The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges 6443 to rent trucks plus an additional fee of 175.75 for each ton of sugar. The second company charges 7500 to rent trucks plus an additional fee of 100.25 for each ton of sugar. For what amount of sugar do the two companies charge the same?

To find the amount of sugar (in tons) for which the two companies charge the same, we need to set up an equation and solve for the number of tons.

Let \( x \) represent the number of tons of sugar.

### Company 1: 
The total cost is given by:
\[
C_1(x) = 6443 + 175.75x
\]
where 6443 is the fixed truck rental fee, and 175.75 is the cost per ton of sugar.

### Company 2:
The total cost is given by:
\[
C_2(x) = 7500 + 100.25x
\]
where 7500 is the fixed truck rental fee, and 100.25 is the cost per ton of sugar.

### Step 1: Set the two cost functions equal to each other.
\[
6443 + 175.75x = 7500 + 100.25x
\]

### Step 2: Solve for \( x \).

First, subtract \( 100.25x \) from both sides:
\[
6443 + 75.5x = 7500
\]

Next, subtract 6443 from both sides:
\[
75.5x = 1057
\]

Now, divide both sides by 75.5:
\[
x = \frac{1057}{75.5} \approx 14
\]

Thus, the two companies will charge the same amount when transporting approximately **14 tons of sugar**.

Would you like more details on any step or have further questions?

Here are five related questions to expand on this topic:
1. How much would each company charge for transporting 14 tons of sugar?
2. For what amount of sugar would Company 1 be cheaper than Company 2?
3. What is the break-even point in terms of total cost for both companies?
4. How does the rental fee difference affect the overall cost as the quantity of sugar increases?
5. What happens to the cost difference if the per-ton fee changes for either company?

**Tip:** In problems like these, it's essential to isolate the variable step by step to maintain accuracy throughout the calculation.

Discover how to solve for the break-even point between two companies using linear equations. Learn how Sugar Sweet Company compares costs between two transport options, with truck rental fees of 6443 and 7500, and per-ton charges of 175.75 and 100.25. Ideal for students in Grades 9-12.

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