Teachers and students at a school are solving problems. There are twic

Question

Teachers and students at a school are solving problems. There are twice as many students as teachers, and each student solves 3 more problems than each teacher. If teachers solve 24 of the 90 total problems solved, how many problems does each teacher solve?

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

(A) 3
(B) 4
(C) 5
(D) 6
(E) 8

GMATinsight · Answer

Let, Students = 2x
then, Teacher = (1/2)*2x = x

Problem solved by each teacher = a = ?  
Problem solved by each Students = (a+3)

Total Problems solved by teacher = a*x
Total Problems solved by Students = (a+3)*2x

a*x = 24
and 
a*x + (a+3)*2x = 90
I.e. ax + 2ax +6x = 90
I.e. 3ax+6x = 90
I.e. 6x = 90 - 3*24 = 18
I.e. x = 3

ax = 24
so a = 24/3 = 8

Answer: Option E

Archit3110 · Answer

let teacher = x and students = 2x
problems solved by teachers = y and students = (y+3)
given x*y = 24
and we are given that
2x*(y+3)+ t*y = 90
2*xy+6x+x*y = 90
48+6x+24 = 90
x= 3 teachers and students = 6
we know 3*y = 24 ; y = 8
IMO E

Kinshook · Answer

Given: 
1. Teachers and students at a school are solving problems. 
2. There are twice as many students as teachers, and each student solves 3 more problems than each teacher.

Asked: If teachers solve 24 of the 90 total problems solved, how many problems does each teacher solve?

Let a teacher solve x problems
Each student solves (x+3) problems

teachers solve 24 problems 
Number of teachers = 24/x 
Number of students = 48/x

Number of problems solved by students = 90- 24 = 66 problems = (48/x)*(x+3) = 48 (1 + 3/x)
1 + 3/x = 66/48 = 33/24 = 1 + 9/24 = 1 + 3/8
x = 8

IMO E

Bambi2021 · Answer

This task is flawed and involves a contradiction.

You cant say that there are 3 teachers that solve 8 problems each, because then there must be twice as many students solving three times as many problems each, just as stated. This amounts to a total of 3*8 + 6*24 = 148 problems.

A_Nishith · Answer

Let, Students = 2x
then, Teacher = x

Problem solved by each teacher = a = ?
Problem solved by each Students = (a+3)

Total Problems solved by teacher = a*x
Total Problems solved by Students = (a+3)*2x

a*x = 24
and
a*x + (a+3)*2x = 90
I.e. ax + 2ax +6x = 90
I.e. 3ax+6x = 90
I.e. 6x = 90 - 3*24 = 18
I.e. x = 3

ax = 24
so a = 24/3 = 8

Answer: Option E

NEYR0N · Answer

a) S = 2T
b) 66 / S = p + 3 
c) 24 / T = p 
System of 3 equations. We are looking for T so :

b) 66/ 2T = p+3 -> 33 = pT + 3T
c) 24 = pT

hence, 9 = 3T - - T = 3. , but we want p, so from c) 24/3 = 8 problems solved by each Teacher

- for me to come back to.

finisher009 · Answer

This method uses the answer choices to work backward, avoiding algebra entirely.

1. Identify the target: The question asks for the number of problems each teacher solves (Pt). The total problems students solved is `90 - 24 = 66`. Our goal is to find which answer choice results in students solving exactly 66 problems.

2. Use logic to eliminate answers: The number of teachers (T) must be a whole number. We know `T = 24 / Pt`. This means Pt must be a factor of 24.
• (A) 3 is a factor.
• (B) 4 is a factor.
• (C) 5 is NOT a factor. Eliminate (C).
• (D) 6 is a factor.
• (E) 8 is a factor.

3. Test a remaining choice (e.g., D):
• Assume each teacher solved 6 problems (Pt = 6).
• Number of teachers = 24 / 6 = 4.
• Number of students = 2 * 4 = 8.
• Problems per student = 6 + 3 = 9.
• Total student problems = 8 students * 9 problems = 72.
• This is too high. We need 66. So (D) is wrong.

4. Test the final choice (E):
• Assume each teacher solved 8 problems (Pt = 8).
• Number of teachers = 24 / 8 = 3.
• Number of students = 2 * 3 = 6.
• Problems per student = 8 + 3 = 11.
• Total student problems = 6 students * 11 problems = 66.
• This is a match. The correct answer is (E).

WhitEngagePrep · Answer

I absolutely agree that Working Backwards from the answer choices is a great method! The clues for that are (a) the clean answer choices and (b) the fact that the question asks for a single, plug-in-able value.

However, I take a bit more of a formulaic approach to all WB problems as follows:

First, copy the answer choices on your page and label what they represent:

Next - ask yourself, "based on the story, what other pieces of information do I know if I know the number of questions that each teacher answered?" In this case, you know the number of questions each student answered (since the students each answered 3 more than each teacher) and you know how many teachers there are (since the teachers answered 24 total questions).

You can eliminate any choices that don't work (like here, answer choice C doesn't work because if teachers write 5 questions each they can't have written 24 all together - it doesn't divide evenly).

Now, what else do you know that you haven't applied? If teachers wrote 24 of the 90, then students wrote the other 66. Use that to compute the number of students the same way you computed the number of teachers.

Again, you can eliminate any choices that don't work (like here, answers B and D don't work because if students write 7 or 9 questions, they can't have written 66 all together - it doesn't divide evenly).

That leaves only choices A and E. So what information have you still not applied? The fact that there are twice as many students as teachers. That is only true in choice E, so that must be the correct answer!

Hope this helps!

Whit

Show Spoiler

Attachment:

GMAT-Club-Forum-nlc5xmcp.png [ 8.65 KiB | Viewed 1038 times ]

Attachment:

GMAT-Club-Forum-bj8wvue2.png [ 18.65 KiB | Viewed 1048 times ]

Attachment:

GMAT-Club-Forum-i1vgxirl.png [ 28.04 KiB | Viewed 1047 times ]

Whit GMAT-Club-Forum-nlc5xmcp.png GMAT-Club-Forum-bj8wvue2.png GMAT-Club-Forum-i1vgxirl.png

Teachers and students at a school are solving problems. There are twic

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

Teachers and students at a school are solving problems. There are twic

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards