Dish Trying With Team Work -- How fast does the daughter dry one dish?

[DanOh62]

Here's the question:

A daughter and her mother dry half the dishes together; then mom rests, while the daughter and her father dry the other half. Drying the dishes this way takes twice as long as when all three work together. If mom takes 2 seconds to dry one dish and dad takes 5 seconds to dry one dish, how long does the daughter take to dry one dish?

Here is my start/guess at a formula:
(2 seconds to dry a dish + x seconds to dry a dish) + (5 seconds to dry a dish + x seconds to dry a dish)=twice as long as (2 seconds to dry + 5 seconds to dry + x seconds to dry)
where x seconds to dry a dish is the rate at which the daughter dries.

Any tips, hints or formulas would be greatly appreciated.

Thanks,
Father of 7th grade math student.

 

[HallsofIvy]

Here's the question:

A daughter and her mother dry half the dishes together; then mom rests, while the daughter and her father dry the other half. Drying the dishes this way takes twice as long as when all three work together. If mom takes 2 seconds to dry one dish and dad takes 5 seconds to dry one dish, how long does the daughter take to dry one dish?

Here is my start/guess at a formula:
(2 seconds to dry a dish + x seconds to dry a dish) + (5 seconds to dry a dish + x seconds to dry a dish)=twice as long as (2 seconds to dry + 5 seconds to dry + x seconds to dry)
where x seconds to dry a dish is the rate at which the daughter dries.

Any tips, hints or formulas would be greatly appreciated.

Thanks,
Father of 7th grade math student.

When you have people (or machines, etc) working together it is their rates of work that add. Adding the times, as you do above is not valid- MORE people working will not take more time! If the mother takes 2 seconds to dry one dish, her rate of work is 1/2 "dish per second". If the father takes 5 seconds to dry one dish, his rate of work is 1/5 "dish per second". Letting x be the number of seconds it takes the daughter to dry one dish, her rate of work is 1/x "dish per second".
Let "N" be the number of dishes to be dried. The daughter and mother working together will dry N/2 dishes in (N/2)/(1/2+ 1/x)= Nx/(x+ 2) seconds. The father and daughter will dry the other N/2 dishes in (N/2)(1/5+ 1/x)= 5Nx/2(x+ 5)= 5Nx/(2x+ 10) seconds. The total time would be Nx/(x+ 2)+ 5Nx/(2x+ 10) seconds. If all three work together they will dry N dishes in N/(1/5+ 1/2+ 1/x)= 10Nx/(7x+ 10). Since the first way takes twice as long as the second, \(\displaystyle Nx/(x+ 2)+ 5Nx/2(x+ 5)= 20Nx/(7x+ 10). I would start solving that equation by multiplying through by (x+2)(2x+10)(7x+10).\)