Fractions - Clarification (Betsy paints a wall in 2.5 hours. Amanda paints a wall...)

[spwittbold]

I have this math problem and was also provided the solution but I'm trying to get clarification on a particular part.

problem:

Betsy paints a wall in 2.5 hours. Amanda paints a wall in 1.25 hours.
How long does it take them to pain one wall if they paint together?

I understand that you can solve this by converting to fractions (Betsy 5/4) and Amanda (5/4).

The solution states this - and what it also states is the following:

If Betsy paints a wall in 5/2 an hour, then she paints 4/5 of a wall in 1 hour.
If Amanda paints a wall in 5/4 an hour, then she paints 4/5 a wall in 1 hour.

I just for some reason am not seeing why this holds true - can you always do this? simply invert the relationship?

i.e. is 5/2 of an hour the inverse of 2/5 of a wall in 1 hour?

The final solution is 50 minutes just in case someone needed.

 

[ksdhart2]

Well, first off I'm assuming you made a minor typo when you said "Betsy 5/4" as 2.5 is actually 5/2 or 10/4. That aside, your interpretation of the solution is correct, in that you can just flip the fraction and take its reciprocal to get the walls each woman paints in 1 hour. To give some insight as to why that works, let's examine what's actually going on. We'll create a variable, b, and let it be the amount of time, in hours, it takes Betsy to paint the wall. The problem text tells us that b = 2.5 = 5/2. From that, we can "clear" the denominator and see that Betsy can paint 2 walls in 5 hours, or 2b = 5. Now, we want to know how many walls Betsy can paint in 1 hour. In other words, we want to create an expression, in terms of b that equals 1. How do you think you might do that?

Another way to think about it is, rather than take the individual steps outlined above, we can go directly to an expression in terms of b that equals 1. To do that, we'd need to multiply both sides of the equation by some amount. Any time you're given a fraction, multiplying by its reciprocal will always give you 1 (excepting cases where the numerator is 0 of course, because you can't divide by 0). If you're unsure about that, maybe play around with it a bit until it "solidifies."

 

[Harry_the_cat]

I have this math problem and was also provided the solution but I'm trying to get clarification on a particular part.

problem:

Betsy paints a wall in 2.5 hours. Amanda paints a wall in 1.25 hours.
How long does it take them to pain one wall if they paint together?

I understand that you can solve this by converting to fractions (Betsy 5/4) and Amanda (5/4).

The solution states this - and what it also states is the following:

If Betsy paints a wall in 5/2 an hour, then she paints 4/5 of a wall in 1 hour.
If Amanda paints a wall in 5/4 an hour, then she paints 4/5 a wall in 1 hour.

I just for some reason am not seeing why this holds true - can you always do this? simply invert the relationship?

i.e. is 5/2 of an hour the inverse of 2/5 of a wall in 1 hour?

The final solution is 50 minutes just in case someone needed.

Think of another scenario.
If your car can go 50 miles on 1 gallon of fuel,
that means it will use \(\displaystyle \frac{1}{50}\) gallon to go one mile.

So you could state your fuel efficiency as 50 miles/gallon OR \(\displaystyle \frac{1}{50}\) gallons/mile. They both mean the same thing.

So, in general:

\(\displaystyle \frac{a}{b}\) unit1/unit2 is the same as \(\displaystyle \frac{b}{a}\) unit2/unit1.