This is a tricky problem that involves setting up a ratio.
Question: Last year the chess team won 7/9 of its matches. If the team won 16 of its first 32 matches, what was the least number of matches it could have played last year?
(A) 40
(B) 54
(C) 56
(D) 63
(E) 72
Solution:
The key to solving this is setting up the ratio correctly.
Let x = number of additional matches.
You need to add this to both the numerator and denominator, giving
(16+x)/(32+x).
Set this equal to the required ratio, 7/9.
The perform the middle school algebra as follows:
(16+x)/(32+x) = 7/9
9(16+x) = 7(32+x)
144+9x = 224+7x
2x = 80
So, x = 40.
Now, add 40 to the number of matches played (32)
The solution is 72.
Answer choice E.
Friday, August 7, 2009
Friday, July 31, 2009
Critical Reasoning
Demographers doing research for an international economics newsletter claim that the average per capita income in the country of Kuptala is substantially lower than that in the country of Bahlton. They also claim, however, that whereas poverty is relatively rare in Kuptala, over half the population of Bahlton lives in extreme poverty. At least one of the demographers’ claims must, therefore, be wrong.
The argument above is most vulnerable to which of the following criticisms?
(A) It rejects an empirical claim about the average per capita incomes in the two countries without making any attempt to discredit that claim by offering additional economic evidence.
(B) It treats the vague term “poverty” as though it had a precise and universally accepted meaning.
(C) It overlooks the possibility that the number of people in the two countries who live in poverty could be the same even though the percentages of the two populations that live in poverty differ markedly.
(D) It fails to show that wealth and poverty have the same social significance in Kuptala as in Bahlton
(E) It does not consider the possibility that incomes in Kuptala, unlike those in Bahlton, might all be very close to the country’s average per capita income.
You need to consider how the incomes are distributed in each country. In Kuptala, most are close to the mean, and few are living in deep poverty, and few are wealthy.
In Bahlton, the distribution is skewed because there are some extremely rich people, who cause the average to be much higher. There are also many living in poverty.
Once you can visualize these, the answer E should be clear.
The argument above is most vulnerable to which of the following criticisms?
(A) It rejects an empirical claim about the average per capita incomes in the two countries without making any attempt to discredit that claim by offering additional economic evidence.
(B) It treats the vague term “poverty” as though it had a precise and universally accepted meaning.
(C) It overlooks the possibility that the number of people in the two countries who live in poverty could be the same even though the percentages of the two populations that live in poverty differ markedly.
(D) It fails to show that wealth and poverty have the same social significance in Kuptala as in Bahlton
(E) It does not consider the possibility that incomes in Kuptala, unlike those in Bahlton, might all be very close to the country’s average per capita income.
You need to consider how the incomes are distributed in each country. In Kuptala, most are close to the mean, and few are living in deep poverty, and few are wealthy.
In Bahlton, the distribution is skewed because there are some extremely rich people, who cause the average to be much higher. There are also many living in poverty.
Once you can visualize these, the answer E should be clear.
Monday, June 8, 2009
Hard problem solving question from OG 12th edition
Problem Solving
OG 12th edition Q163
My students believe this is one of the hardest questions in the new edition of the Offical Guide.
This year Henry will save a certain amount of his income and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1+r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he has available to spend will be equal to half the amount that he spends this year?
(A) 1/(r+2)
(B) 1/(2r+2)
(C) 1/(3r+2)
(D) 1/(r+3)
(E) 1/(2r+3)
Students find it difficult to fathom where to start. Always focus on what you are being asked to solve. Here it is the ratio of savings to income this year. It is helpful to set up a table, inserting the given information and variables, letting i = income for this year, and s = amount saved this year. So, the amount spent this year must be i-s.
This Year Next Year
Spend i-s (1+r)s
Save s 0
Income i 0
The critical cell is Spend Next Year. We are given that for each s, Henry will have 1+r dollars available to spend (You can interpret r as the interest rate). So insert (1+r)s in the critical cell.
The last sentence of the question should be set up algebraically as:
(1+r)s = (i-s)/2
Recall that we need to solve for the ratio of saving to income, that is s/i.
Use your middle school algebra skills to solve for s/i.
2s(1+r) = i-s
2s + 2rs = i-s
3s +2rs = i
s(3+2r) = i
s/i = 1/(3+2r)
Many students will find completing this within two minutes challenging. But if you can do so, you will likely be on your way to a 98 + %ile score.
OG 12th edition Q163
My students believe this is one of the hardest questions in the new edition of the Offical Guide.
This year Henry will save a certain amount of his income and he will spend the rest. Next year Henry will have no income, but for each dollar that he saves this year, he will have 1+r dollars available to spend. In terms of r, what fraction of his income should Henry save this year so that next year the amount he has available to spend will be equal to half the amount that he spends this year?
(A) 1/(r+2)
(B) 1/(2r+2)
(C) 1/(3r+2)
(D) 1/(r+3)
(E) 1/(2r+3)
Students find it difficult to fathom where to start. Always focus on what you are being asked to solve. Here it is the ratio of savings to income this year. It is helpful to set up a table, inserting the given information and variables, letting i = income for this year, and s = amount saved this year. So, the amount spent this year must be i-s.
This Year Next Year
Spend i-s (1+r)s
Save s 0
Income i 0
The critical cell is Spend Next Year. We are given that for each s, Henry will have 1+r dollars available to spend (You can interpret r as the interest rate). So insert (1+r)s in the critical cell.
The last sentence of the question should be set up algebraically as:
(1+r)s = (i-s)/2
Recall that we need to solve for the ratio of saving to income, that is s/i.
Use your middle school algebra skills to solve for s/i.
2s(1+r) = i-s
2s + 2rs = i-s
3s +2rs = i
s(3+2r) = i
s/i = 1/(3+2r)
Many students will find completing this within two minutes challenging. But if you can do so, you will likely be on your way to a 98 + %ile score.
Wednesday, May 20, 2009
Sentence correction - tricky problems
QUESTION
According to medieval monks, the remains of King Arthur and Queen Guinevere were found at Glastonbury Abbey in AD 1191, and Arthur's coffin marked with the inscription (in Latin): "Here Lies Arthur, The Once and Future King."
(A) and Arthur's coffin marked with the inscription
(B) Arthur's coffin marked with the inscription
(C) and the inscription was marked on Arthur's coffin
(D) the inscription that was marked on the coffin of Arthur
(E) the coffin of Arthur had the inscription marked
Solution:
(A) is wrong. To be correct here, the passive voice would have to be used, i.e., “was marked”.
(B) is correct. The phrase acts a modifier.
(C ) is wrong. The word order is clumsy.
(D) is wrong because there is no verb in what should be a verbal phrase.
(D) is wrong. The usage is clumsy.
OG 11th Ed Q136
Joachim Raff and Giacomo Meyerbeer are examples of the kind of composer who receives popular acclaim while living, often goes into decline after death, and never regains popularity again.
(A) often goes into decline after death, and never regains popularity again
(B) whose reputation declines after death and never regains its status again
(C) but whose reputation declines after death and never regains its former status
(D) who declines in reputation after death and who never regained popularity again
(E) then has declined in reputation after death and never regained popularity
Solution:
(A) is wrong. As written it suggests that the composer goes into decline after death, but that is nonsensical. It could only be reputation that goes into decline. “again” is redundant”.
(B) is wrong. The parallel structure is erroneous. The word “whose” is incorrectly placed.
(C) is correct.
(D) is wrong. Verb tenses are confused – “regained” should be “regains”. “again” is redundant”.
(E) is wrong. Verb tenses are confused – “regained” should be “regains”.
According to medieval monks, the remains of King Arthur and Queen Guinevere were found at Glastonbury Abbey in AD 1191, and Arthur's coffin marked with the inscription (in Latin): "Here Lies Arthur, The Once and Future King."
(A) and Arthur's coffin marked with the inscription
(B) Arthur's coffin marked with the inscription
(C) and the inscription was marked on Arthur's coffin
(D) the inscription that was marked on the coffin of Arthur
(E) the coffin of Arthur had the inscription marked
Solution:
(A) is wrong. To be correct here, the passive voice would have to be used, i.e., “was marked”.
(B) is correct. The phrase acts a modifier.
(C ) is wrong. The word order is clumsy.
(D) is wrong because there is no verb in what should be a verbal phrase.
(D) is wrong. The usage is clumsy.
OG 11th Ed Q136
Joachim Raff and Giacomo Meyerbeer are examples of the kind of composer who receives popular acclaim while living, often goes into decline after death, and never regains popularity again.
(A) often goes into decline after death, and never regains popularity again
(B) whose reputation declines after death and never regains its status again
(C) but whose reputation declines after death and never regains its former status
(D) who declines in reputation after death and who never regained popularity again
(E) then has declined in reputation after death and never regained popularity
Solution:
(A) is wrong. As written it suggests that the composer goes into decline after death, but that is nonsensical. It could only be reputation that goes into decline. “again” is redundant”.
(B) is wrong. The parallel structure is erroneous. The word “whose” is incorrectly placed.
(C) is correct.
(D) is wrong. Verb tenses are confused – “regained” should be “regains”. “again” is redundant”.
(E) is wrong. Verb tenses are confused – “regained” should be “regains”.
Problem Solving -- Inequalities
According to many of my students, the solutions in the Official Guide (OG) are often unclear, so I will provide solutions with greater detail for certain problems in this blog.
OG 11th Ed Q114
If x is to be selected at random from set T, what is the probability that x/4 – 5 ≤ 0?
(1) T is a set of 8 integers.
(2) T is contained in the set of integers from 1 to 25, inclusive.
x/4 – 5 ≤ 0
Therefore x/4 ≤ 5.
Therefore x ≤ 20.
Consider (1). We have no information on what integers are in the set T, so it is insufficient. Eliminate A and D.
Consider (2). The set of integers from 1 to 25 is {1,2,3, …, 24, 25}. The members of set T are also in that set, but there are numerous possibilities. T could be {1,2} or it could be {19,20,21} etc. So we cannot fathom what the probability that x/4 – 5 ≤ 0 is. Insufficient. Eliminate B.
Consider (1) and (2) together. T could be for example {1,2,3,4,5,6,7,8} or {1,3,10,11, 15,16,20, 21} or {18,19,20,21,22,23,24,25} or any other set with 8 members. So there is insufficient info to find the probability requested.
The correct answer is E.
OG 11th Ed Q139
If x ≠ -y, is (x-y)/(x+y) > 1?
(1) x > 0
(2) y < 0
As I mentioned last week, a strong strategy is to pick numbers and look for a contradiction.
Consider (1). The only constraint is x>0. Try to pick numbers that demonstrate that the inequality can be either >1 or <1. x="5" y="3," x="5" y="-3,"> 1. Contradiction so (1) is insufficient. Eliminate A and D.
Consider (2). The only constraint is y<0.>1 or <1. x="-5" y="3,"> 1. When x=5 and y=-3, the inequality is (5-3)/(5+3) = ¼ < 1. Contradiction so (2) is insufficient. Eliminate B.
Consider (1) and (2). There are two constraints, x>0 and y<0. I’ll leave it to you to pick numbers and to find contradictions. Eliminate C.
The answer therefore is E.
OG 11th Ed Q114
If x is to be selected at random from set T, what is the probability that x/4 – 5 ≤ 0?
(1) T is a set of 8 integers.
(2) T is contained in the set of integers from 1 to 25, inclusive.
x/4 – 5 ≤ 0
Therefore x/4 ≤ 5.
Therefore x ≤ 20.
Consider (1). We have no information on what integers are in the set T, so it is insufficient. Eliminate A and D.
Consider (2). The set of integers from 1 to 25 is {1,2,3, …, 24, 25}. The members of set T are also in that set, but there are numerous possibilities. T could be {1,2} or it could be {19,20,21} etc. So we cannot fathom what the probability that x/4 – 5 ≤ 0 is. Insufficient. Eliminate B.
Consider (1) and (2) together. T could be for example {1,2,3,4,5,6,7,8} or {1,3,10,11, 15,16,20, 21} or {18,19,20,21,22,23,24,25} or any other set with 8 members. So there is insufficient info to find the probability requested.
The correct answer is E.
OG 11th Ed Q139
If x ≠ -y, is (x-y)/(x+y) > 1?
(1) x > 0
(2) y < 0
As I mentioned last week, a strong strategy is to pick numbers and look for a contradiction.
Consider (1). The only constraint is x>0. Try to pick numbers that demonstrate that the inequality can be either >1 or <1. x="5" y="3," x="5" y="-3,"> 1. Contradiction so (1) is insufficient. Eliminate A and D.
Consider (2). The only constraint is y<0.>1 or <1. x="-5" y="3,"> 1. When x=5 and y=-3, the inequality is (5-3)/(5+3) = ¼ < 1. Contradiction so (2) is insufficient. Eliminate B.
Consider (1) and (2). There are two constraints, x>0 and y<0. I’ll leave it to you to pick numbers and to find contradictions. Eliminate C.
The answer therefore is E.
Wednesday, May 13, 2009
3 Critical Reasoning Problems
1. Civil trials often involve great complexities that are beyond the capacities of jurors to understand. As a result, jurors' decisions in such trials are frequently incorrect. Justice would therefore be better served if the more complex trials were decided by judges rather than juries.
The argument above depends on which of the following assumptions?
(A) A majority of civil trials involve complexities that jurors are not capable of understanding.
(B) The judges who would decide complex civil trials would be better able to understand the complexities of those trials than jurors are.
(C) The judges who would preside over civil trials would disallow the most complex sorts of evidence from being introduced into those trials.
(D) Jurors' decisions are frequently incorrect even in those civil trials that do not involve great complexities.
(E) The sole reason in favor of having juries decide civil trials is the supposition that their decisions will almost always be correct.
SOLUTION:
(A) The given info includes the words “often” and “frequently”. The argument does NOT depend on “the majority”.
(B) is the correct solution.
(C ) is beyond the scope of the argument.
(D) is irrelevant to the argument, which is concerned only with complex cases.
(E) Supposition about juries’ decisions is beyond the scope of the question.
2. The only purpose for which a particular type of tape is needed is to hold certain surgical wounds closed for ten days-the maximum time such wounds need tape. Newtape is a new brand of this type of tape. Newtape's salespeople claim that Newtape will improve healing because Newtape adheres twice as long as the currently used tape does.
Which of the following statements, if true, would most seriously call into question the claim made by Newtape's salespeople?
(A) Most surgical wounds take about ten days to heal.
(B) Most surgical tape is purchased by hospitals and clinics rather than by individual surgeons.
(C) The currently used tape's adhesiveness is more than sufficient to hold wounds closed for ten days.
(D) Neither Newtape nor the currently used tape adheres well to skin that has not been cleaned.
(E) Newtape's adhesion to skin that has been coated with a special chemical preparation is only half as good as the currently used tape's adhesion to such coated skin.
SOLUTION:
(A) It is definitively stated that the maximum time wounds need tape is ten days, so it is irrelevant that most wounds take ten days to heal.
(B) Who is responsible for buying the tape is irrelevant to the claim.
(C ) is the solution
(D) Whether skin has been cleaned is irrelevant to the claim.
(E) The quality of the adhesion/stickiness is not relevant.
3. Demographers doing research for an international economics newsletter claim that the average per capita income in the country of Kuptala is substantially lower than that in the country of Bahlton. They also claim, however, that whereas poverty is relatively rare in Kuptala, over half the population of Bahlton lives in extreme poverty. At least one of the demographers' claims must, therefore, be wrong.
The argument above is most vulnerable to which of the following criticisms?
(A) It rejects an empirical claim about the average per capita incomes in the two countries without making any attempt to discredit that claim by offering additional economic evidence.
(B) It treats the vague term "poverty" as though it had a precise and universally accepted meaning.
(C) It overlooks the possibility that the number of people in the two countries who live in poverty could be the same even though the percentages of the two populations that live in poverty differ markedly.
(D) It fails to show that wealth and poverty have the same social significance in Kuptala as in Bahlton.
(E) It does not consider the possibility that incomes in Kuptala, unlike those in Bahlton, might all be very close to the country's average per capita income.
SOLUTION:
(A) Additional evidence would not be required if
(B) As long as the usage of “poverty” is consistently applied in the cases of Kuptala and Bahlton, there is no need to worry about its meaning being universally accepted.
(C) No, this criticism does not impact the argument because neither claim is about actual numbers.
(D) Social significance is irrelevant as a criticism.
(E) is the solution. Consider the distribution of incomes. In Kuptala, most could be near the mean, but in Bahlton, there are many below the mean and some far above the mean.
The argument above depends on which of the following assumptions?
(A) A majority of civil trials involve complexities that jurors are not capable of understanding.
(B) The judges who would decide complex civil trials would be better able to understand the complexities of those trials than jurors are.
(C) The judges who would preside over civil trials would disallow the most complex sorts of evidence from being introduced into those trials.
(D) Jurors' decisions are frequently incorrect even in those civil trials that do not involve great complexities.
(E) The sole reason in favor of having juries decide civil trials is the supposition that their decisions will almost always be correct.
SOLUTION:
(A) The given info includes the words “often” and “frequently”. The argument does NOT depend on “the majority”.
(B) is the correct solution.
(C ) is beyond the scope of the argument.
(D) is irrelevant to the argument, which is concerned only with complex cases.
(E) Supposition about juries’ decisions is beyond the scope of the question.
2. The only purpose for which a particular type of tape is needed is to hold certain surgical wounds closed for ten days-the maximum time such wounds need tape. Newtape is a new brand of this type of tape. Newtape's salespeople claim that Newtape will improve healing because Newtape adheres twice as long as the currently used tape does.
Which of the following statements, if true, would most seriously call into question the claim made by Newtape's salespeople?
(A) Most surgical wounds take about ten days to heal.
(B) Most surgical tape is purchased by hospitals and clinics rather than by individual surgeons.
(C) The currently used tape's adhesiveness is more than sufficient to hold wounds closed for ten days.
(D) Neither Newtape nor the currently used tape adheres well to skin that has not been cleaned.
(E) Newtape's adhesion to skin that has been coated with a special chemical preparation is only half as good as the currently used tape's adhesion to such coated skin.
SOLUTION:
(A) It is definitively stated that the maximum time wounds need tape is ten days, so it is irrelevant that most wounds take ten days to heal.
(B) Who is responsible for buying the tape is irrelevant to the claim.
(C ) is the solution
(D) Whether skin has been cleaned is irrelevant to the claim.
(E) The quality of the adhesion/stickiness is not relevant.
3. Demographers doing research for an international economics newsletter claim that the average per capita income in the country of Kuptala is substantially lower than that in the country of Bahlton. They also claim, however, that whereas poverty is relatively rare in Kuptala, over half the population of Bahlton lives in extreme poverty. At least one of the demographers' claims must, therefore, be wrong.
The argument above is most vulnerable to which of the following criticisms?
(A) It rejects an empirical claim about the average per capita incomes in the two countries without making any attempt to discredit that claim by offering additional economic evidence.
(B) It treats the vague term "poverty" as though it had a precise and universally accepted meaning.
(C) It overlooks the possibility that the number of people in the two countries who live in poverty could be the same even though the percentages of the two populations that live in poverty differ markedly.
(D) It fails to show that wealth and poverty have the same social significance in Kuptala as in Bahlton.
(E) It does not consider the possibility that incomes in Kuptala, unlike those in Bahlton, might all be very close to the country's average per capita income.
SOLUTION:
(A) Additional evidence would not be required if
(B) As long as the usage of “poverty” is consistently applied in the cases of Kuptala and Bahlton, there is no need to worry about its meaning being universally accepted.
(C) No, this criticism does not impact the argument because neither claim is about actual numbers.
(D) Social significance is irrelevant as a criticism.
(E) is the solution. Consider the distribution of incomes. In Kuptala, most could be near the mean, but in Bahlton, there are many below the mean and some far above the mean.
Monday, May 11, 2009
More data sufficiency - pick numbers carefully!
When you pick numbers, the goal is is often to find a contradiction, i.e., to show that different numbers lead to different answers. In data sufficiency, this can be an extremely effective strategy, as in the following question.
Question
If y doesn't equal 0 and y doesn’t equal -1, which is greater, x/y or x/(y+1)?
(1) x doesn’t equal 0
(2) x > y
Solution
The best approach is to pick numbers with the goal of showing there is insufficient info.
Don’t forget that you can pick negative integers and fractions.
We will have to be careful about the signs of x and y, because signs will determine which is greater.
Consider (1).
If x is 2 and y is 1, then x/y = 2, and x/(y+1) = 1. So x/y is greater.
But if x is -2 and y is 1, then x/y = -2 and x/(y+1) = -1. So x/(y+1) is greater.
There is a contradiction, so there’s insufficient info.
Eliminate A and D.
Consider (2)
Again we could have x=2 and y=1, so x/y is greater
We could not x=-2 and y=1.
However both x and y could be negative, as long as x>y.
If x= -1 and y= -2, then x/y = 1/2 , and x/(y+1) = 1, so x/(y+1) is greater.
Contradiction.
Insufficient info.
Eliminate B.
Consider (1) and (2) together.
We have already done so and found contradiction.
Not enough info.
Answer is E.
Here are two more data sufficiency problems.
Question
1. What fractional part of the total surface area of cube C is red?
(1) Each of 3 faces of C is exactly 1/2 red.
(2) Each of 3 faces of C is entirely white.
Solution
Consider (1).
We don’t know what color the other faces are. They could be all red or not red at all.
So we do not know what proportion of the total surface area is red.
Eliminate A and D.
Consider (2).
Again, we don’t know what color the non-white faces are.
Eliminate B.
Consider (1) and (2) together.
Now we infer that half of three sides are red, and no part of the other three sides are white.
The fractional part that is read is half of half, a quarter.
Answer is C.
Question
S is a set of integers such that
i) if a is in S, then -a is in S, and
ii) if each of a and b is in S, then ab is in S
is -4 in S?
(1) 1 is in S
(2) 2 is in S
Solution
Consider (1).
1 is in S, so -1 is also in S.
Not enough info to determine whether -4 is in S.
Eliminate A and D.
Consider (2).
2 is in S, so -2 is also in S.
2 and -2 are both members of S, so their product,-4, must be in S according to (ii).
Sufficient info.
Answer is B.
Question
If y doesn't equal 0 and y doesn’t equal -1, which is greater, x/y or x/(y+1)?
(1) x doesn’t equal 0
(2) x > y
Solution
The best approach is to pick numbers with the goal of showing there is insufficient info.
Don’t forget that you can pick negative integers and fractions.
We will have to be careful about the signs of x and y, because signs will determine which is greater.
Consider (1).
If x is 2 and y is 1, then x/y = 2, and x/(y+1) = 1. So x/y is greater.
But if x is -2 and y is 1, then x/y = -2 and x/(y+1) = -1. So x/(y+1) is greater.
There is a contradiction, so there’s insufficient info.
Eliminate A and D.
Consider (2)
Again we could have x=2 and y=1, so x/y is greater
We could not x=-2 and y=1.
However both x and y could be negative, as long as x>y.
If x= -1 and y= -2, then x/y = 1/2 , and x/(y+1) = 1, so x/(y+1) is greater.
Contradiction.
Insufficient info.
Eliminate B.
Consider (1) and (2) together.
We have already done so and found contradiction.
Not enough info.
Answer is E.
Here are two more data sufficiency problems.
Question
1. What fractional part of the total surface area of cube C is red?
(1) Each of 3 faces of C is exactly 1/2 red.
(2) Each of 3 faces of C is entirely white.
Solution
Consider (1).
We don’t know what color the other faces are. They could be all red or not red at all.
So we do not know what proportion of the total surface area is red.
Eliminate A and D.
Consider (2).
Again, we don’t know what color the non-white faces are.
Eliminate B.
Consider (1) and (2) together.
Now we infer that half of three sides are red, and no part of the other three sides are white.
The fractional part that is read is half of half, a quarter.
Answer is C.
Question
S is a set of integers such that
i) if a is in S, then -a is in S, and
ii) if each of a and b is in S, then ab is in S
is -4 in S?
(1) 1 is in S
(2) 2 is in S
Solution
Consider (1).
1 is in S, so -1 is also in S.
Not enough info to determine whether -4 is in S.
Eliminate A and D.
Consider (2).
2 is in S, so -2 is also in S.
2 and -2 are both members of S, so their product,-4, must be in S according to (ii).
Sufficient info.
Answer is B.
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