# 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 $3:x:4$.

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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.

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Why don't you multiply (divide)through by all 3 numbers? Why only 2 at a time? – ton Dec 28 '12 at 15:11
@ton: Because I make fewer mistakes this way. If 8 people can make 8 cars in 8 days, it is not true that 1 person can make 1 car in 1 day; instead it is that the making of a car takes 8 person-days, i.e. that 1 person can make $\frac{1}{8}$ of a car in 1 day. – Henry Dec 28 '12 at 15:13
I meant why is it not $1$ person can paint $3/4$ of a house in $3/8$ hour. Or is that correct? I just divided by $8$. – ton Dec 28 '12 at 15:16
@ton: you can only apply the division/multiplication twice. In fact $1$ person can paint $3/4$ of a house in $3$ hours, or $1$ person can paint $3$ houses in $24$ hours (an eighth of the number of people so an eighth of the work done in the same time, or an eighth of the number of people so eight times as long to do the same amount of work) – Henry Dec 28 '12 at 15:26

$8$ 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.

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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.

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@Henry, sorry misread the question. Rectified. – lab bhattacharjee Dec 28 '12 at 15:19