#19-12: During the rainy season, 4/9 of a foot of rain was collected in 2/3 of a day. At that rate, how much rain would be collected in one day?

Accepted Solution

Answer:2/3 ft rainStep-by-step explanation:You could setup a proportion:2/3 day gives you 4/9 ft of rain 1     day gives you    x  ft of rainThis information is organized days in 1 column with rain fall in another column.  Even the rows are organized.  The 4/9 rain goes with the 2/3 day and the x rain goes with the 1 day.I organized is so it is easy to setup the proportion:[tex]\frac{\frac{2}{3}}{1}=\frac{\frac{4}{9}}{x}[/tex]Cross multiply:[tex](\frac{2}{3})x=(\frac{4}{9})(1)[/tex]Simplifying right hand side:[tex]\frac{2}{3}x=\frac{4}{9}[/tex]To solve this equation we can multiply the reciprocal of the coefficient of x on both sides to get x by itself:[tex]\frac{3}{2}\cdot\frac{2}{3}x=\frac{3}{2}\cdot\frac{4}{9}[/tex]The reason we do that is because when you multiply reciprocals you get 1. We wanted to obtain x by itself.  We got 1x which is still just x:[tex]1x=\frac{3}{2} \cdot \frac{4}{9}[/tex][tex]x=\frac{3}{2} \cdot \frac{4}{9}[/tex]Now we can put this last part in your calculator.You could multiply the fractions by multiplying straight across on top and bottom. Then reduce.You could actually reduce then multiply. I'm going to do the last option.3/9=1/3 and 4/2=2 so we have:[tex]x=\frac{3}{9} \cdot \frac{4}{2}[/tex][tex]x=\frac{1}{3} \cdot 2[/tex][tex]x=\frac{2}{3}[/tex]There are other ways people like to think about this.Maybe you are the type who likes this:2/3 day ->4/9 ft rainThe remaining part of the day is 1/3.Half of 2/3 is 1/3 since 1/3+1/3=2/3.So if I half both sides I get:1/3 day=2/9 ft rainSo (2/3+1/3) day=1 day implies we have (4/9+2/9) ft rain.Simplifying:1 day=6/9 ft rainSimplifying the 6/9 by dividing top and bottom by 3 gives you:1 day=2/3 ft rainYou could also think about units rate here because they are asking for one since they want to know what happens in 1 day.So you do (4/9) divided by (2/3).You can change division to multiplication by flipping the 2nd number (the one after the division sign):(4/9) divided by (2/3)(4/9)  times         (3/2)12/1812 and 18 have a common factor of 6 so divide top and bottom by 6:2/3So this unit rate says you get 2/3 ft rain per day.We are looking for the rainfall for one day which would be 2/3 ft here.