Game Show Math Brainteaser (8th Grade Level)

[Otis]

On a game show, boxes sit on elevated platforms above the stage. Each game begins and ends on the stage. Each platform has its own lift, for transporting contestants between stage and box. Each box contains either a dollar sign or nothing at all. A contestant must pay the show $20 before riding a lift up or down.

If a contestant opens a box containing a dollar sign, then the show immediately triples the amount of money they had when they opened the box. For example, if a contestant had $50 when opening a box with a dollar sign in it, then they receive $100 for a total of $150 -- which is triple what they had when opening the box.

Wendy begins the game with a certain amount of money, and she chooses to visit three platforms. Fortunately, she finds a dollar sign in each box, but unfortunately she has just $13 after finishing the game. How much money did Wendy start with?

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[imath]\;[/imath]

 

[AvgStudent]

I skipped 8th grade, so this problem is a challenge for me ?.
Why is it UNFORTUNATE that Wendy won 3 rounds?

 

[Dr.Peterson]

I skipped 8th grade, so this problem is a challenge for me ?.
Why is it UNFORTUNATE that Wendy won 3 rounds?

Because it costs money to go up to get money, and it costs again to get back down. She has less than she started with, even though she did the best she could. Don't play this game unless you can afford it!

(By the way, that money to get back down explains my initial confusion: why $13 is not a multiple of 3.)

Here's an interesting follow-up question: How much do you need to start with in order to gain money if you win all three times?

Answers:

Spoiler

Solving 3(3(3(x-20)-40)-40)-20 = 13, the answer to the problem is x=39; if you start with $40, you break even, and with more you gain.

You can also solve this without algebra, by working backwards, very carefully. I drew a diagram of three trips up and down, and wrote in how much she had at each point, starting with the $13 at the end.

 

[Steven G]

I skipped 8th grade, so this problem is a challenge for me ?.
Why is it UNFORTUNATE that Wendy won 3 rounds?

It say that unfortunately she has just $13 after finishing the game. It never said that it was unfortunate that she won 3 games--although it was!
Actually it Fortunately, she finds a dollar sign in each box.

 

[Otis]

follow-up question: How much do you need to start with in order to gain money if you win all three times?

I'll look at your spoiler, right after I post my answer to the follow-up question:

Spoiler

$40.01 (gains 27¢)

27x - 1040 > x

For each penny above $40 that Wendy starts with, she would gain 27¢.

[imath]\;[/imath]

 

[Otis]

this problem is a challenge for me

Excellent. Are you up for the challenge?

Let x = starting dollar amount

What expression could you write to represent the amount Wendy has when she opens the first box?

Then, what happens to that amount?

Continue ...

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[imath]\;[/imath]

 

[AvgStudent]

Excellent. Are you up for the challenge?

Let x = starting dollar amount

What expression could you write to represent the amount Wendy has when she opens the first box?

Then, what happens to that amount?

Continue ...

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[imath]\;[/imath]

Because it costs money to go up to get money, and it costs again to get back down. She has less than she started with, even though she did the best she could. Don't play this game unless you can afford it!

(By the way, that money to get back down explains my initial confusion: why $13 is not a multiple of 3.)

Here's an interesting follow-up question: How much do you need to start with in order to gain money if you win all three times?

Answers:

Spoiler

Solving 3(3(3(x-20)-40)-40)-20 = 13, the answer to the problem is x=39; if you start with $40, you break even, and with more you gain.

You can also solve this without algebra, by working backwards, very carefully. I drew a diagram of three trips up and down, and wrote in how much she had at each point, starting with the $13 at the end.

I did get the same answer, but I was confused because a logical person would calculate that before playing the game. So I didn't understand why Wendy would participate knowing that she'll lose money.

 

[Otis]

I was confused because a logical person would [have determined they'd lose money] before playing the game. So I didn't understand why Wendy would participate

Ah, but the puzzle never told us that Wendy is a logical person.

With brainteasers in general, readers are expected to accept the givens at face-value. A lot of math puzzles are formed by starting with a mathematical relationship or pattern and then fabricating some "real-world" scenario around it, to create context. That's more interesting than a collection of arithmetic statements about numbers A, B, C, D and E, I suppose.

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[AvgStudent]

Ah, but the puzzle never told us that Wendy is a logical person.

With brainteasers in general, readers are expected to accept the givens at face-value. A lot of math puzzles are formed by starting with a mathematical relationship or pattern and then fabricating some "real-world" scenario around it, to create context. That's more interesting than a collection of arithmetic statements about numbers A, B, C, D and E, I suppose.

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Fair enough?