HELP !!! algebra SAT question

Question

Two pipes can fill a tank in 64 minutes if both are turned on. If only one is used, it would take 54 minutes longer for the smaller pipe to fill the tank than the larger pipe. How long will it take for the smaller pipe to fill the tank? (Round your answer to the nearest tenth.)

Answer 1

The key to problems like this is that rates add.

Suppose the volume of the tank is 64 gal. and the rate (gal/min) of the smaller pipe is rs and the rate of the larger pipe is rl (gal/min) Then

64 = (rs + rl) 64 so rs + rl = 1

The time it would take with the small pipe alone is 64 / rs and the time it would take with the larger pipe along is 64 / rl So

54 = 64 / rs - 64/ rl so 54/64 = 1/rs - 1/rl.

This is now a set of two equations and two unknowns. Using the method of substitution results in

54/64 = 1/rs - 1/(1- rs)

I used a graphing TI-84 graphing calculator to solve this equation .

The result was rs = .3988 and so rl = .6012

Thus the time required using the small pipe alone is 64/.3988 = 160.5 min

The same answer would be obtained no matter what volume of the tank was assumed to have.

The choice of 64 gal just makes the equations look neater.

Answer 2

The combined work rate is (1/64) of the tank per minute.

The larger pipe can fill the tank in x minutes.

The smaller pipe can fill the tank in x+54 minutes.

the combined work rate is 1/x + 1/(x+54) = 1/64

Multiplies by 64 x ( x+ 54)

64(x+54) + 64x = x(x+54)

64x + 3456 + 64x = x^2 + 54x

128x + 3456 = x^2 + 54x

0 = x^2 + 54x - 128x - 3456

0 = x^2 - 74x - 3456

quadratic formula says:

x = 106.46221994724903

the smaller pipe will fill the tank is 160.46221994724903 min

Note that the sum of the reciprocals of these values is 0.01562020095651792374182791452766‬

Multiplying that by 64 is approximately 1