Rates and Ratio work problem
You are given that:
It takes 60 minutes for 7 people to paint 5 walls.
Consider: Rate $\times$ Time $=$ Output.
Let us consider "Rate" to mean the rate at which one person works.
And rather than writing out the whole word, let us just write $R$.
In the given: $7R \times 60$ minutes $=$ $5$ walls.
Omitting units and rearranging, we have: $7R = 5/60 = 1/12$.
Dividing both sides by $7$, we get $R = 1/84$.
Then you ask:
How many minutes does it take 10 people to paint 10 walls?
Set up similarly, we have: $10R \times t$ minutes $= 10$ walls.
Dividing both sides by $10$ and omitting units, we have: $R \times t = 1$.
But we know $R = 1/84$, so this says: $t/84 = 1$.
To finish off matters, multiply both sides by $84$ to obtain: $t = 84$.
It takes eighty four minutes.
Here is an alternative approach, just for fun. Omitting units throughout:
In both scenarios, the rate of work is the same; we will use this to solve the problem.
Note: Rate $=$ Output $\div$ Time.
In scenario one, the rate of one person working is: $5/(60\cdot 7).$
In scenario two, the rate of one person working is: $10/(t\cdot 10).$
We need to solve for $t$, but these expressions are equal. Let us simplify the resulting equation:
$$\frac{5}{60 \cdot 7} = \frac{10}{t \cdot 10} \implies \frac{1}{12 \cdot 7} = \frac{1}{t}$$
Equating denominators (or "cross multiplying") we find $t = 12 \cdot 7 = 84$.
There is a formula you may find useful:
If it takes time $T_1$ for $X_1$ people to do $Y_1$ things, and time $T_2$ for $X_2$ people to do $Y_2$ things, then $$\boxed{\frac{X_1T_1}{Y_1}\ =\ \frac{X_2T_2}{Y_2}}$$ In this particular problem, $T_1=60$, $X_1=7$, $Y_1=5$, $X_2=10$, $Y_2=10$; substituting in the formula gives $T_2=84$.
Proof of the formula:
If $X_1$ people can do $Y_1$ things, then 1 person can do $\dfrac{Y_1}{X_1}$ things in the same time, and so $X_2$ people can do $\dfrac{X_2Y_1}{X_1}$ in this time. Say, this time is $T_1$. Then in time 1, the same number of people $X_2$ can do $\dfrac{X_2Y_1}{X_1T_1}$ things so in time $T_2$ they can do $\dfrac{X_2Y_1T_2}{X_1T_1}$ things. Hence $$Y_2\ =\ \dfrac{X_2Y_1T_2}{X_1T_1}$$ which can be rearranged to the formula above.
Here's a similar problem. You can look at this then work out yours.
Question: It takes 42 minutes for 7 people to paint 6 walls. How many minutes does it take 8 people to paint 8 walls?
Solution: It takes 42 minutes for 7 people to paint 6 walls 42/6=7 minutes per wall
It takes 7 people 7 minutes to paint 1 wall
Each person paints 1/7 of the wall in 7 minutes
Each person paints 1/49 of the wall in 1 minute at the same rate...
8 people paint 8/49 of 1 wall in 1 minute
How many minutes does it take 8 people to paint 1 wall ? 49/8=6 1/8
It takes 8 people 6 1/8 minutes to paint 1 wall
It takes 8 people 8*(6 1/8) minutes to paint 8 walls
8*(6 1/8)=49 minutes
It takes 49 minutes for 8 people to paint 8 walls