Simple Equation

A man bicycles half the distance from  one town to the another at 24 km/hr, and the other half at 16 km/hr. A second man bicycles all the way at 22.5 km/hr. If the difference in the time taken is 5 1/2 min, then what is the whole distance?





Answer:

Let y be the distance between the two towns.
Journey 1st man:
 

T₁   = ^y/_{2x 24}  = ^y/₄₈

T₂ = ^y/_{2x 16}  = ^y/₃₂

Therefore,   total time taken = T₁ + T₂ = ^y/₄₈ + ^y/₃₂

For the Journey by the second man;

T = = ^y/_22.5 =  ^2y/₄₅

Now, given that ( ^y/₄₈ + ^y/₃₂) -  ^2y/₄₅  =  ¹¹/_{2x 60}

Solving the equation we get,

Y = 12 km

Solve for x:

^x/_{(x + a)} + ^x/_{(x + b)} = 2

Answer:

^x/_{(x + a)} + ^x/_{(x + b)} = 2

or, 2x² + x(a + b) = 2[x² + x(a + b) + ab]

or, 2x² + x(a + b) = 2x² + x(a + b) + x(a + b) + 2ab

or, 0 = x(a + b) + 2ab

or, x = ^-2ab/_{(a + b)}