English exam and Math exam were conducted separately for a class of 120 students. The number of students who did not appear for the English exam is twice the number of students who did not appear for the Math exam. The number of students who passed the Math exam is twice the number of students who appeared but failed the English exam. If the number of students who passed the English exam is twice the number of students who appeared but failed the Math exam, then the number of students who appeared but failed the English exam is ________
Show Hint
Correct Answer: 40
Solution and Explanation
This problem involves interpreting logical statements and setting up a system of linear equations. It's a word problem that can be solved using algebraic manipulation. It doesn't require Venn diagrams as there's no information about students taking both exams.
Step 2: Key Formula or Approach:
Define variables for each category of students mentioned. Translate the given sentences into equations and solve the system.
Step 3: Detailed Explanation:
Let the total number of students be \(T = 120\).
Let \(N_M\) and \(N_E\) be the number of students who did not appear for Math and English exams, respectively. Given: \(N_E = 2 N_M\).
Let \(A_M\) and \(A_E\) be the number of students who appeared for Math and English exams. \(A_M = T - N_M = 120 - N_M\). \(A_E = T - N_E = 120 - 2N_M\).
For those who appeared, let \(P_M, F_M\) be the number who passed and failed Math, and \(P_E, F_E\) be the number who passed and failed English. So, \(A_M = P_M + F_M\) and \(A_E = P_E + F_E\).
We are given two more conditions: 1. The number of students who passed the Math exam is twice the number of students who appeared but failed the English exam: \(P_M = 2F_E\). 2. The number of students who passed the English exam is twice the number of students who appeared but failed the Math exam: \(P_E = 2F_M\).
Now let's write the equations for the number of students who appeared: \[ A_M = P_M + F_M \implies 120 - N_M = 2F_E + F_M \quad \text{(Equation 1)} \] \[ A_E = P_E + F_E \implies 120 - 2N_M = 2F_M + F_E \quad \text{(Equation 2)} \] We have a system of two equations with three variables (\(N_M, F_M, F_E\)). Let's try to eliminate variables. Let's rearrange Equation 2 to express \(F_E\): \[ F_E = 120 - 2N_M - 2F_M \] Now, substitute this expression for \(F_E\) into Equation 1: \[ 120 - N_M = 2(120 - 2N_M - 2F_M) + F_M \] \[ 120 - N_M = 240 - 4N_M - 4F_M + F_M \] \[ 120 - N_M = 240 - 4N_M - 3F_M \] Now, move all terms with variables to one side and constants to the other: \[ 4N_M - N_M + 3F_M = 240 - 120 \] \[ 3N_M + 3F_M = 120 \] Divide by 3: \[ N_M + F_M = 40 \] This gives a direct relationship between \(N_M\) and \(F_M\).
The question asks for the number of students who appeared but failed the English exam, which is \(F_E\). Let's use Equation 2 again: \[ 120 - 2N_M = 2F_M + F_E \] Rearrange to solve for \(F_E\): \[ F_E = 120 - 2N_M - 2F_M \] Factor out -2: \[ F_E = 120 - 2(N_M + F_M) \] We found that \(N_M + F_M = 40\). Substitute this into the equation for \(F_E\): \[ F_E = 120 - 2(40) \] \[ F_E = 120 - 80 = 40 \] Step 4: Final Answer:
The number of students who appeared but failed the English exam is 40.
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