Algebra Math Word Problem

[Kristina123]

A team of employees completed a large project. Their manager gave them a big thumbs up. Later they were recognized more tangibly with a monetary bonus to share among themselves. The ages of the team members are consecutive integers and no one on the team has the same age. The oldest team member is 45.

The bonus was paid out as follows:
(i) $1000 to the oldest member of the project team plus 1/10 of what remains, then
(ii) $2000 to the second oldest member of the project team plus 1/10 of what then remains, then
(iii) $3000 to the third oldest member of the project team plus 1/10 of what then remains, and so on.

After all of the bonus money had been distributed, each member of the project team had received the same amount. What is the age of the youngest member of the project team?

 

[Deleted member 4993]

A team of employees completed a large project. Their manager gave them a big thumbs up. Later they were recognized more tangibly with a monetary bonus to share among themselves. The ages of the team members are consecutive integers and no one on the team has the same age. The oldest team member is 45.

The bonus was paid out as follows:
(i) $1000 to the oldest member of the project team plus 1/10 of what remains, then
(ii) $2000 to the second oldest member of the project team plus 1/10 of what then remains, then
(iii) $3000 to the third oldest member of the project team plus 1/10 of what then remains, and so on.

After all of the bonus money had been distributed, each member of the project team had received the same amount. What is the age of the youngest member of the project team?

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[Dr.Peterson]

A team of employees completed a large project. Their manager gave them a big thumbs up. Later they were recognized more tangibly with a monetary bonus to share among themselves. The ages of the team members are consecutive integers and no one on the team has the same age. The oldest team member is 45.

The bonus was paid out as follows:
(i) $1000 to the oldest member of the project team plus 1/10 of what remains, then
(ii) $2000 to the second oldest member of the project team plus 1/10 of what then remains, then
(iii) $3000 to the third oldest member of the project team plus 1/10 of what then remains, and so on.

After all of the bonus money had been distributed, each member of the project team had received the same amount. What is the age of the youngest member of the project team?

One way to start would be to call the total amount x, and write an expression for the amount each of the first few will get. I would hope to see a pattern; if one were not obvious (and then provable), I would back up a bit and look for a pattern in how the expressions are formed.

We'll also want you to tell us the context of the problem. The fact that this is algebra is obvious; but are you learning about series (which kinds?) or some other topic that might be of use? We can't help effectively without knowing where you need help, and what knowledge we can assume as we help.

 

[Kristina123]

One way to start would be to call the total amount x, and write an expression for the amount each of the first few will get. I would hope to see a pattern; if one were not obvious (and then provable), I would back up a bit and look for a pattern in how the expressions are formed.

We'll also want you to tell us the context of the problem. The fact that this is algebra is obvious; but are you learning about series (which kinds?) or some other topic that might be of use? We can't help effectively without knowing where you need help, and what knowledge we can assume as we help.

I don't know how to answer the problem. I've tried converting the word problem into a math equation but don't know how to.

 

[tkhunny]

I don't know how to answer the problem. I've tried converting the word problem into a math equation but don't know how to.

You answered none of the questions or clarifications, above.

1) If you cannot learn to communicate you efforts at all, you're toast - not just in mathematics, but in life. Let's assume that's not the case. What have you tried? What did your efforts look like? Why do you think these efforts were less than productive?

2) As suggested, WRITE DOWN, "x = original sum of money."
If the first individual receives $1,000, how much money is left in the pot?
----- WRITE an expression that means, "The original sum of money less $1,000".
----- WRITE an expression that means, "1/10 of what remains".
----- WRITE an expression that means, "The amount of money remaining after the first is paid".

Get a handle on the process. We can go back and worry about the ages, later.

3) Can you set up an example in a spreadsheet? That might help you see more clearly. Maybe not. This depends on your skill at the spreadsheet.

 

[Kristina123]

It's so confusing. I don't even know if I'm doing it right. I don't know what the next step is.

 

[Romsek]

\(\displaystyle \text{The key to the problem is that they all get the same amount as a bonus}\)

\(\displaystyle \text{Suppose there is a total of $N$ dollars of bonus money}\\~\\
\text{The 45 yr old gets bonus $b_{45} = 1000 + \dfrac{(N-1000)}{10} = 900+\dfrac{N}{10}$}\\~\\

\text{The 44 yr old gets bonus $b_{44} = 2000 + \dfrac{(N-2000-b_{45})}{10}$}\\
b_{45} = 1800 + \dfrac{N}{10} - \dfrac{900+\frac{N}{10}}{10} = 1710 + \dfrac{9N}{100}\\~\\

\text{We're told that everyone's bonus is identical. Thus $b_{45}=b_{44}$}\\
900+\dfrac{N}{10} = 1710 + \dfrac{9N}{100}\\
\dfrac{N}{100}=810\\
N=81000
\)

\(\displaystyle b_{45}=b_{44} = 900 + \dfrac{81000}{10} = 9000\\
\text{Thus it must be that $\dfrac{81000}{9000} = 9$ employees got bonuses}\\
\text{These have ages 45 through 37}\\
\text{37 is the age of the youngest employee to get a bonus}
\)

At some point you just need to show the student how it's done.
There's no point in harassing them until they leave.

 

[Steven G]

\(\displaystyle \text{The key to the problem is that they all get the same amount as a bonus}\)

\(\displaystyle \text{Suppose there is a total of $N$ dollars of bonus money}\\~\\
\text{The 45 yr old gets bonus $b_{45} = 1000 + \dfrac{(N-1000)}{10} = 900+\dfrac{N}{10}$}\\~\\

\text{The 44 yr old gets bonus $b_{44} = 2000 + \dfrac{(N-2000-b_{45})}{10}$}\\
b_{45} = 1800 + \dfrac{N}{10} - \dfrac{900+\frac{N}{10}}{10} = 1710 + \dfrac{9N}{100}\\~\\

\text{We're told that everyone's bonus is identical. Thus $b_{45}=b_{44}$}\\
900+\dfrac{N}{10} = 1710 + \dfrac{9N}{100}\\
\dfrac{N}{100}=810\\
N=81000
\)

\(\displaystyle b_{45}=b_{44} = 900 + \dfrac{81000}{10} = 9000\\
\text{Thus it must be that $\dfrac{81000}{9000} = 9$ employees got bonuses}\\
\text{These have ages 45 through 37}\\
\text{37 is the age of the youngest employee to get a bonus}
\)

At some point you just need to show the student how it's done.
There's no point in harassing them until they leave.

To OP, In Romsek's 5th line there is a typo, it should be b₄₄, not b₄₅. I hope that everything written is now clear.