Bus interval frequency

If the interval between bus arrivals is shortened by one minute, there will be 5 more buses on this route per hour.
How many buses ran on the original route?

(Recommended: solve with an experiment or a table where we look for the dependence between the length of the interval and the number of buses on the route. )

Final Answer:

x =  15

Step-by-step explanation:

$$\begin{gathered} 60/(T-1)=5+60/T \\ 60\cdot T=5\cdot T\cdot (T-1)+60\cdot (T-1) \\ 60\cdot T=5\cdot T\cdot (T-1)+60\cdot (T-1) \\ -5T^2+5T+60=0 \\ 5T^2-5T-60=0 \\ 5...\text{ prime number} \\ 60=2^2\cdot 3\cdot 5 \\ \text{GCD}(5,5,60)=5 \\ {T}^2-T-12=0 \\ a=1;b=-1;c=-12 \\ D=b^2-4ac=1^2-4\cdot 1\cdot (-12)=49 \\ D>0 \\ {T}_{1,2}=\frac{-b\pm \sqrt{D}}{2a}=\frac{1\pm \sqrt{49}}{2} \\ {T}_{1,2}=\frac{1\pm 7}{2} \\ {T}_{1,2}=0.5\pm 3.5 \\ {T}_1=4 \\ {T}_2=-3 \\ T=T_1=4\text{min} \\ x=\frac{60}{T}=\frac{60}{4}=15 \end{gathered}$$

Our quadratic equation calculator calculates it.

Try another example