Two jet planes travelling in opposite directions were 3800 miles apart at the end of 4 hours. if one traveled 80 miles per hour faster than the other, find the rate of each plane.

Their total speed is (3800 mi)/(4 h) = 950 mi/h. This lets you write two equations for the speeds (x, y) of the two planes:

... x + y = 950

... x - y = 80 . . . . . . one travels 80 mi/h faster than the other

Adding the two equations gives

... 2x = 1030

... x = 515

... y = 515 -80 = 435

The speeds of the planes are 515 miles per hour and 435 miles per hour.

Comment on the problem

This is a problem of the generic type "sum and difference problem." You are given the total of the two speeds and the difference between them. The higher speed is the average of those two numbers: (950+80)/2 = 515.

This is the generic solution for any "sum and difference problem:" the larger value is the average of the sum and difference.

Explanation:

The problem we are given is

We can write the square root of a fraction as a fraction with a separate radical for the numerator and denominator; this gives us

We can write the whole number 4 as the fraction 4/1; this gives us

We now need to "rationalize the denominator."  This means we need to cancel the square root in the denominator.  In order to do this, we multiply both numerator and denominator by √(2x); this is because squaring a square root will cancel it:

When multiplying radicals, we can extend the radical over both factors: