Question 53·Medium·Ratios, Rates, Proportional Relationships, and Units
A chemical process produces grams of a substance each second. After seconds, the mass produced is grams.
Which of the following expressions gives the mass, in grams, that will be produced in seconds?
For rate problems like this, first translate the situation into the basic formula "amount = (rate) × (time)." Identify how the time changes (for example, from to means multiplying by 7), then apply the same factor to the expression for amount. Look for an answer that multiplies the entire original expression by that factor, and be careful not to confuse "7 times " with "" or to add incompatible quantities like seconds and grams.
Hints
Focus on what means
comes from "rate × time." The rate is grams per second, and the time is seconds. Think about how this changes if the time changes.
Compare seconds and seconds
Is seconds just 7 more seconds than , or is it a multiple of ? Decide whether the new time is "" or "7 times ."
How should the mass change?
If you run the process for 7 times as long, should you add something to , or should you multiply by something? Look for an expression that is a constant multiple of .
Check units (grams vs. seconds)
Remember is in grams. Expressions that add a bare or may be mixing units (seconds) with grams, which is not correct.
Desmos Guide
Represent the original situation
In Desmos, type f(s) = 0.8s to represent the mass (in grams) after seconds.
Represent the mass after seconds
On the next line, type g(s) = 7*f(s) to represent 7 times the original mass expression, which corresponds to the mass after seconds.
Graph each answer choice as a function of s
Add four more lines for the choices, for example: A(s) = 0.8(s + 7), B(s) = 0.8s + 7, C(s) = 7(0.8s), and D(s) = 7s + 0.8s.
Compare with the correct relationship
Either use a table (click the gear icon and enable a table for each function) or look at the graphs. The correct choice will be the one whose values always match g(s) for every tested value of .
Step-by-step Explanation
Interpret the given expression
The process produces grams every second.
After seconds, the mass produced is given by the expression grams. This already uses the idea mass = (rate) × (time).
Relate seconds to seconds
The new time, seconds, is 7 times as long as seconds.
At a constant rate, if you run the process for 7 times as much time, you get 7 times as much mass.
So the mass after seconds should be 7 times the mass after seconds, meaning we need an expression that is 7 times (without yet writing it out explicitly).
Write and match the new expression
Seven times is written by multiplying 7 and :
- The expression is .
Looking at the choices, this matches option C, so the correct answer is .