Question 27·Hard·Ratios, Rates, Proportional Relationships, and Units
A machine fills identical bottles with a total of liters of liquid in minutes. The machine operates at a constant rate. Which expression represents the number of minutes it would take the machine to fill bottles of the same size?
For constant-rate word problems like this, avoid getting lost in the symbols: focus on units and proportionality. First, decide what single unit you want (here, minutes per bottle), then compute it by dividing total time by number of bottles. Next, scale that unit up to the new quantity (multiply time per bottle by the new number of bottles). Finally, simplify the algebraic fraction—factoring common terms in the denominator—so you can match it cleanly to one of the answer choices. Keeping track of whether a variable should be in the numerator or denominator by checking how time should change when that variable increases is a quick sanity check.
Hints
Use the idea of a constant rate
Since the machine operates at a constant rate and all bottles are identical, ask yourself: if you know the time for a certain number of bottles, how can you find the time for one bottle?
Compute minutes per bottle
You are told the machine fills bottles in minutes. What do you get if you divide the total time by the number of bottles?
Scale up to bottles
Once you know the time for one bottle, how can you use that to find the time for bottles? After you write that expression, simplify the fraction by factoring the denominator.
Desmos Guide
Assign sample values to the variables
In Desmos, pick convenient positive values for the variables that make the denominators nonzero, for example:
- Type
p = 2 - Type
r = 3 - Type
q = 10(any positive value works, since it will cancel)
You can instead create sliders for , , and if you like.
Compute the actual time for bottles from the situation
Using the problem setup, enter the step-by-step time expression:
- First, time per bottle:
t_per = 4r/(6p - 2) - Then, time for bottles:
t_ref = t_per*(9p + 1)
This t_ref is the “reference” time calculated directly from the word problem for your chosen values of and .
Compare each answer choice numerically
In Desmos, define each option using the same and values:
A = 4r*(9p + 1)/(6p + 2)B = (9p + 1)/(2r*(3p - 1))C = (3p - 1)*(9p + 1)/(2r)D = 2r*(9p + 1)/(3p - 1)
Compare the numerical values of A, B, C, and D with t_ref. The expression whose value always matches t_ref (for any reasonable and you try) is the correct choice.
Step-by-step Explanation
Relate time to number of bottles
The machine runs at a constant rate and all bottles are the same size. That means:
- More bottles of the same size require more time in direct proportion.
- So, time per bottle is constant, and
- total time = (time per bottle)(number of bottles).
Find the time per bottle from the given information
We are told that the machine fills bottles in minutes.
So the time per bottle is
Write an expression for the time to fill bottles
To fill bottles, multiply the time per bottle by :
This matches the situation but is not yet simplified to match an answer choice.
Simplify the expression to match an answer choice
Simplify the denominator of
Factor 2 from the denominator:
Substitute this in and simplify:
So the correct expression for the number of minutes is
which corresponds to choice D.