   SEARCH HOME Math Central Quandaries & Queries  Question from Kenneth: Hello: Here is my question: Four carpenters can build eight houses in 10 days. Two carpenters can build how many houses in 15 days? I have determined the solution to the above by using the following calculation; (2 X 15 X 8)/(4 X 10) that is divide (2 X 15 X 8) by (4 X 10) Answer: 6 houses Can someone explain why or how this solution will provide the answer? What is the mathematical reasoning for this calculation providing the missing amount? I thank you for your reply. Hi Kenneth,

First let me show you how I would solve this problem and then try to explain why the "formula you used" gives the same solution.

4 carpenters can build 8 houses in 10 days.

If you have half as many carpenters they can only build half as many houses in 10 days. Hence

2 carpenters can build 4 houses in 10 days.

If these 2 carpenters only have 5 days they can only build half as many houses. Thus

2 carpenters can build 2 houses in 5 days.

But if they have 15 days then can build 3 times as many houses so

2 carpenters can build 6 houses in 15 days.

Look at the number of houses in the 4 statements. The number of houses started at 8, then I multiplies by $\large \frac12.$ I then multiplied by $\large \frac12$ again and finally I multiplies by $3.$ Thus my calculation was

$8 \times \frac12 \times \frac12 \times3.$

But $\large \frac24 = \frac12$ and $\large \frac{15}{10} = \frac12 \times 3.$ so my calculation can be written

$8 \times \frac24 \times \frac{15}{10} = \frac{8 \times 2 \times 15}{4 \times 10}$

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.