Question

The plane flew straight a wind between two towns in $84$ minutes and returned with that wind in $9$ minutes less than it would take in still air. The number of minutes (two answers) for the return trip was.

• A. $54$ or $18$
• B. $60$ or $15$
• C. $63$ or $12$
• D. $72$ or $16$

Answer 1

The correct option is C $63$ or $12$

Let $d$ be the distance between the two towns,

$p$ be the speed of the plane, and $w$ be the speed of the wind.

Since the time it took to fly against the wind towards the other town is $84$ minutes,

$\frac{d}{p-w}=84$

$d=84p-84w$

Since flying with the wind on the return trip takes $$9$$ minutes less compared to the return trip without wind,

$\frac{d}{p+w}+9=\frac{d}{p}$

Substitute $d$ and substitute $p$ and $w$

$⟹\frac{84p-84w}{p+w}+9=\frac{84p-84w}{p}$

$⟹\frac{93p-75w}{p+w}=\frac{84p-84w}{p}$

$⟹93p^2-75pw=84p^2+84pw-84pw-84w^2$

$⟹9p^2-75pw+84w^2=0$

$⟹3p^2-25pw+28w^2=0$

$⟹(3p-4w)(p-7w)=0$

That means $p=\frac{4}{3}w$ or $p=7w$.

If $p=\frac{4}{3}w$, then $\frac{112w-84w}{\frac{7}{3}w}=12$.

If $p=7w$, then $\frac{84\left(7w\right)-84w}{8w}=63$.

The number of minutes for the return trip can be $63,12$.