Question 26·Medium·Linear Equations in Two Variables
During a charity run, participants could register for either a 5-kilometer race or a 10-kilometer race. The registration fee for the 5-kilometer race is dollars and the fee for the 10-kilometer race is dollars. If 18 people registered for the 5-kilometer race and 12 people registered for the 10-kilometer race, and together they paid $780 in registration fees, which equation represents this situation?
For word problems that ask for an equation, first clearly label what each variable stands for, then write a separate expression for each part of the situation (for example, “number of items” × “price per item” for each group). Finally, add these parts together and set the sum equal to the given total. On the SAT, avoid guessing from the answer choices’ appearance; instead, systematically match each coefficient to the quantity it should represent in the story.
Hints
Match the numbers in the story to the variables
Which numbers in the word problem tell you how many people registered for each race, and which symbols represent how much each person in that race pays?
Write the total from each race separately
For the 5-kilometer race, 18 people each pay dollars. How do you write the total money from that race as an algebraic expression? Do the same for the 10-kilometer race with 12 people and dollars.
Combine the totals and use the overall amount
Once you have expressions for the money from each race, remember that together they add up to $780. How do you write an equation that shows “money from 5-km race + money from 10-km race = 780”?"}],
Compare with the story’s total for your test values
Using your chosen and values, compute the total registration money based on the story: 18 people pay and 12 people pay . Then see which answer choice’s expression in Desmos gives the same total for any reasonable and you try; that option matches the situation.
Desmos Guide
Represent the fees with sliders
In Desmos, type f = 10 and g = 20, then click the slider icons so that and become sliders. You can adjust these to test different possible fee amounts for the two races.
Test each answer choice as an expression
For each option, type just its left-hand side into Desmos (for example, for choice A type 12f + 18g, for choice B type 30f + g, etc.). For a chosen pair of and values, Desmos will show the numerical output of each expression.
Compare with the story’s total for your test values
Using your chosen and values, compute the total registration money based on the story: 18 people pay and 12 people pay . Then see which answer choice’s expression in Desmos gives the same total for any reasonable and you try; that option matches the situation.
Step-by-step Explanation
Identify what the variables represent
From the problem:
- is the registration fee (in dollars) for the 5-kilometer race.
- is the registration fee (in dollars) for the 10-kilometer race.
Also, there are 18 people in the 5-kilometer race and 12 people in the 10-kilometer race, and the total money collected is $780.
Write an expression for the money from each race
Money from each group is:
- 5-kilometer race: 18 people, each paying dollars, so this group pays dollars total.
- 10-kilometer race: 12 people, each paying dollars, so this group pays dollars total.
These are the two parts that must be added together to get the total registration money.
Add the parts and write the equation
The total money collected is the sum of the amounts from the two races, and we are told this total is dollars.
So we add the two expressions and set them equal to :
This is the equation that represents the situation, so the correct choice is .