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Kaylis [27]
3 years ago
9

Carmen recently rode her bicycle to visit her friend who lives 12 miles away. On her way there, her average speed was 15 miles p

er hour faster than on her way home. If Carmen spent a total of 3 hours bicycling, find the two rates.
Mathematics
1 answer:
Ket [755]3 years ago
5 0

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.

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