Algebra word problem

Question

Roshini cycles 2 km at ykm/hr then runs 4km at (y-4) the whole journey takes 40mins. write an equation for y that shows it simplifies to y^2-13y+12=0

Answer 1

We are working with the distance formula:  d = rt

distance = rate * time

 

In this case, we need to solve that for time so:  t = d/r

time = distance/rate

 

1) cycling:

    d=2 km

    r = y km/hr

    t = d/r = 2/y

 

2)  running

    d = 4 km

    r = y-4 km/hr

    t = 4/(y-4)

 

Note that our total time is time spent cycling plus time spent running.

That total time is 40 mins....

BUT:  our units are in km per hour so we need our time units to

be in hours.  40 mins = 40/60 hours = 2/3 hours

 

The initial equation is:

 

2/y + 4/(y-4) = 2/3

Get a common denominator on left of y(y-4)

 

[2(y-4) + 4y]/y(y-4) = 2/3

 

We can make that denominator y²-4y

by multiplying the y through the parenthesis.

Also multiply the 2 through the parenthesis in

the numerator.

 

(2y-8+4y)/(y²-4y) = 2/3

(6y-8)/(y²-4y)=2/3

Multiply both sides by 3

3(6y-8)/(y²-4y) = 2

Multiply both sides by (y²-4y)

3(6y-8) = 2(y²-4y)

18y-24 = 2y²-8y

subtract 18y from both sides

-24 = 2y²-8y-18y

-24 = 2y²-26y

Add 24 to both sides

0 = 2y²-26y+24

 

This is the same as:

 

2y²-26y+24 = 0

Factor a 2 from each term

2(y²-13y+12) = 0

Divide both sides by 2

 

y²-13y+12 = 0

 

Note:  total distance travelled was 2km on bike + 4 km running = 6 km

 

Note: the units for (y-4) is km/hr