Question

Driving at a constant speed, Sharon usually takes $180$ minutes to drive from her house to her mother’s house. One day Sharon begins the drive at her usual speed, but after driving $\dfrac{1}{3}$ of the way, she hits a bad snowstorm and reduces her speed by $20$ miles per hour. This time the trip takes her a total of $276$ minutes. How many miles is the drive from Sharon’s house to her mother’s house?

$132$

$135$

$138$

$141$

$144$

Answer: B

Concepts:distance rate and time system of equations

Difficulty rating: 1660

Small Hint:

Her usual drive is $3$ hours, so the distance equals $3$ times her usual speed

Big Hint:

The final two-thirds takes $276-\dfrac13\cdot180=216$ minutes at the reduced speed

Solution:

Let the distance be $d$ miles and the usual speed $r$ mph. Since the usual trip is $3$ hours, $d=3r.$

The first $\dfrac{1}{3}$ of the drive takes $\dfrac{1}{3}\cdot180=60$ minutes at speed $r,$ so the remaining $\dfrac{2}{3}$ takes $276-60=216$ minutes $=\dfrac{18}{5}$ hours at speed $r-20.$

That final portion covers $\dfrac{2}{3}d=2r$ miles, so $2r=(r-20)\cdot\dfrac{18}{5}.$ Solving gives $10r=18r-360,$ so $r=45$ and $d=3\cdot45=135.$

Thus, the correct answer is B.

2017 AMC 12A Problem 13

Answer: B

Difficulty rating: 1660

Small Hint:

Big Hint:

Solution:

Problem 13 in Other Years