Question

Answer the question on the basis of the information given below.

\[ P \# Q = \frac{P+PQ}{Q} \] \[ P \, \$ \, Q = \frac{P-PQ}{Q} \] \[ P \; ? \; Q = \frac{Q+PQ}{P} \] \[ P \, @ \, Q = \frac{Q-PQ}{P} \]

Find \(a \# b\), if \(a = (P \# Q + P \; ? \; Q)\), \(b = (P \, \$ \, Q + P \, @ \, Q)\) and \(P = 1\) and \(Q = 3\).

• A. \( \dfrac{5}{3} \)
• B. \( \dfrac{1}{11} \)
• C. \( -\dfrac{22}{3} \)
• D. \( -\dfrac{11}{3} \)

Hint:

First find the numeric values of a and b separately by plugging P = 1 and Q = 3 into the four given formulas, then apply the # rule a second time, this time treating a as P and b as Q.

Answer 1

The Correct Option is D

Step 1: Write down the four operator rules and the given values.
The operators are defined as \( P \# Q = \dfrac{P+PQ}{Q} \), \( P \, \$ \, Q = \dfrac{P-PQ}{Q} \), \( P \; ? \; Q = \dfrac{Q+PQ}{P} \) and \( P \, @ \, Q = \dfrac{Q-PQ}{P} \), with \(P=1\) and \(Q=3\), so \(PQ = 1 \times 3 = 3\).

Step 2: Work out \(a = P \# Q + P \; ? \; Q\).
\[ P \# Q = \frac{1+3}{3} = \frac{4}{3} \] \[ P \; ? \; Q = \frac{3+3}{1} = 6 \] \[ a = \frac{4}{3} + 6 = \frac{4}{3} + \frac{18}{3} = \frac{22}{3} \]

Step 3: Work out \(b = P \, \$ \, Q + P \, @ \, Q\).
\[ P \, \$ \, Q = \frac{1-3}{3} = -\frac{2}{3} \] \[ P \, @ \, Q = \frac{3-3}{1} = 0 \] \[ b = -\frac{2}{3} + 0 = -\frac{2}{3} \]

Step 4: Apply the \(\#\) rule again, this time to \(a\) and \(b\).
The rule \(P \# Q = \dfrac{P+PQ}{Q}\) is a general rule for any two numbers, so with \(a\) in place of \(P\) and \(b\) in place of \(Q\):
\[ a \# b = \frac{a+ab}{b} \]
First find \(ab\): \[ ab = \frac{22}{3} \times \left(-\frac{2}{3}\right) = -\frac{44}{9} \]
Then find \(a+ab\): \[ a+ab = \frac{22}{3} - \frac{44}{9} = \frac{66}{9} - \frac{44}{9} = \frac{22}{9} \]
Finally divide by \(b\): \[ a \# b = \frac{22/9}{-2/3} = \frac{22}{9} \times \left(-\frac{3}{2}\right) = -\frac{66}{18} = -\frac{11}{3} \]

Step 5: Why the other options are wrong.
Option (a), \(5/3\), and option (b), \(1/11\), come from mixing up which operator symbol pairs with which formula while combining \(a\) and \(b\). Option (c), \(-22/3\), is simply the value of \(a\) itself, mistaken for the final answer instead of computing \(a \# b\) fully.

Final Answer:
\[ \boxed{a \# b = -\frac{11}{3}} \]