For a car moving at a constant speed, the distance traveled varies directly with the time spent driving. If such a car travels 180 miles in 9 hours, how long does it take to travel 140 miles?
The Correct Answer and Explanation is:
To solve this, we can use the concept of direct variation, which states that if two quantities vary directly, their ratio remains constant. In this case, the distance traveled (D) varies directly with time (T). Mathematically, we can express this relationship as:D1T1=D2T2\frac{D_1}{T_1} = \frac{D_2}{T_2}T1D1=T2D2
Where:
- D1=180 milesD_1 = 180 \, \text{miles}D1=180miles is the distance traveled in the first case,
- T1=9 hoursT_1 = 9 \, \text{hours}T1=9hours is the time spent driving for the first case,
- D2=140 milesD_2 = 140 \, \text{miles}D2=140miles is the distance we want to calculate the time for,
- T2T_2T2 is the time we need to find.
We can rearrange the formula to solve for T2T_2T2:T2=D2×T1D1T_2 = \frac{D_2 \times T_1}{D_1}T2=D1D2×T1
Now, plug in the known values:T2=140 miles×9 hours180 milesT_2 = \frac{140 \, \text{miles} \times 9 \, \text{hours}}{180 \, \text{miles}}T2=180miles140miles×9hours
Simplifying the calculation:T2=1260180=7 hoursT_2 = \frac{1260}{180} = 7 \, \text{hours}T2=1801260=7hours
So, it will take the car 7 hours to travel 140 miles.
Explanation:
In a situation like this, where the distance traveled and time are directly related, we can use the constant ratio of distance to time to set up a proportion. This allows us to solve for the unknown time when the distance changes but the speed stays the same. The proportional relationship helps simplify the calculation, giving us the time required for the car to travel a different distance at the same constant speed.
