If N = K/(T + x/y), where T = K/5 and x = 5 - T, which of the following...

1. Translate the problem requirements: We need to express y in terms of N and T by working backwards from the given equation \(\mathrm{N = K/(T + x/y)}\), using the relationships \(\mathrm{T = K/5}\) and \(\mathrm{x = 5 - T}\)
2. Substitute known relationships: Replace K and x in the original equation using the given relationships to eliminate variables we don't need
3. Isolate the fraction containing y: Rearrange the equation to get x/y by itself on one side
4. Solve for y: Since we know what x equals in terms of T, we can substitute and solve for y in terms of N and T

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by understanding what we have and what we need to find. We're given an equation with several variables, and we need to rearrange it to express one specific variable (y) in terms of two others (N and T).

We have:
• Main equation: \(\mathrm{N = K/(T + x/y)}\)
• Relationship 1: \(\mathrm{T = K/5}\)
• Relationship 2: \(\mathrm{x = 5 - T}\)

Our goal is to find y in terms of N and T only.

Think of this like solving a puzzle - we have three pieces of information, and we need to combine them cleverly to get what we want.

Process Skill: TRANSLATE - Converting the problem requirements into a clear mathematical goal

2. Substitute known relationships

Now let's eliminate the variables we don't want (K and x) by using our given relationships.

From \(\mathrm{T = K/5}\), we can find that \(\mathrm{K = 5T}\).
We also know that \(\mathrm{x = 5 - T}\).

Let's substitute \(\mathrm{K = 5T}\) into our main equation:
\(\mathrm{N = K/(T + x/y)}\)
\(\mathrm{N = 5T/(T + x/y)}\)

Next, let's substitute \(\mathrm{x = 5 - T}\):
\(\mathrm{N = 5T/(T + (5-T)/y)}\)

Now our equation only contains N, T, and y - exactly what we want!

3. Isolate the fraction containing y

To solve for y, we need to rearrange this equation step by step. Let's work backwards from N to isolate the part with y.

Starting with: \(\mathrm{N = 5T/(T + (5-T)/y)}\)

First, let's flip both sides (take reciprocals):
\(\mathrm{1/N = (T + (5-T)/y)/(5T)}\)

Multiply both sides by 5T:
\(\mathrm{5T/N = T + (5-T)/y}\)

Subtract T from both sides:
\(\mathrm{5T/N - T = (5-T)/y}\)

Simplify the left side:
\(\mathrm{(5T - NT)/N = (5-T)/y}\)
\(\mathrm{T(5-N)/N = (5-T)/y}\)

Process Skill: MANIPULATE - Systematic algebraic rearrangement to isolate the target variable

4. Solve for y

Now we have the equation: \(\mathrm{T(5-N)/N = (5-T)/y}\)

To solve for y, we cross-multiply:
\(\mathrm{y × T(5-N)/N = 5-T}\)

Divide both sides by \(\mathrm{T(5-N)/N}\):
\(\mathrm{y = (5-T) × N/[T(5-N)]}\)
\(\mathrm{y = N(5-T)/[T(5-N)]}\)

Let's verify this matches our answer choices. Looking at choice A: \(\mathrm{N(5-T)/[T(5-N)]}\)

This exactly matches what we derived!

Final Answer

The answer is A. \(\mathrm{N(5-T)/[T(5-N)]}\)

We can verify this makes sense: when we substitute our expression for y back into the original equation, we should get N = N, which confirms our algebra is correct.

Common Faltering Points

Errors while devising the approach

1. Misunderstanding the goal of variable isolation
Students may attempt to solve for K or x first instead of recognizing that the goal is to express y in terms of only N and T. This leads them down unnecessary algebraic paths and creates confusion about which variables to eliminate.

2. Incorrect substitution sequence
Students might try to substitute \(\mathrm{x = 5 - T}\) before dealing with \(\mathrm{K = 5T}\), or attempt to work with all variables simultaneously. The key insight is recognizing that K must be eliminated first using \(\mathrm{T = K/5}\), then x can be substituted.

3. Missing the connection between given relationships
Students may treat the three given equations as separate, unrelated pieces rather than understanding that \(\mathrm{T = K/5}\) and \(\mathrm{x = 5 - T}\) are meant to be substituted into the main equation \(\mathrm{N = K/(T + x/y)}\) to create a solvable relationship.

Errors while executing the approach

1. Algebraic manipulation errors with complex fractions
When working with \(\mathrm{N = 5T/(T + (5-T)/y)}\), students often make errors in taking reciprocals or clearing denominators. The step from \(\mathrm{N = 5T/(T + (5-T)/y)}\) to \(\mathrm{1/N = (T + (5-T)/y)/(5T)}\) is particularly error-prone.

2. Sign errors during rearrangement
Students frequently make sign errors when rearranging \(\mathrm{T(5-N)/N = (5-T)/y}\), especially when cross-multiplying or when dealing with the \(\mathrm{(5-N)}\) and \(\mathrm{(5-T)}\) terms. They might incorrectly write \(\mathrm{(N-5)}\) instead of \(\mathrm{(5-N)}\).

3. Fraction simplification mistakes
In the final steps, students may incorrectly simplify \(\mathrm{y = N(5-T)/[T(5-N)]}\), particularly confusing which terms belong in the numerator versus denominator, or incorrectly canceling terms that cannot be cancelled.

Errors while selecting the answer

1. Confusing similar-looking answer choices
Answer choices A and B are very similar, differing only in the sign: A has \(\mathrm{N(5-T)}\) while B has \(\mathrm{N(T-5)}\). Students who made a sign error during execution might select B thinking it's equivalent, not recognizing that \(\mathrm{(5-T) ≠ (T-5)}\).

2. Not verifying the final form matches answer choices exactly
Students might arrive at a correct expression but write it in a different form (like rearranging numerator/denominator) and then incorrectly match it to a wrong answer choice, particularly confusing the placement of N in the numerator versus other positions.

Alternate Solutions

Smart Numbers Approach

Step 1: Choose Smart Numbers

Let's choose \(\mathrm{K = 10}\). This gives us clean values to work with.

From \(\mathrm{T = K/5}\), we get: \(\mathrm{T = 10/5 = 2}\)

From \(\mathrm{x = 5 - T}\), we get: \(\mathrm{x = 5 - 2 = 3}\)

Step 2: Choose a Target Value for N

Let's say we want \(\mathrm{N = 1}\). Using the equation \(\mathrm{N = K/(T + x/y)}\):

\(\mathrm{1 = 10/(2 + 3/y)}\)

Solving for y: \(\mathrm{2 + 3/y = 10}\), so \(\mathrm{3/y = 8}\), therefore \(\mathrm{y = 3/8}\)

Step 3: Test Answer Choices

We now have: \(\mathrm{N = 1, T = 2}\), and \(\mathrm{y = 3/8}\)

Let's check which answer choice gives us \(\mathrm{y = 3/8}\):

Choice A: \(\mathrm{N(5-T)/[T(5-N)] = 1(5-2)/[2(5-1)] = 1(3)/[2(4)] = 3/8}\) ✓

Choice B: \(\mathrm{N(T-5)/[T(5-N)] = 1(2-5)/[2(5-1)] = 1(-3)/[2(4)] = -3/8}\) ✗

Choice C: \(\mathrm{(5-T)/[T(5-N)] = (5-2)/[2(5-1)] = 3/[2(4)] = 3/8}\)

Wait, this also gives \(\mathrm{3/8}\). Let's try different smart numbers to distinguish.

Step 4: Try Different Smart Numbers

Let's use \(\mathrm{K = 15}\), so \(\mathrm{T = 3}\), and \(\mathrm{x = 2}\). Let's target \(\mathrm{N = 2}\):

\(\mathrm{2 = 15/(3 + 2/y)}\), so \(\mathrm{3 + 2/y = 7.5}\), therefore \(\mathrm{2/y = 4.5}\), so \(\mathrm{y = 2/4.5 = 4/9}\)

Choice A: \(\mathrm{N(5-T)/[T(5-N)] = 2(5-3)/[3(5-2)] = 2(2)/[3(3)] = 4/9}\) ✓

Choice C: \(\mathrm{(5-T)/[T(5-N)] = (5-3)/[3(5-2)] = 2/[3(3)] = 2/9}\) ✗

This confirms that Choice A is correct.