Linear equation word problems — Harder example
About this video
A tax problem with two brackets is solved by building one equation from the given information: 10 percent of the first 13,000 dollars, plus 15 percent of the income above 13,000, set equal to the 2,290 dollars of tax paid. The video works by ruling out choices that charge the wrong rate, such as applying 15 percent to total income, and it selects the expression that correctly models the bracket structure.
You'll be able to
- translate a two-bracket tax rule into an equation in terms of income x
- express income above a threshold as x minus the threshold
- rule out equations that apply a rate to the wrong base
Key words
- Tax bracket
- A band of income taxed at a particular rate, here 10 percent up to 13,000 dollars and 15 percent above it.
- Marginal rate
- The rate applied only to the income within a bracket, not to a person's whole income.
- Excess income
- The part of income above a threshold, written as x minus the threshold.
- Income x
- The unknown total income, which is the variable the equation is solved for.
Transcript
- We're told "In 2017, Celeste's income tax was $2,290. "Her income tax rate was 10% for the first $13,000 "of her income and 15% for her income in excess of $13,000. "Which of the following equations best models x, "Celeste's income in dollars in 2017?" So pause this video and see if you can work through this.
All right, now let's do it together. And rather than going choice by choice, I would just like to construct an expression that involves her income x, and then gets us to what her taxes should be and then equates that to $2,290. So they tell us that the first $13,000...
Or her income tax rate was 10% for the first $13,000 of her income. So one way to do that, we could say that 10% times $13,000. So I could write it like that, maybe. So that's going to be her first $13,000. And then it says 15% for her income in excess of $13,000.
So it's going to be plus 15% times her income in excess of $13,000. Well how could we express that? Well her income in excess of $13,000 would be x minus $13,000. And that all of that, this would be her total income tax, all of that should be equal to $2,290. Okay, let's see which of these choices describe that.
All right, this one is adding the income tax rates and then multiplying it times x, so it's really charging 25% of all the taxes then subtracting out $13,000. No, that would be... That is not going to be the same as that. Let's see, this is 15% of her total income and then 10% on the $13,000.
No, cause this isn't the 15% on the excess above $13,000. Rule that out. This is 10% on the first $13,000; and so this and this are definitely equivalent. And then 15% on her total income. No, it should be on the excess. So this one is not right, rule this one out.
Let's hope this one works. So 10% of the first $13,000, that's equivalent to that. Plus 15% of the excess. So that is equivalent to this. And that should be equal to $2,290. So I like that choice.
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