To determine after how many kilometers the costs from both car hire companies would be the same, we need to set up an equation where the total costs from both companies are equal.
Let [tex]\( km \)[/tex] be the number of kilometers traveled.
For Car Hire A:
- Car Hire A charges a deposit of R270.
- Additionally, it charges R12 per kilometer.
So, the total cost (Cost\_A) for Car Hire A after traveling [tex]\( km \)[/tex] kilometers can be expressed as:
[tex]\[ \text{Cost\_A} = 270 + 12 \times km \][/tex]
For Car Hire B:
- Car Hire B does not charge a deposit.
- It charges R15 per kilometer.
So, the total cost (Cost\_B) for Car Hire B after traveling [tex]\( km \)[/tex] kilometers can be expressed as:
[tex]\[ \text{Cost\_B} = 15 \times km \][/tex]
We need to find the point at which these two costs are equal:
[tex]\[ 270 + 12 \times km = 15 \times km \][/tex]
Now, we can solve for [tex]\( km \)[/tex]:
1. Subtract [tex]\( 12 \times km \)[/tex] from both sides of the equation:
[tex]\[ 270 + 12 \times km - 12 \times km = 15 \times km - 12 \times km \][/tex]
[tex]\[ 270 = 3 \times km \][/tex]
2. Divide both sides by 3 to solve for [tex]\( km \)[/tex]:
[tex]\[ \frac{270}{3} = km \][/tex]
[tex]\[ km = 90 \][/tex]
Therefore, the rates from Car Hire A and Car Hire B are the same after traveling 90 kilometers.
So, the answer is:
[tex]\[ \boxed{90.0 \text{ km}} \][/tex]
