Question

Latoya can paint five 10-foot-by-14-foot rooms by herself in 14 hours. Lisa can paint five 10-foot-by-14-foot rooms by herself in 10 hours. Working together, how long would it take to paint 5 10-foot-by-14 foot rooms?

(Please show the work! I really want to understand this problem.)

Answer 1

let's say if both work together, they can do it in "t" hrs

so, if Latoya can do it by herself in 14hrs, how much can she do in 1hr? well, she can do only 1/14 of the total

if Lisa can do it in 10hrs by herself, how much can she do in 1hr? well, 1/10 of the total

now, let's add their rate together, keeping in mind, their rate added, will be doing in 1hr only 1/t of the total work

thus $\bf \cfrac{1}{14}+\cfrac{1}{10}=\cfrac{1}{t}\implies \cfrac{5+7}{70}=\cfrac{1}{t}\implies \cfrac{12}{70}=\cfrac{1}{t}\implies \cfrac{6}{35}=\cfrac{1}{t} \\\\\\ t=\cfrac{35\cdot 1}{6}\implies t=\cfrac{35}{6}\implies t=5\frac{5}{6}$

so, they'd do it in 5hrs and 5/6 of an hour, which is just 50mins, so, 5hrs and 50mins