Linear equation word problems — Harder example | Math | SAT | Khan Academy

Watch Sal work through a harder Linear equation word problem.

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[jzyhere]

FOR BEGINNERS: Guys, he wrote x-13000 because x is supposed to be the total income, and in the question it's asked about the income in excess whichs not the actual income so if we subtract total total income from 13000 we get income in excess. and the reason he wrote 0.15 instead of 15% is because of the division. we can write 15% as 15/100 too.

[TasyaAdzkiya]

This is my take on another way to view the 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 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])

[loepesci]

He doesn't define what X is in the question so how do we assume that X is the excess income

[rellfull]

Why (x-13, 000), can anyone explain me?

[danialkhan]

Why we cannot denote x to excess income and make option c becomes right one