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PSAT Math Practice: Word Problems & Multiple Equations

While solving systems of equations can be relatively straightforward once you get the hang of it, sometimes you’ll encounter a complex word problem and need to translate it into a system of equations and then solve. It sounds a lot scarier than it actually is. Remember to use the Kaplan Strategy for Translating English into Math to set up your equations, and then solve using either substitution or combination.

 

PSAT Math Practice Question: Word Problems with Multiple Equations

At a snack stand, hot dogs cost $3.50 and hamburgers cost $5. If the snack stand sold 27 snacks and made $118.50 in revenue, how many hot dogs were sold? How many hamburgers?

A. 16 hot dogs; 11 hamburgers

B. 16 hot dogs; 16 hamburgers

C. 11 hot dogs; 14 hamburgers

D. 11 hot dogs; 16 hamburgers

 

 

Work through  the Kaplan Method for Math to solve this question step-by-step. The following table shows Kaplan’s strategic thinking on the left, along with suggested math scratchwork on the right.

 

Strategic Thinking Math Scratchwork
Step 1: Read the question, identifying and organizing important information as you go
You need to find the number of hot dogs and hamburgers sold.
Step 2: Choose the best strategy to answer the question
Use the Kaplan Strategy for Translating English into Math.
What variables should you use?
Because both snacks start with h, that’s likely to be a confusing choice. Instead, use d for hot dogs and b for hamburgers.
How do you break apart the question into smaller phrases?
Break off each piece of relevant information into a separate phrase.
What should you do with the phrases?
Translating each phrase into a math expression will create the components of a system of equations.
What is the system of equations that will get you to the answer?
Assemble the math expressions you have to get the system of equations needed.
How can you best solve this system of equations?
In this case, you can use either substitution or combination to arrive at the numbers of hot dogs and hamburgers. To use combination, multiply the first equation by –5 to set it up:
Continuing to solve, you see that d = 11.
What does this information enable you to do?
That immediately eliminates A and B. You can plug 11 back into the first equation to get b.
  d = hot dogs sold b = hamburgers sold  
hot dogs cost $3.50 → 3.5d
hamburgers cost $5 → 5b
snack stand sold 27 snacks → d + b = 27
made $118.50 in revenue → Total $ = 118.5 d + b = 27 3.5d + 5b = 118.5   −5d − 5b = −135+   3.5d + 5b = 118.5 −1.5d + 0b = −16.5                   
d = 11  
d + b = 27
11 + b = 27
b = 16
Step 3: Check that you answered the right question The only answer choice that meets these criteria is (D).

Always choose variable names that make sense to you. Countless students struggle on multi-part problems due to disorganized notes. Don’t let that happen to you. Move beyond x and y when selecting variable names.

Other questions of this type will simply ask you to choose from a series of answer choices that describes the system of equations—they won’t actually ask you to calculate a solution! These questions can be great time-savers.