MHB Rate Problem: How many minutes does it take 14 people to paint 14 walls?

[bp05528]

It takes 28 minutes for 7 people to paint 7 walls.
How many minutes does it take 14 people to paint 14 walls?

 

[skeeter]

It takes 28 minutes for 7 people to paint 7 walls.
How many minutes does it take 14 people to paint 14 walls?

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.

 

[Olinguito]

Hi bp05528.

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

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?

Answers: (a) $5$ days, (b) $5$ hens, (c) $20$ eggs. You can either work the answers out by simple logic, or use the formula above, where the “producers” are hens and the “products” are eggs.

In this case of your problem:

It takes 28 minutes for 7 people to paint 7 walls.
How many minutes does it take 14 people to paint 14 walls?

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 gives
$$\frac{7\cdot28}7\ =\ \frac{14\cdot T_2}{14}$$
$\implies\ T_2=28$ 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.