A building project went over budget by 25%. The tools went overbudget by 19% and the rest by 28%. What was the original portion of the tools of the original budget?

Let’s define the key variables to solve the problem:

- Let \( T \) be the original cost of the tools.
- Let \( R \) be the original cost of the rest of the budget (non-tools).
- Let the original total budget be \( B = T + R \).

The project went over budget by 25%, meaning the final cost was \( 1.25B \).

The tools went over budget by 19%, so their final cost is \( 1.19T \).  
The rest of the project went over budget by 28%, so the final cost of the rest is \( 1.28R \).

Thus, the total final cost of the project is:
\[
1.19T + 1.28R = 1.25B
\]
Since \( B = T + R \), substitute this into the equation:
\[
1.19T + 1.28R = 1.25(T + R)
\]
Now, expand both sides:
\[
1.19T + 1.28R = 1.25T + 1.25R
\]
Now, move all terms involving \( T \) and \( R \) to one side:
\[
1.19T - 1.25T = 1.25R - 1.28R
\]
Simplify the equation:
\[
-0.06T = -0.03R
\]
Now divide both sides by -0.06 to isolate \( T \):
\[
T = \frac{-0.03}{-0.06}R = \frac{1}{2}R
\]
Thus, \( T = \frac{1}{2}R \), meaning the tools originally made up half the cost of the rest of the budget.

To find the portion of the tools in the total original budget \( B \), we calculate:
\[
\frac{T}{B} = \frac{T}{T + R}
\]
Since \( T = \frac{1}{2}R \), substitute this into the formula:
\[
\frac{T}{T + R} = \frac{\frac{1}{2}R}{\frac{1}{2}R + R} = \frac{\frac{1}{2}R}{\frac{3}{2}R} = \frac{1}{3}
\]

So, the original portion of the tools in the total budget was **one-third (1/3)**.

Would you like more details on any of the steps?

Here are 5 related questions:
1. How would the problem change if the tools went over budget by a different percentage?
2. How do you calculate the total percentage over budget when both tools and the rest are given different percentages?
3. Can this method apply if there are more than two categories of expenses in a budget?
4. What if the project had gone under budget? How would the equation change?
5. What happens if both categories go over budget by the same percentage?

**Tip:** Always break down a word problem into smaller, manageable equations to simplify the solution.

Solve the budget overrun problem where tools went over by 19% and the rest by 28%, with the overall project exceeding by 25%. Learn how to calculate the original portion of the tools in the total budget using algebraic equations and percentages. Suitable for students in Grades 9-12.

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