Question

DIRECTIONS for questions 42 and 43: Read the information below and answer the question that follows.
A truck travelled from town A to town B over several days. During the first day, it covered \(\frac{1}{p}\) of the total distance, where p is a natural number. During the second day, it travelled \(\frac{1}{q}\) of the remaining distance, where q is a natural number. During the third day, it travelled \(\frac{1}{p}\) of the distance remaining after the second day, and during the fourth day, \(\frac{1}{q}\) of the distance remaining after the third day. By the end of the fourth day, the truck had travelled \(\frac{3}{4}\) of the distance between A and B.

42. The value of p + q is:

• A. 4
• B. 5
• C. 6
• D. 7

Hint:

Track the fraction of distance left after each day; after four days the leftover fraction is the square of \(\frac{(p-1)(q-1)}{pq}\), and it must equal \(\frac{1}{4}\).

Answer 1

The Correct Option is D

Step 1: Write the remaining fraction after 4 days.
Take the total distance as 1. After day 1, the fraction left is $\frac{p-1}{p}$. After day 2, multiply by $\frac{q-1}{q}$: what is left is $\frac{(p-1)(q-1)}{pq}$. Days 3 and 4 repeat the same two steps, so the fraction left after day 4 is the square of this value:
\[ \text{fraction left after day 4} = \left(\frac{(p-1)(q-1)}{pq}\right)^2 \]

Step 2: Match this to the given data.
The truck has covered $\frac{3}{4}$ of the distance in 4 days, so $\frac{1}{4}$ is left:
\[ \left(\frac{(p-1)(q-1)}{pq}\right)^2=\frac{1}{4} \implies \frac{(p-1)(q-1)}{pq}=\frac{1}{2} \]
(the negative square root is dropped because p and q are natural numbers, which keeps the ratio zero or positive).

Step 3: Test small natural numbers directly.
Rearranging gives $2(p-1)(q-1)=pq$. Instead of factoring, just try small values of p and see what q comes out to be:
If $p=2$: $2(1)(q-1)=2q \implies 2q-2=2q$, no solution.
If $p=3$: $2(2)(q-1)=3q \implies 4q-4=3q \implies q=4$, a natural number, so this works.
If $p=4$: $2(3)(q-1)=4q \implies 6q-6=4q \implies q=3$, also a natural number (this is just the $p=3$ case with the roles swapped).
If $p=5$: $2(4)(q-1)=5q \implies 8q-8=5q \implies 3q=8$, not a whole number, so no solution.
Trying bigger p only pushes q below 1 or gives a fraction, so $(p,q)=(3,4)$ and $(4,3)$ are the only natural number solutions that work.

Step 4: Final Answer.
Either way, $p+q=3+4=7$.
\[ \boxed{p+q=7} \]