The given equation is:

\[
30\left(T + \frac{10}{60}\right) = 40\left(T - \frac{5}{60}\right)
\]

Let's solve this step-by-step:

### Step 1: Simplify the fractions
\[
\frac{10}{60} = \frac{1}{6}, \quad \frac{5}{60} = \frac{1}{12}
\]
Now the equation becomes:
\[
30\left(T + \frac{1}{6}\right) = 40\left(T - \frac{1}{12}\right)
\]

### Step 2: Distribute the constants
Distribute the 30 and 40 on both sides:
\[
30T + 30 \times \frac{1}{6} = 40T - 40 \times \frac{1}{12}
\]
\[
30T + 5 = 40T - \frac{40}{12}
\]

### Step 3: Simplify the right-hand side fraction
\[
\frac{40}{12} = \frac{10}{3}
\]
So the equation becomes:
\[
30T + 5 = 40T - \frac{10}{3}
\]

### Step 4: Multiply through by 3 to eliminate the fraction
Multiply every term by 3:
\[
3(30T + 5) = 3\left(40T - \frac{10}{3}\right)
\]
\[
90T + 15 = 120T - 10
\]

### Step 5: Solve for \(T\)
Move the terms involving \(T\) to one side and the constants to the other:
\[
90T - 120T = -10 - 15
\]
\[
-30T = -25
\]

### Step 6: Divide by -30
\[
T = \frac{-25}{-30} = \frac{5}{6}
\]

Thus, \(T = \frac{5}{6}\).

Would you like more details or further explanation?

Here are 5 related questions you might find useful:
1. How do we simplify fractions in equations?
2. What is the importance of distributing constants across terms?
3. How can multiplying by a common denominator simplify equations with fractions?
4. How do you handle terms with variables on both sides of the equation?
5. What strategies can be used to check if a solution is correct?

**Tip:** Always simplify fractions early in the process to make solving equations easier.

Learn how to solve the linear equation 30(T + 10/60) = 40(T - 5/60) with a step-by-step solution. This problem covers key algebraic concepts, including distributive property, fraction simplification, and solving linear equations. Suitable for students in Grades 6-8.

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