Let's solve this step-by-step by understanding the problem and analyzing each option given:
1. Understanding the problem:
- Hiroshi spends 30 minutes on history homework.
- Hiroshi spends 60 minutes on English homework.
- Hiroshi spends [tex]\( x \)[/tex] minutes on math homework.
- One fourth of Hiroshi's total homework time is spent on math homework.
2. Writing the total homework time:
- The total homework time [tex]\( T \)[/tex] consists of the sum of history, English, and math homework times.
- [tex]\( T = 30 + 60 + x \)[/tex]
- Simplify this to:
[tex]\[
T = 90 + x
\][/tex]
3. Expressing the given condition in mathematical terms:
- One fourth of the total homework time [tex]\( T \)[/tex] is spent on math homework.
- The math homework time is [tex]\( x \)[/tex].
- Therefore, we can write:
[tex]\[
\frac{1}{4} \cdot (90 + x) = x
\][/tex]
Now let's compare this derived equation with the given options:
### Option 1: [tex]\(\frac{1}{4}(x + 30 + 60) = x\)[/tex]
- Simplify the equation inside the parenthesis:
[tex]\[
\frac{1}{4}(x + 30 + 60) = \frac{1}{4}(x + 90)
\][/tex]
- This simplifies to:
[tex]\[
\frac{1}{4} \cdot (90 + x) = x
\][/tex]
- This matches our derived equation.
### Option 2: [tex]\(\frac{1}{4}(x) = x(30 + 60)\)[/tex]
- Simplify the equation:
[tex]\[
\frac{1}{4} \cdot x = x \cdot 90
\][/tex]
- This means:
[tex]\[
\frac{x}{4} = 90x
\][/tex]
- This equation does not make sense considering the problem statement. It implies that [tex]\( x \)[/tex] should be both large and small simultaneously.
### Option 3: [tex]\(\frac{1}{4} x(30 + 60) = x\)[/tex]
- Simplify the equation inside the parenthesis:
[tex]\[
\frac{1}{4} \cdot x \cdot 90 = x
\][/tex]
- This means:
[tex]\[
\frac{90x}{4} = x \quad \text{or} \quad 22.5x = x
\][/tex]
- This equation does not make logical sense.
### Option 4: [tex]\(\frac{1}{4}(x) = (30 + 60)\)[/tex]
- Simplify the equation inside the parenthesis:
[tex]\[
\frac{x}{4} = 90
\][/tex]
- This means:
[tex]\[
x = 360
\][/tex]
- This is a numerical solution but does not frame the correct time relationship described in the problem.
### Conclusion:
The only equation that matches our derived condition is:
[tex]\[\boxed{\frac{1}{4}(x + 30 + 60) = x}\][/tex]
