Question

Carmen recently rode her bicycle to visit her friend who lives 12 miles away. On her way there, her average speed was 15 miles per hour faster than on her way home. If Carmen spent a total of 3 hours bicycling, find the two rates.

Answer 1

Answer:

Carmen rode her bicycle at 20 miles per hour while riding to her friend house and while returning home she rode at speed of 5 miles per hour.

Step-by-step explanation:

Given:

Distance traveled = 12 miles

Total number of hours spent on bicycling = 3 hrs.

We need to find the speed while riding her friends house and rate while riding home.

Solution:

Let the speed while riding home be 'x'.

Now given:

On her way there, her average speed was 15 miles per hour faster than on her way home.

Speed while riding to her friends house = $x+15$

Now we know that;

Time is equal to distance divided by speed.

Time required to visit her friend house  $t_1=\frac{12}{x+15}$

Time required while returning home $t_2=\frac{12}{x}$

Now we know that;

Total time she spent on bicycling is equal to sum of Time required to visit her friend house and Time required while returning home.

framing in equation form we get;

$t_1+t_2 =3$

Substituting the values of $t_1$ and $t_2$ we get;

$\frac{12}{x+15}+\frac{12}{x} = 3$

Now we will use LCM to make the denominator common we get;

$\frac{12x}{x(x+15)}+\frac{12(x+15)}{x(x+15)} = 3$

Now the denominator are common so we will solve the numerator.

$\frac{12x+12x+180}{x(x+15)}=3\\\\\frac{24x+180}{x(x+15)}=3$

By cross multiplication we get;

$24x+180=3x(x+15)\\\\24x+180=3x^2+45x\\\\3x^2+45x-24x-180=0\\\\3x^2+21x-180=0$

taking 3 common we get;

$3(x^2+7x-60)=0$

Dividing both side by 3 we get;

$\frac{3(x^2+7x-60)}{3}=\frac{0}{3}\\\\x^2+7x-60=0$

Now by factorizing the equation to find the roots we get;

$x^2+12x-5x-60=0\\\\x(x+12)-5(x+12)=0\\\\(x+12)(x-5)=0$

Now we will solve separately to find the value of x we get;

$x+12=0 \ \ \ \ Or \ \ \ \ x-5=0\\\\x=-12 \ \ \ \ \ \ \ Or \ \ \ \ \ x=5$

Now we have got 2 values of x one positive and one negative.

we know that time cannot be negative and hence we will discard the negative value of x and consider the positive value of x.

speed while riding home = $5\ mi/hr$

Speed of bicycle while visiting her friend house = $x+15 = 5+15 =20\ mi/hr$

Hence Carmen rode her bicycle at 20 miles per hour while riding to her friend house and while returning home she rode at speed of 5 miles per hour.