###### Example12.10.1

Joaquin and Abraham co-own a bicycle repair shop. On average, Joaquin can swap out both tires on a bicycle in \(26\) minutes while, on average, it takes Abraham \(32\) minutes to complete the same task. Determine the amount of time it would take the two of them to swap out both tires on 40 bicycles. Assume that the rate at which they complete the task when they work together is the sum of the rates each owner averages on his own.

Let \(t\) represent the amount of time it takes them to swap out the tires on \(40\) bicycles when they work together. The information given in the problem is summarized in Table 12.10.2. In each row, the rate expression was derived using:

rate (swaps/hr) | time (minutes) | number of swaps completed | |

Joaquin alone | \(\frac{1}{26}\) | \(26\) | \(1\) |

Abraham alone | \(\frac{1}{32}\) | \(32\) | \(1\) |

working together | \(\frac{40}{t}\) | \(t\) | \(40\) |

The equation we need to solve comes from the fact that the two individual rates sum to the rate at which they work together. To wit:

We begin solving the equation by noting that \(26=2 \times 13\) and \(32=2 \times 16\text{,}\) so the LCD of the fractions in the equation is \(2 \times 13 \times 16t\) which simplifies to \(416t\text{.}\) We need to multiply both sides of the equation by the LCD, distribute, simplify each term, and solve the resultant equation.

That's a whole lot of minutes. Let's convert \(573.8\) minutes to hours to get a better sense of the amount of time we're looking at here.

So when they work together, it takes Abraham and Joaquin approximately \(9.6\) hours to swap out both tires on \(40\) bicycles.