Ratio Problem Technique 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 $3:x:4$. 
 A: 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.
A: 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.
A: $8$ 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.
A: Let's compare amount of work!
Painting 6 houses takes 8 people's work for 3 hours, i.e. 24 man-hours.
Painting X houses takes 3 people's work for 4 hours, i.e. 12 man-hours.
So $$X : 6 = 12 : 24$$ hence $$X = \frac {6\times 12} {24} = 3.$$
