Rational Word Problem Help!

Question

This is just one of those pesky work problems:

Formula: $(rate\space of\space work)(time\space worked) = (part\space of\space job\space done)$

Rosa can mow the lawn in 20 minutes using a power mower. Her brother, Fidel, can mow the same lawn in 30 minutes using a hand mower. If they work together, how long will it take them to complete the job?

I decided to organise my work into a table, solving the equation on the side:

Obviously... my final answer isn't right. I don't understand, since I'm pretty sure the fractions I added were right and my equation was correct. I believe it must've been the way I isolated for $x$? Otherwise, what did I do wrong?

Answer 1

You seem to be using a common denominator of $600$ by mistake, not $60$. You're multiplying $30$ and by $20$, and vice versa. You should be multiplying $30$ by $2$ and $20$ by $3$. You should get:

$$\frac x{20} + \frac x{30} = 1$$ $$\color{red}{\frac 33} \cdot \frac x{20} + \color{red}{\frac 22} \cdot \frac x{30} = \frac {60}{60}$$ $$3x + 2x = 60$$ $$x = 12$$

The other option would be to set the other side to a denominator of $600$ to get $30x + 20x = 600$ which yields the same result.

Answer 2

$$30x + 20x = 60\color{blue}0$$

$$50x=600$$

$$x = 12$$

Answer 3

In one minute, Rosa can mow $1/20$ of the yard and Fidel can mow $1/30$ of the yard. So together, in one minute, they can mow $1/20+1/30 = 5/60 =1/12$ of the yard. So $12$ minutes for the whole yard.

Answer 4

It could be simpler to think of this as a speed distance and time question.Rosa can mow one field in 20 minutes and therefore 3 fields in an hour. Fidel can do 2 fields in an hour. That means that together, a total of 5 fields can be done in an hour. So we can make a ratio for minutes to fields:

60:5

Since we are only working out one field this scales down to:

12:1

So the answer is 12 minutes