Just a Quick Note on Word Problem Vocabulary

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Hi Everyone:

Most of you probably know how to convert English words to math symbols. For example, “gained,” “got,” “increased” all mean “add” and “of” means multiply.” For a nice review of these ideas, see Section 6-H “Word Problems” of the Barron’s New SAT review book, or click here.

However, standardized test (ACT, SAT, PSAT, SSAT, etc.) problem writers are usually adults who have had more life experience than the average high school student, so they may use vocabulary that most high school students aren’t familiar.  I stumbled into this realization when I was working with a 16-year-old student who got a question about “net pay” and “gross pay” wrong.

The problem was something like “During his second week on the job, Jim earned $160.00 net pay after a deduction of 18% for payroll taxes and deductions, and an additional deduction of 22% for withholding taxes. Those are the only deductions from his pay. Jim’s gross pay for his second week on the job was 100% more than his first week’s gross pay. How much was Jim’s first-week gross pay?”

The student was confused about the difference between net and gross pay, and miscalculated as a result. He let me know that he wasn’t familiar with how deductions were taken from paychecks, since he’d never had a job with actual payroll checks before. I started my first such job shortly before my 18th birthday, so I understand.

“Gross pay” is the amount of money you made, which you can calculate by multiplying the number of hours you worked by your hourly pay rate*. “Net pay” is the amount written your paycheck, or more likely, directly deposited into your bank account. In other words, it’s your “take-home pay.” Your net pay is almost always less than your gross pay, since you almost certainly have taxes withheld by the federal, state, and/or local government, you may pay for health care or retirement plans, or may pay for other benefits (e.g., transit passes) from your gross pay.  So the net pay is the pay after ALL deductions are made.

How does that work in the problem of Jim’s pay, above? It’s simple.
We know that Jim’s second-week net pay was $160.00, and his only deductions are 22% + 18% = 40%.
So $160 = (Gross)(0.6) -->  Gross = $160/0.6 = $266.67.

Since $266.67 is a 100% increase over the first week’s gross pay, we have to use the percent increase/decrease formula, which is [(New Value – Old Value)/(Old Value)] * 100%

We can write this as (SW – FW)/FW * 100% = 100%, where SW is the second week’s gross pay, and FW is the first week’s gross pay. In this case, ($266.67 – FW)/FW * 100% = 100%.
Dividing by 100%, we get ($266.67 – FW)/FW = 1 , which leads to $266.67 – FW = FW,
which gives us $266.67 = FW, so FW = $133.34 (after rounding up). There's your answer. :-) 

I hope this helps, and good luck if you’re taking the June SAT. :-)  

*Note: You can also earn 1.5 times your rate for any hours over 40 worked in a week in all US jurisdictions, and sometimes even when you work more than an 8-hour day in some areas – this is not legal advice and I am not your lawyer- check with an attorney or your state labor department if you need legal advice regarding wage and hour law.

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