Question

An animal shelter is building new cages for 26 dogs and 34 cats. All the animals are kept separate. The first crew can build a cage in 1 hour and the second crew, which is smaller, takes 2 hours to build a cage. How many hours will it takes to build the cages using both crews if they do not take a break?

Answer 1

The animal shelter is building cages for 26 dogs and 34 cats.

26 + 34 = 60

It needs to build 60 cages.

The first crew builds 1 cage in 1 hour. Rate: 1 cage/hour

The second crew builds 1 cage in 2 hours, or 1/2 cage in 1 hour. Rate: 1/2 cage/hour

They work together for x hours and build all 60 cages.

In x hours, the first crew builds 1 cage/hour * x hours = x cages.

In x hours, the second crew builds 1/2 cage/hour * x hours = 1/2 x cages.

Combined, they build 60 cages.

x + 1/2 x = 60

x + 0.5x = 60

1.5x = 60

x = 40

Answer: It will take 40 hours.

Check: In 40 hours, the first crew builds 40 cages.

In 40 hours, the second crew builds 20 cages.

40 + 20 = 60

In 40 hours, the two crews build 60 cages.

Our answer above is correct.

Answer 2

All together, the shelter needs (26 + 34) = 60 new cages built.

Crew #1 builds a cage in 1 hour.  They finish 1 cage/hour.

Crew #2 builds a cage in 2 hours.  They finish  1/2 cage/hour.

When both crews work, they finish (1 + 1/2) = 1.5 cage/hour.

To build 60 cages, it will take them

(60 cages) / (1.5 cage/hour) =

(60 cages) ÷ (1.5 cage/hour) =

(60 cages) x (hour / 1.5 cage) =

(60 cage-hour) / (1.5 cage) =

(60 / 1.5) (cage·hour/cage) =

40 hours