Ratio Problem Technique

Question

Suppose $8$ people can paint $6$ houses in $3$ hours. How many houses can $3$ people paint in $4$ hours?

So it seems that $1$ person can paint $3/4$ of a house in $3/8$ of an hour. Then this implies that $3$ people can paint $9/4$ of a house in $9/8$ of an hour. Is there any easy way to convert this to the desired result?

Or maybe we should look at the fixed ratios: $8:6:3$ versus $3:x:4$.

Answer 1

The dreaded 19th century "rule of three".

• $8$ people can paint $6$ houses in $3$ hours
• $8$ people can paint $2$ houses in $1$ hour
• $1$ person can paint $\frac{1}{4}$ of a house in $1$ hour
• $1$ person can paint $1$ house in $4$ hours
• $3$ person can paint $3$ houses in $4$ hours

Aim for the middle bullet and the rest becomes easy.

Answer 2

8 people in 3 hours, can paint 6 houses.

8 people in 1 hours, can paint $\frac63$ houses. (the number of houses(quantity of task) is directly proportional to the time).

1 people in 1 hour can paint $\frac6{3\cdot8}$ houses(the number of houses(quantity of task) is directly proportional to the man-power)

3 people in 4 hours in can paint $\frac{6\cdot 3\cdot 4}{8\cdot3}=3$ houses.

Answer 3

$8$ people can paint $6$ houses in $3$ hours.

$\iff$ Hence, $8$ people can paint $2$ houses in an hour.

$\iff$ Hence, $4$ people can paint a house in an hour.

$\iff$ Hence, $1$ person can paint a quarter in an hour.

$\iff$ Hence, $3$ persons can paint $\dfrac{3}{4}$ in an hour.

$\iff$ Hence, $3$ people can paint $\underline{}$ houses in $4$ hours.

This approach is logically easy, just keep going step-by-step as shown.