A cyclist travels for 60km at x km/h and 180km at y km/h and takes 10 hours altogether for the journey. If the speed are interchange, the journey takes 8.6 (8 whole number 2 over 3) hours. Find x and y.
60 + 180 = 140 km journey length
time at x = 60/x
time at y = 180/y
total time = 60/x + 180/y = 10
now interchange
total time = 180/x + 60/y = 8 2/3 = 26/3
two equations
60 y + 180 x = 10 x y
180 y + 60 x = 26 x y /3
multiply bottom equation by 3
60 y + 180 x = 10 x y and
540y + 180 x = 26 x y
---------------------------subtract
-480 y = -16 x y
x = 30
etc
Thank you very much Damon :)
The ages of a man and a woman are ratio 5:4.In 12 years time the ratio of their ages will be 19:16.How old are they now
A woman borrows £180,000 from a bank at 10% per annum compound interest. She repays 50,000 at the end of each year. How much does she still owe at the end of 4 years?
To solve this problem, we can use the formula Time = Distance / Speed.
Let's consider the first part of the journey, where the cyclist travels 60 km at a speed of x km/h. According to the given information, the time taken for this part is given by:
Time1 = 60 km / x km/h
Similarly, for the second part of the journey where the cyclist travels 180 km at a speed of y km/h, the time taken is:
Time2 = 180 km / y km/h
According to the problem, the total time taken for the journey is 10 hours. So, we can write the first equation:
Time1 + Time2 = 10
Now, if the speeds are interchanged, the total time taken for the journey becomes 8 2/3 hours. In this case, the time taken for the first part is:
Time1' = 60 km / y km/h
And the time taken for the second part is:
Time2' = 180 km / x km/h
We can write the second equation based on this information:
Time1' + Time2' = 8 2/3
Now, we have two equations with two unknowns (x and y). We can solve these equations using substitution or elimination method:
Equation 1: Time1 + Time2 = 10
Equation 2: Time1' + Time2' = 8 2/3
Substituting the values of Time1, Time2, Time1', and Time2' derived earlier:
(60/x) + (180/y) = 10
(60/y) + (180/x) = 26/3
To simplify the equations, we can multiply both sides of Equation 1 by xy and Equation 2 by 3xy:
180y + 60x = 10xy ........(3)
180x + 60y = 26xy ........(4)
Now, we can solve these two equations simultaneously to find the values of x and y.
Multiplying Equation 4 by 3 and Equation 3 by 26 gives:
540x + 180y = 78xy ........(5)
1560x + 360y = 260xy ........(6)
Subtracting Equation 5 from Equation 6:
(1560x + 360y) - (540x + 180y) = (260xy - 78xy)
1020x + 180y = 182xy
Dividing both sides of the equation by 2:
510x + 90y = 91xy ........(7)
Substituting Equation 3 into Equation 7:
510x + 90(10xy - 60x) = 91xy
510x + 900xy - 540x = 91xy
-30x = -809xy
Simplifying and rearranging:
30x = 809xy
Dividing both sides by x:
30 = 809y
Solving for y:
y = 30 / 809
Similarly, substituting y back into Equation 3:
60 / x + 180 / (30 / 809) = 10
Multiply both sides by x:
60 + (180 * (809 / 30)) = 10x
Simplify:
60 + 4,852 = 10x
4,912 = 10x
Divide both sides by 10:
491.2 = x
Therefore, the values of x and y are approximately x = 491.2 and y = 0.0371.
Man's age presently =5a
Woman's age presently = 4a
In 12yrs time :
Man's age = 5a+12
Woman's age = 4a+12
...5a+12:4a+12= 19: 16
5a+12/4a+12=19/16
...a = 9
....Man's present age = 5a = 5(9)=45yrs
.....Woman's present age = 4a=4(9)=36yrs