A boat takes 5 hours more to travel 24 km upstream than to travel 24 km downstream. If the downstream speed of the boat is 2 km/h more than twice the upstream speed of the boat, then, which of the following statements are correct?
(A) Speed upstream of the boat is 3 km/h.
(B) Speed downstream of the boat is 8 km/h.
(C) Speed upstream of the boat is 3.5 km/h
(D) Speed downstream of the boat is 8.5 km/h
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → (A) and (B) only
Let the upstream speed be u km/h. Then the downstream speed is 2u + 2 km/h. Time = distance ÷ speed:
\( \displaystyle \frac{24}{u}-\frac{24}{2u+2}=5 \)
Multiplying by \( u(2u+2) \):
\( \displaystyle 24(2u+2)-24u=5u(2u+2) \)
\( \displaystyle 24u+48=10u^2+10u \)
\( \displaystyle 10u^2-14u-48=0 \)
\( \displaystyle 5u^2-7u-24=0 \)
\( \displaystyle (u-3)(5u+8)=0 \)
\( \displaystyle u=3\quad(\text{speed cannot be negative}) \)
\( \displaystyle \text{Downstream speed}=2(3)+2=8 \)
Check: 24 ÷ 3 = 8 h upstream and 24 ÷ 8 = 3 h downstream, a difference of 5 h.
So the upstream speed is 3 km/h and the downstream speed is 8 km/h.
Checking the statements
(A) Speed upstream of the boat is 3 km/h. — Correct. The upstream speed is 3 km/h.
(B) Speed downstream of the boat is 8 km/h. — Correct. The downstream speed is 2(3) + 2 = 8 km/h.
(C) Speed upstream of the boat is 3.5 km/h — Incorrect. The upstream speed is 3 km/h, not 3.5 km/h.
(D) Speed downstream of the boat is 8.5 km/h — Incorrect. The downstream speed is 8 km/h, not 8.5 km/h.
Shortcut: Test the given speeds instead of solving the quadratic. An upstream speed of 3 km/h gives a downstream speed of 2(3) + 2 = 8 km/h, and 24 ÷ 3 − 24 ÷ 8 = 8 − 3 = 5 h, as required.