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### Course: College Algebra>Unit 8

Lesson 2: Modeling with rational expressions

# Rational equation word problem

Sal models a context that concerns the number and price of pizza slices. The model turns out to be a rational equation. Created by Sal Khan.

## Want to join the conversation?

• at how do I know which set of numbers to add/subtract from each side? Real confused about it :/ and how do you know weather to add/subtract?
• It doesn't really matter. Just get everything on one side however you want to do it. Go back and view the "Quadratic equations" playlist before these, if you need review.
• In the upcoming practice section, Modeling with one-variable equations and inequalities, I stumbled across a problem which, given the following equation, required us to solve for t:

1024*(1/2)^(t/29)=32
solve for t.
The solution given in the hints went:
(1/2)^(t/29)=32/1024
(1/2)^(t/29)=1/32
(1/2)^(t/29)=(1/2)^5
therefore
t/29=5
t=29*5=145

Now, although I fully understand the logic when I see it, the passage that went from (1/2)^(t/29)=1/32 to (1/2)^(t/29)=(1/2)^5 would never have occurred to me on my own, not in a million years. Is there any previous section I should study harder, any method of handling bases and exponents I should assimilate perhaps, or am I just not smart enough to see what should be obvious at this stage? Thanks in advance for any answer.
• If you have a variable or constant in an exponent that you need to solve for, you only have a few options available to you at this level of study (there are some much more advanced techniques, but those a few years ahead of Algebra 2). And, actually, the two methods are really just different applications of the same method.

Method 1: use logarithms to bring the constant or variable out of the exponent, so you can solve for it.
Method 2: get both sides of the equation to be the same base raised to some exponent. If you can do that, then you know that the two exponents must be equal to each other.

You've seen method 2, but it is not always particularly easy to use (and is just a different way of doing method 1 anyway). So, let us look at how to do this same problem with method 1:
1024(½)^(t/29)=32
(½)^(t/29)=32/1024
(½)^(t/29)= 1 /32
log [(½)^(t/29)] = − log (32) ←This uses the property log (1/a) = − log (a)
(t/29) log (½) = − log (32) ← This uses the property log (a^b) = b log (a)
(t/29)[− log 2] = − log (32)
(t/29) = − log (32)] / (− log 2)
t/29 = 5
t = 29*5
t = 145
• Why is it that making more pizzas per day would result in her expenses not changing when the ingredients started costing more? Maybe I'm just overthinking it but I don't really get it.
• Her total expenses will go up, since the ingredients cost more and she is using more of them. But the expense per pizza is the same. Expense per pizza is (total expenses)/(number of pizzas). Total expenses have increased the numerator, so we must also increase the denominator to keep expense per pizza the same.
• A slow train traveling from Tashkent to Samarkand arrives 9 minutes late when traveling at 36km/h. If it travels at 27km/h it arrives 39 minutes late. What is the distance between Tashkent and Samarkand?
Help me, please, simulate this problem, if anyone can!
• Why wasn't she baking 8 more pizzas already if she had customers trying to buy 8 more pizzas? How is she selling these extra pizzas?
• That is more an economics question than a math question. With math problems, we don't usually assume the world of the problem is entirely logical.
• At , for the equation on the right (purple), can't it be simplified to 8+2?

My reasoning: 8+2(p+8)/p+8, the p+8 cancel out and left with 8+2.

Also at , does the coefficient must be 1?
• No becasue it is (8+2(p+8)/(p+8), so you are not canceling equivalents. If you broke this fraction down into 2 parts, it would be 8/p+8 + 2. This makes it worst than where we started.
To your second question, a coefficient of 1 can become invisible, so p^2 and 1 p^2 are actually the same thing. If you see a variable without a coefficient, get used to the idea that there is an invisible 1 in front that will almost never show up in final answers.
• why could we not solve this by only equating the before and after total cost?
8+1.5p = 8 +2(p+8)
why cant we solve for p with just the above equation?
• No, you can't just equate the above equations. This is because the denominators aren't the same.

Why are they fractions, anyway? Because they are rates. The problem is talking about price per pizza. So, the above equations represent the prices, and the bottom equations represent the amount of pizzas.

Hope this helps!
(1 vote)
• So if ingrident price increased why would she increase the amount of pizzas maid to maintain the same expense cost and not make less pizzas instead to keep the expense price the same.
• She just wants to get the same expense per pizza and not just to use less money.
(It is a math question so it might not make realistic sense)
(1 vote)
• Intuitively I feel like there should be a variable for # of days in this model. I know it would cancel out somehow, but I can't figure out where.
(1 vote)
• Since the calculations are for only one day, the number of days does not need to be included as part of the equation.