Work Together

Part of the Polynomials & Rational Expressions section of Coddy's Algebra 2 journey. Lesson 48 of 54.

A hose fills a pool in 3 hours, a second hose in 6. How long do they take together? Think in rates. In one hour the first hose fills 1/3 of the pool and the second 1/6. If together they take t hours, they fill 1/t of it each hour, so 1/3 + 1/6 = 1/t.

Add the left side: 1/3 + 1/6 is 1/2, so 1/2 = 1/t. Half the pool in one hour means the whole pool in 2 hours: t is the reciprocal of 1/2, so t = 2.

Rates add; times do not. 3 hours plus 6 hours is 9, slower than either hose alone. Working together has to beat the faster hose, so the answer is always below the smaller time.

Challenge

One pipe fills a tank in 4 hours and a second pipe fills it in 6 hours. Running together, they fill it in t hours: each hour the first fills a quarter of the tank and the second a sixth, so 1/4 + 1/6 = 1/t. Solve the equation on the board for t, leaving the answer as a fraction of hours.

Try it yourself

1/4 + 1/6 = 1/t

Solve for t