January 2017 Common Core Algebra I Regents, Part 4

The following are the questions and answers (and commentary) for part of the New York State Algebra Regents exam. If you have any questions or comments (or corrections), please add them in the Comments section.
Part II was posted here. Part III was posted here.

January 2017, Algebra I (Common Core), Part IV

This question was worth 6 credits

37.Ian is borrowing $1000 from his parents to buy a notebook computer. He plans to pay them back at the rate of $60 per month. Ken is borrowing $600 from his parents to purchase a snowboard. He plans to pay his parents back at the rate of $20 per month.
Write an equation that can be used to determine after how many months the boys will owe the same amount.

Determine algebraically and state in how many months the two boys will owe the same amount.
State the amount they will owe at this time.

Ian claims that he will have his loan paid off 6 months after he and Ken owe the same amount.
Determine and state if Ian is correct. Explain your reasoning.

First part, the two equations:
Ian: y = 1000 - 60x, where x is the number of months and y is the remaining balance owed.
Ken: y = 600 - 20x.

They will be equal at 1000 - 60x = 600 - 20x

Second part: Solve the equation. Note, if you're equation is incorrect, solve it anyway because you will still get some credit!

1000 - 60x = 600 - 20x
1000 = 600 + 40x
400 = 40x
x = 10 months
y = 600 - 20(10) = 600 - 200 = $400 still owed.

Third part: Solve Ian's equation for y = 0, when no money is owed.

1000 - 60x = 0
1000 = 60x
x = 16.66666.... or 17 months.
Ian will not be paid off 6 months after he and Ken owe the same amount. He will still owe money.

Alternatively, six months after 10 is 16 months,
and y = 1000 - 60(16) = 1000 - 960 = 40. He will still owe $40 at 16 months, so he is not paid off.

End of Part IV

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