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

• MHB
• bp05528
In summary, it takes 28 minutes for 7 people to paint 7 walls. Using the formula for problems of this kind, it can be determined that it also takes 28 minutes for 14 people to paint 14 walls, as the amount of time is directly proportional to the number of people and walls. This can also be seen by applying simple logic. Additionally, if 10 hens can lay 10 eggs in 10 days, it would take 5 days for 5 hens to lay 5 eggs and 5 hens can lay 20 eggs in 10 days. This formula can be proven by breaking down the number of producers and products and using basic algebraic manipulation.
bp05528
It takes 28 minutes for 7 people to paint 7 walls.
How many minutes does it take 14 people to paint 14 walls?

bp05528 said:
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.

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:
bp05528 said:
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.

## 1. How do you calculate the rate for painting 14 walls in a certain amount of time?

The rate for painting 14 walls can be calculated by dividing the number of walls (14) by the time it takes to paint them. In this case, the rate would be 1 wall per person per minute.

## 2. How many people are needed to paint 14 walls in 30 minutes?

To calculate the number of people needed, divide the number of walls (14) by the time (30 minutes). In this case, it would take 14 people to paint 14 walls in 30 minutes.

## 3. How long will it take 8 people to paint 14 walls?

To calculate the time, divide the number of walls (14) by the number of people (8). In this case, it would take 1.75 minutes (or 1 minute and 45 seconds) for 8 people to paint 14 walls.

## 4. How does the rate change if the number of walls or people is increased or decreased?

The rate remains constant as long as the ratio of walls to people stays the same. For example, if the number of walls is increased to 20, but the number of people also increases to 20, the rate remains 1 wall per person per minute. However, if the number of people decreases to 10, the rate would increase to 2 walls per person per minute.

## 5. Is the rate affected by the size or complexity of the walls being painted?

The rate is not affected by the size or complexity of the walls being painted, as long as the number of walls and people remains the same. However, if the walls are larger or more complex, it may take longer for each person to paint their assigned wall, resulting in a longer overall time to paint all 14 walls.

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