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07 May 2026, 08:43
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8. Word Problems
8.1. A training coach conducted two types of practice sessions on one day: advanced sessions and standard sessions. If the total practice time for the sessions was 210 minutes that day, how many standard sessions were conducted?
(1) Each advanced session lasted 49 minutes, and each standard session lasted 28 minutes.
(2) There were 6 sessions in total.
SOLUTION:
A training coach conducted two types of practice sessions on one day: advanced sessions and standard sessions. If the total practice time for the sessions was 210 minutes that day, how many standard sessions were conducted?
This is a classic C-trap question. In C-trap questions, the two statements together clearly appear sufficient, but in reality, one statement alone may already be enough. Whenever that happens, we have to proceed very carefully. In this question, once the two statements are combined, it is obvious that we have all the information there could be about the problem, so the combination is clearly sufficient. The real issue is whether either statement, or both, is already sufficient on its own.
Let the number of advanced sessions be a and the number of standard sessions be s. The question is: what is the value of s?
(1) Each advanced session lasted 49 minutes, and each standard session lasted 28 minutes.
Because the total practice time was 210 minutes, we get:
49a + 28s = 210
Reducing by 7 gives:
7a + 4s = 30
In general, a linear equation of the form ax + by = c has infinitely many solutions. But here a and s represent numbers of sessions, so they must be non-negative integers. That means we have an equation whose solutions must be integers only. In such cases, we must check directly whether there is just one integer combination that satisfies the equation.
It is not hard to test whether 7a + 4s = 30 has one solution or more than one. Rearranging gives:
4s = 30 - 7a
So 30 minus a multiple of 7 must be divisible by 4:
• 30 is not divisible by 4
• 23 is not
• 16 is divisible by 4 (a = 2, s = 4)
• 9 is not
• 2 is not
So there is only one pair of values of a and s that satisfies 7a + 4s = 30, namely a = 2 and s = 4. Sufficient.
(2) There were 6 sessions in total.
This gives:
a + s = 6
As above, this is also an equation whose solutions must be integers only. But unlike statement (1), this equation has several integer solutions for a and s. Therefore, it is clearly not sufficient to determine a single value of s.
Answer: A.
Takeaway
In Data Sufficiency, you should be alert to C-trap questions. These are questions in which the two statements together clearly seem sufficient, but in fact, one statement alone may already be enough. When that happens, we should proceed carefully and not fall for the C-trap.
Also, for a linear equation in two variables, we should not automatically conclude that it has infinitely many solutions. We must pay attention to the constraints, especially the domain of the variables, because those constraints can sharply reduce the number of valid solutions. In a question like this, the variables represent counts, so they must be non-negative integers. Because of that restriction, a linear equation that would otherwise have infinitely many solutions can end up having only one valid pair of values.
What This Question Tests
This question tests word problems, specifically a special type of Data Sufficiency word problem known as a C-trap question, as well as linear equations with integer-restricted solutions. It also tests whether you notice that the variables represent counts, so the non-negative integer restriction is essential to determining whether a statement is sufficient.
8.1. A training coach conducted two types of practice sessions on one day: advanced sessions and standard sessions. If the total practice time for the sessions was 210 minutes that day, how many standard sessions were conducted?
(1) Each advanced session lasted 49 minutes, and each standard session lasted 28 minutes.
(2) There were 6 sessions in total.
SOLUTION:
A training coach conducted two types of practice sessions on one day: advanced sessions and standard sessions. If the total practice time for the sessions was 210 minutes that day, how many standard sessions were conducted?
This is a classic C-trap question. In C-trap questions, the two statements together clearly appear sufficient, but in reality, one statement alone may already be enough. Whenever that happens, we have to proceed very carefully. In this question, once the two statements are combined, it is obvious that we have all the information there could be about the problem, so the combination is clearly sufficient. The real issue is whether either statement, or both, is already sufficient on its own.
Let the number of advanced sessions be a and the number of standard sessions be s. The question is: what is the value of s?
(1) Each advanced session lasted 49 minutes, and each standard session lasted 28 minutes.
Because the total practice time was 210 minutes, we get:
49a + 28s = 210
Reducing by 7 gives:
7a + 4s = 30
In general, a linear equation of the form ax + by = c has infinitely many solutions. But here a and s represent numbers of sessions, so they must be non-negative integers. That means we have an equation whose solutions must be integers only. In such cases, we must check directly whether there is just one integer combination that satisfies the equation.
It is not hard to test whether 7a + 4s = 30 has one solution or more than one. Rearranging gives:
4s = 30 - 7a
So 30 minus a multiple of 7 must be divisible by 4:
• 30 is not divisible by 4
• 23 is not
• 16 is divisible by 4 (a = 2, s = 4)
• 9 is not
• 2 is not
So there is only one pair of values of a and s that satisfies 7a + 4s = 30, namely a = 2 and s = 4. Sufficient.
(2) There were 6 sessions in total.
This gives:
a + s = 6
As above, this is also an equation whose solutions must be integers only. But unlike statement (1), this equation has several integer solutions for a and s. Therefore, it is clearly not sufficient to determine a single value of s.
Answer: A.
Takeaway
In Data Sufficiency, you should be alert to C-trap questions. These are questions in which the two statements together clearly seem sufficient, but in fact, one statement alone may already be enough. When that happens, we should proceed carefully and not fall for the C-trap.
Also, for a linear equation in two variables, we should not automatically conclude that it has infinitely many solutions. We must pay attention to the constraints, especially the domain of the variables, because those constraints can sharply reduce the number of valid solutions. In a question like this, the variables represent counts, so they must be non-negative integers. Because of that restriction, a linear equation that would otherwise have infinitely many solutions can end up having only one valid pair of values.
What This Question Tests
This question tests word problems, specifically a special type of Data Sufficiency word problem known as a C-trap question, as well as linear equations with integer-restricted solutions. It also tests whether you notice that the variables represent counts, so the non-negative integer restriction is essential to determining whether a statement is sufficient.
07 May 2026, 08:43
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8. Word Problems
8.2. During a long-jump drill, Elena completed three jumps. If exactly two of the jumps were the same length and the third jump had a different length, was Elena’s second jump longer than her third jump?
(1) Elena’s first jump was longer than her third jump.
(2) Elena’s first jump was longer than her second jump.
SOLUTION:
During a long-jump drill, Elena completed three jumps. If exactly two of the jumps were the same length and the third jump had a different length, was Elena’s second jump longer than her third jump?
Let the lengths of Elena’s first, second, and third jumps be x, y, and z, respectively.
We are told that exactly two of x, y, and z are equal, and the third is different.
Question: Is y > z?
(1) Elena’s first jump was longer than her third jump.
So x > z.
Now consider the possible equal pairs.
If x = y, then y > z. YES.
If y = z, then y = z, so y is not greater than z. NO.
Since the two possible cases give different answers, statement (1) alone is not sufficient.
(2) Elena’s first jump was longer than her second jump.
So x > y.
Now consider the possible equal pairs.
If x = z, then z > y, so y is not greater than z. NO.
If y = z, then y = z, so y is not greater than z. NO.
These are the only possible cases, and both give the same answer - NO. Sufficient.
Answer: B.
Takeaway
In a yes-no Data Sufficiency question, a definite no answer is just as sufficient as a definite yes answer. If a statement allows you to conclude with certainty that the answer to the question is no, then that statement is sufficient.
What This Question Tests
This question tests a yes-no Data Sufficiency setup involving case analysis. It tests whether you can translate the question precisely into a mathematical comparison, test all valid cases under the given restriction, and determine whether every valid case leads to the same answer. It also tests whether you recognize that a definite no answer is sufficient in Data Sufficiency.
8.2. During a long-jump drill, Elena completed three jumps. If exactly two of the jumps were the same length and the third jump had a different length, was Elena’s second jump longer than her third jump?
(1) Elena’s first jump was longer than her third jump.
(2) Elena’s first jump was longer than her second jump.
SOLUTION:
During a long-jump drill, Elena completed three jumps. If exactly two of the jumps were the same length and the third jump had a different length, was Elena’s second jump longer than her third jump?
Let the lengths of Elena’s first, second, and third jumps be x, y, and z, respectively.
We are told that exactly two of x, y, and z are equal, and the third is different.
Question: Is y > z?
(1) Elena’s first jump was longer than her third jump.
So x > z.
Now consider the possible equal pairs.
If x = y, then y > z. YES.
If y = z, then y = z, so y is not greater than z. NO.
Since the two possible cases give different answers, statement (1) alone is not sufficient.
(2) Elena’s first jump was longer than her second jump.
So x > y.
Now consider the possible equal pairs.
If x = z, then z > y, so y is not greater than z. NO.
If y = z, then y = z, so y is not greater than z. NO.
These are the only possible cases, and both give the same answer - NO. Sufficient.
Answer: B.
Takeaway
In a yes-no Data Sufficiency question, a definite no answer is just as sufficient as a definite yes answer. If a statement allows you to conclude with certainty that the answer to the question is no, then that statement is sufficient.
What This Question Tests
This question tests a yes-no Data Sufficiency setup involving case analysis. It tests whether you can translate the question precisely into a mathematical comparison, test all valid cases under the given restriction, and determine whether every valid case leads to the same answer. It also tests whether you recognize that a definite no answer is sufficient in Data Sufficiency.
14 May 2026, 07:31
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BATCH #7 (10 QUESTIONS)
9. Work and Rate Problems
9.1. Two conveyor belts, Belt R and Belt S, working together at their constant rates, can load a shipment of boxes onto a truck in 1/4 hour. How many minutes would Belt S, working alone at its constant rate, take to load the entire shipment?
(1) The shipment contains 720 boxes.
(2) Belt R loads 32 boxes per minute.
SOLUTION:
Two conveyor belts, Belt R and Belt S, working together at their constant rates, can load a shipment of boxes onto a truck in 1/4 hour. How many minutes would Belt S, working alone at its constant rate, take to load the entire shipment?
To solve this problem, we use the idea rate * time = work. Since the two belts together complete the shipment in 1/4 hour, that is 15 minutes, we can write the relationship as
(r + s) * 15 = t
where r and s are the loading rates of Belts R and S, in boxes per minute, and t is the total number of boxes in the shipment.
The question asks for the time Belt S alone would need to load the entire shipment.
(1) The shipment contains 720 boxes.
This gives t = 720, so
(r + s) * 15 = 720
From this, we get
r + s = 48
However, knowing only the sum of the two rates is not enough to determine the value of s.
Not sufficient.
(2) Belt R loads 32 boxes per minute.
This gives r = 32, so the equation becomes
(32 + s) * 15 = t
With two unknowns, s and t, and only one equation, we still do not have enough information. For example, if t = 900, then s = 28. But if t = 720, then s = 16.
Not sufficient.
(1) + (2) From the two statements, we know that r = 32 and r + s = 48. So s = 16. Since Belt S loads 16 boxes per minute and the shipment contains 720 boxes, the time Belt S alone would need is 720/16 = 45 minutes to load the shipment. Sufficient.
Answer: C.
Takeaway
In work-rate questions, it is often helpful to translate the situation into the basic relationship rate * time = work.
What This Question Tests
This question tests work and rate reasoning, algebraic translation, and handling combined rates versus individual rates. It also tests whether you can distinguish between finding a rate and finding the time needed to complete the whole job.
9. Work and Rate Problems
9.1. Two conveyor belts, Belt R and Belt S, working together at their constant rates, can load a shipment of boxes onto a truck in 1/4 hour. How many minutes would Belt S, working alone at its constant rate, take to load the entire shipment?
(1) The shipment contains 720 boxes.
(2) Belt R loads 32 boxes per minute.
SOLUTION:
Two conveyor belts, Belt R and Belt S, working together at their constant rates, can load a shipment of boxes onto a truck in 1/4 hour. How many minutes would Belt S, working alone at its constant rate, take to load the entire shipment?
To solve this problem, we use the idea rate * time = work. Since the two belts together complete the shipment in 1/4 hour, that is 15 minutes, we can write the relationship as
(r + s) * 15 = t
where r and s are the loading rates of Belts R and S, in boxes per minute, and t is the total number of boxes in the shipment.
The question asks for the time Belt S alone would need to load the entire shipment.
(1) The shipment contains 720 boxes.
This gives t = 720, so
(r + s) * 15 = 720
From this, we get
r + s = 48
However, knowing only the sum of the two rates is not enough to determine the value of s.
Not sufficient.
(2) Belt R loads 32 boxes per minute.
This gives r = 32, so the equation becomes
(32 + s) * 15 = t
With two unknowns, s and t, and only one equation, we still do not have enough information. For example, if t = 900, then s = 28. But if t = 720, then s = 16.
Not sufficient.
(1) + (2) From the two statements, we know that r = 32 and r + s = 48. So s = 16. Since Belt S loads 16 boxes per minute and the shipment contains 720 boxes, the time Belt S alone would need is 720/16 = 45 minutes to load the shipment. Sufficient.
Answer: C.
Takeaway
In work-rate questions, it is often helpful to translate the situation into the basic relationship rate * time = work.
What This Question Tests
This question tests work and rate reasoning, algebraic translation, and handling combined rates versus individual rates. It also tests whether you can distinguish between finding a rate and finding the time needed to complete the whole job.
