,{\rm{in}}\,\,{\rm{each}}\,\,{\rm{type}})$$
$$B + C + D + G = 100$$
$$B,C,D,G\,\,\, \le \,\,40$$
$$? = \min \left\{ {B,C,D,G} \right\}$$
$$\left( 1 \right)\,C = 2G\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {10,40,30,20} \right)\,\,\, \Rightarrow \,\,\,? = 10 \hfill \cr
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {5,40,35,20} \right)\,\,\, \Rightarrow \,\,\,? = 5 \hfill \cr} \right.$$
$$\left( 2 \right)\,\,B,C,D,G\,\,\, \ge \,\,20\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {20,25,25,30} \right)\,\,\, \Rightarrow \,\,\,? = 20 \hfill \cr
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {21,22,23,34} \right)\,\,\, \Rightarrow \,\,\,? = 21 \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\left\{ \matrix{
G \ge 20\,\,\, \Rightarrow \,\,\,C \ge 40 \hfill \cr
B \ge 20 \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,D \le 100 - \left( {20 + 20 + 40} \right) = 20$$
$${\rm{Hence}}:\,\,\left( {B,C,D,G} \right) = \left( {20,40,20,20} \right)\,\,\,\,\,\, \Rightarrow \,\,\,? = 20$$

The correct answer is therefore (C).


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

Regards,
Fabio.

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Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or https://GMATH.com.br (Portuguese version)
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BTGmoderatorLU wrote:
Source: Veritas Prep

Albert spent $100 in total on four different types of scratch tickets, and he did not spend more than $40 on any one type of scratch ticket. How much did he spend on the scratch ticket on which he spent the least?

1) Albert spent twice as much on C-type tickets as he did on G-type tickets.
2) Albert spent at least $20 on each type of ticket.
We need to determine the least amount Albert spent on one type of scratch tickets, given that he spent a total of $100 on 4 different types of tickets, and he spent no more than $40 on any one type of ticket.

Statement One Alone:

Albert spent twice as much on C-type tickets as he did on G-type tickets.

If he spent $40 on the C-type tickets, then he spent $20 on the G-type tickets. He could have spent $20 and $20 on the other two types of tickets OR $30 and $10 on the other two types of tickets. In the former scenario, the least he spent on any type of ticket is $20. In the latter scenario, the least he spent is $10. Statement one alone is not sufficient.

Statement Two Alone:

Albert spent at least $20 on each type of ticket.

If he spent at least $20 on each type of ticket and we are given that he spent at most $40 on any one type of ticket. It is possible that he spent $40, $20, $20 and $20 on the 4 types of tickets OR $25 on each of the 4 types of tickets. In the former scenario, the least he spent on any type of ticket is $20. In the latter scenario, the least he spent is $25. Statement two alone is not sufficient.

Statements One and Two Together:

With the two statements together we see that he must have spent $40 on the C-type tickets, $20 on the G-type tickets and also $20 on each of the other 2 types of tickets. Therefore, the least he spent on any type of ticket is $20.

Answer: C

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Scott Woodbury-Stewart Founder and CEO

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