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- Mar 1, 2012

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If the walls are the same size and each individual paints at an equal rate (1 person paints 1 wall in 28 minutes), then it takes the same amount of time ... 28 minutes.It takes 28 minutes for 7 people to paint 7 walls.

How many minutes does it take 14 people to paint 14 walls?

- Apr 22, 2018

- 251

There is a very useful formula for problems of this kind:

If $X_1$ “producers” can make $Y_1$ “products” in time $T_1$ and $X_2$ “producers” can make $Y_2$ “products” in time $T_2$ at the same rate, then

$$\boxed{\frac{X_1T_1}{Y_1}\ =\ \frac{X_2T_2}{Y_2}}.$$

$$\boxed{\frac{X_1T_1}{Y_1}\ =\ \frac{X_2T_2}{Y_2}}.$$

Example: If $5$ hens can lay $5$ eggs in $5$ days …

- how long will it take $10$ hens to lay $10$ eggs?
- how many hens can lay $10$ eggs in $10$ days?
- how many eggs will $10$ hens lay in $10$ days?

In this case of your problem:

the “producers” are the wall painters and “products” are painted walls. Substituting $X_1=7$, $Y_1=7$, $T_1=28$, $X_2=14$, $Y_2=14$ into the formula givesIt takes 28 minutes for 7 people to paint 7 walls.

How many minutes does it take 14 people to paint 14 walls?

$$\frac{7\cdot28}7\ =\ \frac{14\cdot T_2}{14}$$

$\implies\ T_2=28$ minutes. (In other words, it takes the same time for twice the number of people to do twice the amount of work – which makes sense, doesn’t it?)

Here is the proof of the formula above.

$X_1$ producers make $Y_1$ products in time $T_1$

$\implies$ $1$ producer makes $\dfrac{Y_1}{X_1}$ products in time $T_1$

$\implies$ $X_2$ producers make $\dfrac{X_2Y_1}{X_1}$ products in time $T_1$

$\implies$ $X_2$ producers make $\dfrac{X_2Y_1}{X_1T_1}$ products in time $1$

$\implies$ $X_2$ producers make $\dfrac{X_2Y_1T_2}{X_1T_1}$ products in time $T_2$.

That is to say,

$$Y_2\ =\ \frac{X_2Y_1T_2}{X_1T_1}$$

which can be rearranged to the formula above.