I've encountered this problem
It takes $60$ minutes for $7$ people to paint $5$ walls. How many minutes does it take $10$ people to paint $10$ walls.
The answer to this one is $84$ minutes. However, How did it come up to this answer? can someone explain to me why?
Can anyone give a step by step method on how did we lead to this/solve this kind of problems?
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
Seven people working for 60 minutes amounts to a total of 420 minutes of work to paint 5 walls.
So one wall takes $\frac{420}{5}=84$ minutes of work to paint.
So 10 walls requires 840 minutes to paint. And if you have 10 people working together, that comes to 84 minutes each.
Exploit linear dependence.
You have a relation between a number P of persons, a number M of minutes and a number W of walls painted. You know that P=7, M=60 gives W=5 and you are asked what is M such that P=10 gives W=10. So you should move from P=7 to P=10 and from W=5 to W=10 and keep track of the changes of M.
What happens if you double the number of people P and keep the same amount of time M? Of course, the walls painted W gets doubled. The same is true if you multiply P by any number k: W gets multiplied by the same number. So from P=7, M=60, W=5 you multiply both P and W by 10/7 and find P=10, M=60, W=(10/7)5=50/7.
Now what happens if you double the time M keeping P fixed? Of course the number of walls painted W also doubles. The same is true if you multiply M by any number, you get W multiplied by the same number. You need to find anumber k such that W goes from 50/7 to 10, so (50/7)k=10 which means k=(7/50)10 = 7/5. The time M then goes from 60 to (7/5)60 which is 84.
These problems are not difficult to solve in your head, even without an explicit algebraic formula:
It takes $60$ minutes for $7$ people to paint $5$ walls. How many minutes does it take $10$ people to paint $10$ walls?
Original time: $60$ minutes.
Double the count of walls from $5$ to $10$ -> double the time to $120$ minutes.
Pretend only $1$ person instead of the current $7$ -> multiply the time by $7$.
($7\times120 = 840$ minutes.)
Now with $10$ times as many people ($10$ instead of $1$), it will take a tenth of that amount of time, which is $84$ minutes.
Of course I advise you to understand the formula. This method is the exact same steps as you will take by computing the formula algebraically, but I have found students who trip on the algebraic manipulation of variables are able to pick up this approach quite easily, and then become able to make sense of the algebra when they couldn't before. Hence I feel it is a valuable approach worth mentioning.
The "formula" is, of course, that "walls per person per minute" is a constant value.
Actually answering such questions, I factually don't use any "method" personally as the calculations are done too fast to illustrate, whether through formulas or sentences. How do you answer the question, "What is $5+5$?" If you actually count $5$ and then $5$ more I feel bad for you. But, steps can be illustrated and followed through until you achieve the ability to just instantly state the answer.
