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## Digital SAT Math

### Course: Digital SAT Math > Unit 2

Lesson 2: Linear equation word problems: foundations# Linear equation word problems — Harder example

Watch Sal work through a harder Linear equation word problem.

## Want to join the conversation?

- What is income in excess?(49 votes)
- it is the excess income which is greater than 13000 so it will be x-13000 if x is total income(32 votes)

- The questions in the practice have different ideas and are harder..(32 votes)
- "q" ~Fark Fensly(0 votes)

- Is there a easier way this is kink of comusing.(5 votes)
- With this sort of problem you're pretty much supposed to turn or condense the word problem directly into a sort of equation/model--English into math. I guess there isn't really an easier way to do it. Practice helps a lot though (https://www.khanacademy.org/mission/sat/practice/math). If what you're looking for is another explanation for this particular problem:

The question here ("Which of the following equations best models x, Celeste's income in dollars in 2017?") is confusing as it's not really asking for a particular equation, but some sort of equation will be your answer. Modeling x means that the equation will operate on the variable x (income) using some of the info from the word problem, and end up with another bit of info from the word problem. Here, if you glance at the answers you'll see that all four end up with $2,290, the income tax, so you have to find out what else to use to equal that--the total income, tax rates, and the variable x. And essentially, the different answers are just trying to show mathematically how a certain income, x, was taxed a total of $2,290.

This particular problem says the tax rate for the first $13,000 out of the total income was 10%. 10% of $13,000 = (0.10)($13,000).

However, the total income was greater than $13,000 and the portion over $13,000 was taxed 15%. Since the total income is x, this portion (over $13,000) is x - $13,000. Apply the tax rate, and (0.15)(x - $13,000).

The two different parts of the total income, multiplied by their tax rates then added, equals the total income tax.

PartA(tax) + PartB(tax) = total income tax

(0.1)($13,000) + (0.15)(x - $13,000) = $2,290.(17 votes)

- My questions is why does the SAT uses sometimes 1.15 or 0.15 for percent. How do you know when to use each?(5 votes)
- 1.15 is larger than 1, and 0.15 is less than 1. So, we can use 1.15 when the value increases, and 0.15 when the value is decreasing. For example, if you have a diamond ring that increases in value by 15% every year, to find its actual value you take the 15% that it increased by, and add this to the original value of the ring. In effect, you're just multiplying by 1.15. To restate it, if you have a percent increase from a value, multiply by 1 + the percent, and otherwise just multiply by the decimal less than 1.(15 votes)

- please tell me why i should care about celeste's income tax. just commit tax fraud(11 votes)
- This is my take on the toddler-level explanation of this problem. Please correct me if I'm wrong and sorry for the bad grammar. Sorry if it's more hairy and not-so-clear! Sal did a great job explaining it in 2 minutes, I just wrote this because I think of this problem this way the first time I tried it before I continued to watch the video.

Information we get from the word problems:

• The total income tax is $2,290.

• The income tax rate was 10% (or 0.1 because 10% is 10/100) for the first $13,000 of her income.

• The income tax rate was 15% (or 0.15 because 15% is 15/100) for her income in excess of $13,000.

Let's say her income is x and her income in excess of $13,000 (it means that her income subtracted by $13,000) is y. Now let's write all of that again but with variables.

Information we get from the word problems:

• The total income tax is $2,290.

• Income tax for the first $13,000 of her income = 10% ⋅ (x - y) = 0.1 ⋅ (x - y).

Let's first make it clear why we multiply the rate to the amount:

Say your sister has 20 apples. If you were to ask your sister for half of those apples, it means that you are asking for 50% of the amount of apples that she has. 50% = 50/100 (to simplify it, divide both the numerator and the denominator by 50) = 1/2.

We all know (without calculating it) that half of 20 apples is 10 apples. So, mathematically, how do we get 10 apples if we have 20 apples and the rate of 1/2 (or 50%)? By multiplying the 20 by 1/2. It means that 1/2 of the apples = 1/2 ⋅ 20 = 10. Same goes with 10% and 15% in this case. What differs is that the rate of which we take the tax is different between the first $13,000 of her income and her income in excess of $13,000. However, in this apple example, the 50% rate is for the whole thing; notice that we don't separate the apples and just multiply the whole thing by 50%.

If you want to imagine an example that is more similar to the tax problem (where we separate the amount of things), let's say that your sister has 12 green apples and 8 red apples. If you were to ask your sister for 25% of the green apples, it would be 12 ⋅ 25% or 12 ⋅ 1/4 (because 25% is 25/100 and if we simplify it by dividing the numerator and the denominator by 25, it would be 1/4). It means you'll get 3 green apples. And if you were to ask your sister for 50% of the red apples, it means 8 ⋅ 50% = 8 ⋅ 1/2 = 4 red apples. So, the total number of apples you get is equal to the number of green apples you get and the number of red apples you get. Hence, the total number of apples you get = 3 + 4 = 7 apples.

Let's move back to the problem. If you were to ask, why does the first $13,000 of her income is denoted by (x - y)? Well, her total income (x) is equal to the first $13,000 of her income + her income in excess of $13,000 (y). So, the first $13,000 of her income is equal to her total income (x) - her income in excess of $13,000 (y).

Now, it means that her income in excess of $13,000 (y) is equal to her total income (x) - the first $13,000 of her income, which if we denote it with variables, it would be y = x - $13,000. Let's substitute y into "income tax for the first $13,000 of her income" equation.

=> Income tax for the first $13,000 of her income = 0.1 ⋅ (x - y).

=> Income tax for the first $13,000 of her income = 0.1 ⋅ (x - [x - $13,000]). Distribute the -1 because (x - [x - $13,000]) is the same as (x + -1⋅ [x - $13,000]).

=> Income tax for the first $13,000 of her income = 0.1 ⋅ (x - x + $13,000).

=> Income tax for the first $13,000 of her income = 0.1 ⋅ $13,000. Let's just ignore the dollar symbol for a second so that you can see that it is the exact same equation as Sal's equation.

=> Income tax for the first $13,000 of her income = 0.1 ⋅ 13,000.

• Income tax for her income in excess of $13,000 = 0.15 ⋅ y. We know y = x - $13,000. So, income tax for her income in excess of $13,000 = 0.15 ⋅ (x - $13,000).

Let's put it all together:

=> The total income tax = income tax for the first $13,000 of her income + income tax for her income in excess of $13,000.

=> $2,290 = (0.1 ⋅ $13,000) + (0.15 ⋅ [x - $13,000])

=> 2,290 = (0.1 ⋅ 13,000) + (0.15 ⋅ [x - 13,000])(10 votes) - but the question here is what if the income x was less than 13000 then the equation does not work. for example x=8000 we can calculate:

(0.1)(13000)+0.15(8000-13000)=1300-750=550 but, since x is less than 13000, we can use this expression:

(0.1)(8000)=800

so the option D is not correct. Is that right?

Please, I need help.(6 votes)- The question tells us to expect that the total income >13000. It specifically says that the income tax was calculated as 10% for the first 13,000$ of the income and 15% for the income which is greater than 13,000$. This wording tells us to assume that the income is greater than 13,000$, so option D would be the best answer. But, if the question doesn't mention this statement, then option D would be wrong.(3 votes)

- Does anyone know the formula for this specific problem? (I want to practice memorizing this formula for future problems.)(4 votes)
- I don't think there's a formula for this, since you're using what the problem says to make a formula on your own... Writing down the info you get from the question while solving might help.(4 votes)

- At1:27, i did not get the logic "x-13,000$" why can't it be "x+13,000$" ?(2 votes)
- We subtract 13000 instead of adding it because we're calculating the tax on the income in excess of, or greater than, $13,000. This means that we need to offset the "x" in that statement, as it only counts the portion of x that is greater than $13,000. How do we get that? Well, it's just x - $13,000.

You can verify this with a quick example. If you had $14,000, your income in excess of $13,000 would be $1,000. So subtracting $13,000 makes logical sense here, while adding just wouldn't. Creating hypothetical scenarios like that can be very helpful on SAT math, just to double check your work and make sure that equations do what they actually should.(6 votes)

- The parenthesis around 0.1 aren't necessary, right? Why is it that in option D we have parenthesis around 0.1 but not around 0.15?

Isn't the point of parenthesis to group operations together in order to define a clear order of operation? If so, why would they use parenthesis around a single term?(3 votes)- Parenthesis isn't necessary, you could say 0.1x0.15 for example, but writing 0.15(0.1) isn't wrong. Both of them mean the same thing. :)(3 votes)

## Video 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.