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05 Apr 2022, 01:30
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Students were given tokens to spend at a fair with various tents to visit. Each student was given 16 tokens. Some tents cost 3 tokens to enter, and some, 4 tokens. How many tents did Amelia visit?
(1) Amelia spent all of her tokens.
(2) Not all of the tents Amelia visited were the same token-price.
(1) Amelia spent all of her tokens.
(2) Not all of the tents Amelia visited were the same token-price.
05 Apr 2022, 20:22
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Students were given tokens to spend at a fair with various tents to visit. Each student was given 16 tokens. Some tents cost 3 tokens to enter, and some, 4 tokens. How many tents did Amelia visit?
Stat1: Amelia spent all of her tokens.
3x + 4y= 16, x = nos. of tents cost 3 tokens and y= nos. of tents cost 4 tokens.
We can have two solutions, x = 0 and y=4 or x =4, y= 1. Not sufficient
Stat2: Not all of the tents Amelia visited were the same token-price.
But, did she spend all tokens, we don't know it. Not sufficient
Combining both, x or y can't be zero. We have only option left as x = 4, y= 1. Sufficient
So, I think C.
Stat1: Amelia spent all of her tokens.
3x + 4y= 16, x = nos. of tents cost 3 tokens and y= nos. of tents cost 4 tokens.
We can have two solutions, x = 0 and y=4 or x =4, y= 1. Not sufficient
Stat2: Not all of the tents Amelia visited were the same token-price.
But, did she spend all tokens, we don't know it. Not sufficient
Combining both, x or y can't be zero. We have only option left as x = 4, y= 1. Sufficient
So, I think C.
11 Apr 2022, 11:50
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This is a question on Special linear equations, where, although there aren’t as many independent equations as the number of unknowns, the equation can still be solved for unique values of the unknowns, taking advantage of the additional constraints provided.
Step 1: Analyse Question Stem
In this question, ‘tokens’ and ‘tents’ are ‘countable’ items; hence, they need to be integral values, and non-negative values at that.
Now, that’s a very valuable piece of information, which should not be missed out on.
Some tents cost 3 tokens to enter; let the number of such tents be x.
Some other tents cost 4 tokens to enter; let the number of such tents be y.
We know that each student was given 16 tokens (and so was Amelia); therefore, for each student,
3x + 4y = 16.
This is where one needs to spend a little more time to analyse the equation above, to narrow down the possible values for the unknowns.
We have already deduced that x and y have to be non-negative integers (remember that Amelia or any other student for that matter, need not necessarily enter each type of tent; one of the types that they enter could be ZERO)
Because y is an integer, 4y will definitely be even. Additionally, 16 is also even. Therefore,
3x = 16 – 4y = even – even
3x = even. This can only happen when x is even.
Therefore, x can only take values like 0, 2, 4, 6,.. and so on.
Some more analysis will tell us that x cannot take values of 6 or above; if we do so, 3x will become more than 16 and then 4y will have to be negative, which is not possible.
Also, x = 0 and y = 4 satisfy the given equation; so does x = 4 and y = 1.
Now, all that is left is to use the statement data to figure out the exact case that matches all the data.
Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE
Statement 1: Amelia spent all of her tokens.
We only know that she spent all 16 of the tokens. However, this does not tell us about which one of the two cases we have to pick.
We are still left with both the cases.
Statement 1 alone is sufficient. Answer options A and D can be eliminated.
Statement 2: Not all of the tents Amelia visited were the same token-price.
We know that she visited both types of tents. However, we do not know if she exhausted all her tokens.
Answer option B is a trap here. If you end up using the data given in statement 1 with statement 2 (which violates the rules of Data sufficiency), you will have fallen for this trap.
Step 3: Analyse Statements by combining
From statement 1: Amelia spent all of her tokens.
From statement 2: Not all of the tents Amelia visited were the same token-price.
When we combine the information given in the statements, we can eliminate the case where x = 0. Therefore, x = 4 and y = 1.
Amelia visited 5 tents in total.
The combination of statements is sufficient. Answer option E can be eliminated.
The correct answer option is C.
Step 1: Analyse Question Stem
In this question, ‘tokens’ and ‘tents’ are ‘countable’ items; hence, they need to be integral values, and non-negative values at that.
Now, that’s a very valuable piece of information, which should not be missed out on.
Some tents cost 3 tokens to enter; let the number of such tents be x.
Some other tents cost 4 tokens to enter; let the number of such tents be y.
We know that each student was given 16 tokens (and so was Amelia); therefore, for each student,
3x + 4y = 16.
This is where one needs to spend a little more time to analyse the equation above, to narrow down the possible values for the unknowns.
We have already deduced that x and y have to be non-negative integers (remember that Amelia or any other student for that matter, need not necessarily enter each type of tent; one of the types that they enter could be ZERO)
Because y is an integer, 4y will definitely be even. Additionally, 16 is also even. Therefore,
3x = 16 – 4y = even – even
3x = even. This can only happen when x is even.
Therefore, x can only take values like 0, 2, 4, 6,.. and so on.
Some more analysis will tell us that x cannot take values of 6 or above; if we do so, 3x will become more than 16 and then 4y will have to be negative, which is not possible.
Also, x = 0 and y = 4 satisfy the given equation; so does x = 4 and y = 1.
Now, all that is left is to use the statement data to figure out the exact case that matches all the data.
Step 2: Analyse Statements Independently (And eliminate options) – AD / BCE
Statement 1: Amelia spent all of her tokens.
We only know that she spent all 16 of the tokens. However, this does not tell us about which one of the two cases we have to pick.
We are still left with both the cases.
Statement 1 alone is sufficient. Answer options A and D can be eliminated.
Statement 2: Not all of the tents Amelia visited were the same token-price.
We know that she visited both types of tents. However, we do not know if she exhausted all her tokens.
Answer option B is a trap here. If you end up using the data given in statement 1 with statement 2 (which violates the rules of Data sufficiency), you will have fallen for this trap.
Step 3: Analyse Statements by combining
From statement 1: Amelia spent all of her tokens.
From statement 2: Not all of the tents Amelia visited were the same token-price.
When we combine the information given in the statements, we can eliminate the case where x = 0. Therefore, x = 4 and y = 1.
Amelia visited 5 tents in total.
The combination of statements is sufficient. Answer option E can be eliminated.
The correct answer option is C.
