- Define the total homework time as the sum of history, English, and math times: \(30 + 60 + x\).
- State the condition that math time (\(x\)) is one-fourth of the total time: \(x = \frac{1}{4}(30 + 60 + x)\).
- Identify that this matches the first given option.
- The correct equation is \(\boxed{\frac{1}{4}(x + 30 + 60) = x}\).
Explanation
- Analysis of the Problem
The problem provides information about the time Hiroshi spends on different school subjects. We are given:
- Time spent on history homework: \(30\) minutes.
- Time spent on English homework: \(60\) minutes.
- Time spent on math homework: \(x\) minutes.
The total time Hiroshi spends on all his homework is the sum of these durations:
\[\text{Total Time} = 30 + 60 + x\]
We are told that math homework accounts for one fourth of his total homework time. This relationship can be expressed as:
\[\text{Math Time} = \frac{1}{4} \times (\text{Total Time})\]
Substituting the known values into this equation gives:
\[x = \frac{1}{4}(30 + 60 + x)\]
This is equivalent to the expression \(\frac{1}{4}(x + 30 + 60) = x\).
- Evaluating the Options
We need to identify which of the provided options matches the equation we derived.
The equation we found is:
\[\frac{1}{4}(x + 30 + 60) = x\]
Looking at the options:
- \(\frac{1}{4}(x + 30 + 60) = x\) - This matches our derivation.
- \(\frac{1}{4}(x) = x(30 + 60)\) - This is not the correct relationship.
- \(\frac{1}{4}x(30 + 60) = x\) - This is not the correct relationship.
- \(\frac{1}{4}(x) = (30 + 60)\) - This is not the correct relationship.
Therefore, the first equation is the one that correctly represents the problem.
- Final Conclusion
The correct equation is \(\frac{1}{4}(x + 30 + 60) = x\). This equation captures that one fourth of the sum of all subjects (history, English, and math) equals the time spent on math homework. The answer is the first option provided.
\[\boxed{\frac{1}{4}(x + 30 + 60) = x}\]
Examples
Understanding how to build equations from real-world conditions is an essential skill for managing daily resources. For instance, if you have a total budget for the week and you want to ensure that your grocery spending is exactly one-third of your total expenses including rent and utilities, you can set up an algebraic equation similar to this one. By defining your variables and establishing relationships based on proportions, you can solve for unknown values and plan your finances effectively, ensuring that your spending stays within the desired limits.