To celebrate a colleague's retirement, the T coworkers in an office agreed to share equally the cost of a catered...

1. Translate the problem requirements: We need to find the additional amount each remaining coworker must pay when some coworkers fail to contribute to a shared lunch cost
2. Calculate the original individual share: Determine how much each person was supposed to pay when all T coworkers were contributing
3. Determine the new payment structure: Figure out how much each remaining coworker must pay in total when only (T-S) people are paying
4. Find the additional burden: Calculate the difference between the new individual payment and the original individual payment

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what's happening in plain English. We have a group lunch situation where everyone was supposed to split the cost equally, but some people backed out and didn't pay. This means the people who do pay have to cover more than their original share.

Specifically, we need to find: How much EXTRA does each paying person have to contribute beyond what they originally planned to pay?

The key variables are:

• T = total number of coworkers originally agreeing to pay
• x = total cost of the lunch in dollars
• S = number of coworkers who fail to pay their share
• (T-S) = number of coworkers who actually end up paying

Process Skill: TRANSLATE - Converting the word problem into clear mathematical relationships

2. Calculate the original individual share

Let's think about this with a concrete example first. Suppose the lunch costs \(\$120\) and there are 10 coworkers total. Originally, each person was supposed to pay \(\$120 \div 10 = \$12\).

In general terms: When all T coworkers were supposed to pay equally, each person's share would be the total cost divided by the number of people.

Original individual share = \(\frac{\mathrm{x}}{\mathrm{T}}\) dollars per person

3. Determine the new payment structure

Now let's continue our example. Suppose 2 people back out, so only 8 people end up paying for the \(\$120\) lunch. Each of the 8 paying people must now cover \(\$120 \div 8 = \$15\).

In general terms: When only (T-S) coworkers actually pay, they must split the entire cost x among themselves.

New individual payment = \(\frac{\mathrm{x}}{\mathrm{T-S}}\) dollars per person

4. Find the additional burden

Back to our example: Each person was supposed to pay \(\$12\), but now must pay \(\$15\). The additional amount is \(\$15 - \$12 = \$3\) per person.

In general terms: The additional amount each remaining coworker must pay is the difference between what they actually pay and what they were originally supposed to pay.

Additional amount = New payment - Original payment

Additional amount = \(\frac{\mathrm{x}}{\mathrm{T-S}} - \frac{\mathrm{x}}{\mathrm{T}}\)

To subtract these fractions, we need a common denominator of T(T-S):

\(\frac{\mathrm{x}}{\mathrm{T-S}} = \frac{\mathrm{xT}}{\mathrm{T(T-S)}}\)

\(\frac{\mathrm{x}}{\mathrm{T}} = \frac{\mathrm{x(T-S)}}{\mathrm{T(T-S)}}\)

Additional amount = \(\frac{\mathrm{xT}}{\mathrm{T(T-S)}} - \frac{\mathrm{x(T-S)}}{\mathrm{T(T-S)}}\)

Additional amount = \(\frac{\mathrm{xT - x(T-S)}}{\mathrm{T(T-S)}}\)

Additional amount = \(\frac{\mathrm{xT - xT + xS}}{\mathrm{T(T-S)}}\)

Additional amount = \(\frac{\mathrm{xS}}{\mathrm{T(T-S)}}\)

Process Skill: MANIPULATE - Carefully working with fractions to find the difference

Final Answer

The additional amount each remaining coworker must contribute is \(\frac{\mathrm{Sx}}{\mathrm{T(T-S)}}\) dollars.

Let's verify with our example: S=2, x=120, T=10, so (T-S)=8

Additional amount = \(\frac{2 \times 120}{10 \times 8} = \frac{240}{80} = \$3\) ✓

This matches answer choice D: \(\frac{\mathrm{Sx}}{\mathrm{T(T-S)}}\)

Common Faltering Points

Errors while devising the approach

1. Misinterpreting what "additional amount" means

Students often think the question is asking for the total amount each remaining person pays (\(\frac{\mathrm{x}}{\mathrm{T-S}}\)), rather than the EXTRA amount beyond their original share. This leads them to select answer choice B instead of working toward the correct difference calculation.

2. Confusion about who pays what

Students may incorrectly assume that the S people who don't pay still need to be accounted for in the final payment structure, or conversely, forget that the original agreement involved all T people. This can lead to setting up the wrong fractions from the start.

3. Not recognizing this as a "difference of fractions" problem

Students might try to solve this using proportions or ratios instead of recognizing that they need to find: (new individual payment) - (original individual payment). This conceptual error leads to completely different solution approaches.

Errors while executing the approach

1. Common denominator errors when subtracting fractions

When calculating \(\frac{\mathrm{x}}{\mathrm{T-S}} - \frac{\mathrm{x}}{\mathrm{T}}\), students frequently make mistakes finding the common denominator T(T-S). They might incorrectly use T or (T-S) as the common denominator, or make errors when converting each fraction.

2. Algebraic simplification mistakes

After getting \(\frac{\mathrm{xT - x(T-S)}}{\mathrm{T(T-S)}}\), students often make errors when distributing and combining like terms. Common mistakes include: xT - x(T-S) = xT - xT - xS (wrong sign) instead of xT - xT + xS, leading to incorrect final expressions.

3. Rushing through fraction arithmetic

Students may correctly set up \(\frac{\mathrm{x}}{\mathrm{T-S}} - \frac{\mathrm{x}}{\mathrm{T}}\) but then make calculation errors due to working too quickly with the algebraic manipulation, especially when dealing with multiple variables simultaneously.

Errors while selecting the answer

1. Confusing similar-looking answer choices

Answer choices C (\(\frac{\mathrm{Sx}}{\mathrm{T-S}}\)) and D (\(\frac{\mathrm{Sx}}{\mathrm{T(T-S)}}\)) look very similar. Students who get the right numerator (Sx) but forget about the T in the denominator often select C instead of D, especially if they don't verify their answer with a concrete example.

2. Not double-checking with the verification example

Students may arrive at the correct algebraic expression but fail to substitute their example values (S=2, x=120, T=10) to confirm their answer matches the expected result of \(\$3\). This verification step could catch errors in their algebraic work.

Alternate Solutions

Smart Numbers Approach

This problem can be solved efficiently using smart numbers by choosing convenient values for the variables.

Step 1: Choose Smart Numbers

Let's select values that will make our calculations clean:

• Total coworkers: T = 10
• Total lunch cost: x = \(\$120\) (chosen because it's divisible by 10)
• Coworkers who fail to pay: S = 2

Step 2: Calculate Original Individual Share

Originally, each of the 10 coworkers was supposed to pay:

Original share per person = \(\$120 \div 10 = \$12\)

Step 3: Determine New Payment Structure

After 2 coworkers fail to pay, only (10 - 2) = 8 coworkers remain to cover the full cost.

New share per remaining person = \(\$120 \div 8 = \$15\)

Step 4: Calculate Additional Amount

Additional amount each remaining coworker must pay:

Additional amount = New share - Original share = \(\$15 - \$12 = \$3\)

Step 5: Verify Against Answer Choices

Using our smart numbers in each answer choice:

1. \(\frac{\mathrm{x}}{\mathrm{T}} = \frac{120}{10} = 12\) ❌
2. \(\frac{\mathrm{x}}{\mathrm{T-S}} = \frac{120}{10-2} = \frac{120}{8} = 15\) ❌
3. \(\frac{\mathrm{Sx}}{\mathrm{T-S}} = \frac{2 \times 120}{10-2} = \frac{240}{8} = 30\) ❌
4. \(\frac{\mathrm{Sx}}{\mathrm{T(T-S)}} = \frac{2 \times 120}{10 \times (10-2)} = \frac{240}{10 \times 8} = \frac{240}{80} = 3\) ✓
5. \(\frac{\mathrm{x(T-S)}}{\mathrm{T}} = \frac{120 \times (10-2)}{10} = \frac{120 \times 8}{10} = 96\) ❌

Answer choice D matches our calculated result of \(\$3\).

Conceptual Verification

The formula \(\frac{\mathrm{Sx}}{\mathrm{T(T-S)}}\) makes intuitive sense:

• S×x represents the total unpaid amount (2 people × \(\$120\) = unpaid portion)
• T(T-S) represents the original number of people times the remaining people
• This distributes the unpaid portion among the remaining contributors