Understanding the Exam Score Problem

This problem involves finding the lowest score in an examination given specific relationships between the highest and lowest scores. We are told two key pieces of information:

• The difference between the highest score and the lowest score is 55.
• The highest score is \(\frac{9}{4}\) times the lowest score.

We need to use these relationships to set up equations and solve for the value of the lowest score.

Setting Up Equations for Exam Scores

Let's represent the scores using variables:

• Let \(H\) be the highest score.
• Let \(L\) be the lowest score.

From the problem statement, we can write two equations based on the given information:

Equation 1 (Difference): The difference between the highest and lowest score is 55.

\(H - L = 55\)

Equation 2 (Ratio): The highest score is \(\frac{9}{4}\) times the lowest score.

\(H = \frac{9}{4} L\)

Solving for the Lowest Score

We have a system of two linear equations with two variables (\(H\) and \(L\)). We can solve this system using substitution. Since Equation 2 already gives us an expression for \(H\) in terms of \(L\), we can substitute this expression into Equation 1.

Substitute \(H = \frac{9}{4} L\) into the equation \(H - L = 55\):

\(\left(\frac{9}{4} L\right) - L = 55\)

Now, we need to solve this equation for \(L\). To combine the terms involving \(L\), we can write \(L\) as \(\frac{4}{4} L\):

\(\frac{9}{4} L - \frac{4}{4} L = 55\)

Combine the terms on the left side:

\(\left(\frac{9}{4} - \frac{4}{4}\right) L = 55\)

\(\frac{9 - 4}{4} L = 55\)

\(\frac{5}{4} L = 55\)

To isolate \(L\), multiply both sides of the equation by the reciprocal of \(\frac{5}{4}\), which is \(\frac{4}{5}\):

\(L = 55 \times \frac{4}{5}\)

Now, calculate the value of \(L\):

\(L = \frac{55 \times 4}{5}\)

We can simplify by dividing 55 by 5:

\(L = 11 \times 4\)

\(L = 44\)

So, the lowest score is 44.

Verification of the Scores

Let's check if the calculated scores satisfy the original conditions:

• Lowest Score \(L = 44\).
• Highest Score \(H = \frac{9}{4} L = \frac{9}{4} \times 44\).
• \(H = 9 \times \frac{44}{4} = 9 \times 11 = 99\).

Check the difference:

\(H - L = 99 - 44 = 55\)

This matches the first condition (difference is 55). The second condition (\(H = \frac{9}{4} L\)) was used to calculate \(H\), so it is also satisfied.

The calculated lowest score of 44 is consistent with the problem statement.

Conclusion: Finding the Lowest Exam Score

By setting up and solving a system of linear equations based on the given relationships between the highest and lowest exam scores, we found that the lowest score is 44.

Revision Table: Key Concepts

Additional Information: Solving Systems of Equations

Systems of linear equations like the one in this exam score problem can be solved using different methods. The substitution method, as used here, is particularly useful when one equation is already solved for one variable (like \(H = \frac{9}{4} L\)). Other common methods include:

• Elimination Method: This involves adding or subtracting the equations (or multiples of the equations) to eliminate one of the variables.
• Graphical Method: Each equation represents a line; the solution is the point where the lines intersect. This method is less precise for non-integer solutions.
• Matrix Methods: Using matrices and matrix operations (like finding the inverse) to solve the system. This is more advanced and often used for larger systems.

Understanding how to translate word problems into mathematical equations is a fundamental skill in algebra. Always define your variables clearly and ensure your equations accurately represent the relationships described in the problem.