This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 18

2000 Slovenia National Olympiad, Problem 2
Tags: rate problems
Three students start walking with constant speeds at the same time, each along a straight line in the plane. Prove that if the students are not on the same line at the beginning, then they will be on the same line at most twice during their journey.

2020 AMC 12/AHSME, 3
Tags: AMC , AMC 10 , AMC 12 , 2020 AMC , 2020 AMC 12A , 2020 AMC 10A , rate problems
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? $\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$

1958 AMC 12/AHSME, 24
Tags: rate problems
A man travels $ m$ feet due north at $ 2$ minutes per mile. He returns due south to his starting point at $ 2$ miles per minute. The average rate in miles per hour for the entire trip is: $ \textbf{(A)}\ 75\qquad \textbf{(B)}\ 48\qquad \textbf{(C)}\ 45\qquad \textbf{(D)}\ 24\qquad\\ \textbf{(E)}\ \text{impossible to determine without knowing the value of }{m}$

2014 AMC 12/AHSME, 11
Tags: rate problems , AMC
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

2022 AMC 10, 4
Tags: AMC 10 , 2022 AMC 10a , 2022 AMC , AMC , rates , rate problems , algebra
In some countries, automobile fuel efficiency is measured in liters per $100$ kilometers while other countries use miles per gallon. Suppose that $1$ kilometer equals $m$ miles, and $1$ gallon equals $\ell$ liters. Which of the following gives the fuel efficiency in liters per $100$ kilometers for a car that gets $x$ miles per gallon? $\textbf{(A) } \frac{x}{100\ell m} \qquad \textbf{(B) } \frac{x\ell m}{100} \qquad \textbf{(C) } \frac{\ell m}{100x} \qquad \textbf{(D) } \frac{100}{x\ell m} \qquad \textbf{(E) } \frac{100\ell m}{x}$

2018 AMC 10, 2
Tags: AMC , AMC 10 , 2018 AMC 10B , 2018 AMC 12B , AMC 12 , rate problems
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? $\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

2017 AMC 10, 4
Tags: AMC , AMC 10 , AMC 10 A , 2017 AMC 10A , rate problems , rates
Mia is “helping” her mom pick up $30$ toys that are strewn on the floor. Mia’s mom manages to put $3$ toys into the toy box every $30$ seconds, but each time immediately after those $30$ seconds have elapsed, Mia takes $2$ toys out of the box. How much time, in minutes, will it take Mia and her mom to put all $30$ toys into the box for the first time? $\textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 14.5\qquad\textbf{(D)}\ 15\qquad\textbf{(E)}\ 15.5$

2010 AMC 10, 13
Tags: rate problems , AMC
Angelina drove at an average rate of $ 80$ kph and then stopped $ 20$ minutes for gas. After the stop, she drove at an average rate of $ 100$ kph. Altogether she drove $ 250$ km in a total trip time of $ 3$ hours including the stop. Which equation could be used to solve for the time $ t$ in hours that she drove before her stop? $ \textbf{(A)}\ 80t\plus{}100(8/3\minus{}t)\equal{}250 \qquad \textbf{(B)}\ 80t\equal{}250 \qquad \textbf{(C)}\ 100t\equal{}250 \\ \textbf{(D)}\ 90t\equal{}250 \qquad \textbf{(E)}\ 80(8/3\minus{}t)\plus{}100t\equal{}250$

1962 AMC 12/AHSME, 10
Tags: rate problems
A man drives $ 150$ miles to the seashore in $ 3$ hours and $ 20$ minutes. He returns from the shore to the starting point in $ 4$ hours and $ 10$ minutes. Let $ r$ be the average rate for the entire trip. Then the average rate for the trip going exceeds $ r$ in miles per hour, by: $ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 4 \frac{1}{2} \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 1$

1955 AMC 12/AHSME, 10
Tags: calculus , function , analytic geometry , graphing lines , slope , rate problems
How many hours does it take a train traveling at an average rate of $ 40$ mph between stops to travel $ a$ miles it makes $ n$ stops of $ m$ minutes each? $ \textbf{(A)}\ \frac{3a\plus{}2mn}{120} \qquad \textbf{(B)}\ 3a\plus{}2mn \qquad \textbf{(C)}\ \frac{3a\plus{}2mn}{12} \qquad \textbf{(D)}\ \frac{a\plus{}mn}{40} \qquad \textbf{(E)}\ \frac{a\plus{}40mn}{40}$

2018 AMC 12/AHSME, 2
Tags: AMC , AMC 10 , 2018 AMC 10B , 2018 AMC 12B , AMC 12 , rate problems
Sam drove $96$ miles in $90$ minutes. His average speed during the first $30$ minutes was $60$ mph (miles per hour), and his average speed during the second $30$ minutes was $65$ mph. What was his average speed, in mph, during the last $30$ minutes? $\textbf{(A) } 64 \qquad \textbf{(B) } 65 \qquad \textbf{(C) } 66 \qquad \textbf{(D) } 67 \qquad \textbf{(E) } 68$

2014 AMC 10, 15
Tags: rate problems , AMC
David drives from his home to the airport to catch a flight. He drives $35$ miles in the first hour, but realizes that he will be $1$ hour late if he continues at this speed. He increases his speed by $15$ miles per hour for the rest of the way to the airport and arrives $30$ minutes early. How many miles is the airport from his home? $\textbf{(A) }140\qquad \textbf{(B) }175\qquad \textbf{(C) }210\qquad \textbf{(D) }245\qquad \textbf{(E) }280\qquad$

2024 ISI Entrance UGB, P7
Tags: conics , parabola , maxima , rate problems , applications of calculus , ISI UGB 2024
Consider a container of the shape obtained by revolving a segment of parabola $x = 1 + y^2$ around the $y$-axis as shown below. The container is initially empty. Water is poured at a constant rate of $1\, \text{cm}^3$ into the container. Let $h(t)$ be the height of water inside container at time $t$. Find the time $t$ when the rate of change of $h(t)$ is maximum.

1953 AMC 12/AHSME, 17
Tags: rate problems
A man has part of $ \$4500$ invested at $ 4\%$ and the rest at $ 6\%$. If his annual return on each investment is the same, the average rate of interest which he realizes of the $ \$4500$ is: $ \textbf{(A)}\ 5\% \qquad\textbf{(B)}\ 4.8\% \qquad\textbf{(C)}\ 5.2\% \qquad\textbf{(D)}\ 4.6\% \qquad\textbf{(E)}\ \text{none of these}$

2024 AMC 12/AHSME, 2
Tags: AMC , AMC 12 , 2024 AMC 10A , 2024 AMC 12A , rate problems , 2024 AMC , AMC 10
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? $\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$

2022 AMC 10, 2
Tags: AMC 10 , AMC , 2022 AMC , 2022 AMC 10a , rates , rate problems
Mike cycled $15$ laps in $57$ minutes. Assume he cycled at a constant speed throughout. Approximately how many laps did he complete in the first $27$ minutes? $\textbf{(A) } 5 \qquad \textbf{(B) } 7 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 13$

2024 AMC 10, 2
Tags: AMC , AMC 12 , 2024 AMC 10A , 2024 AMC 12A , rate problems , 2024 AMC , AMC 10
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimate it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? $\textbf{(A) } 240 \qquad \textbf{(B) } 246 \qquad \textbf{(C) } 252 \qquad \textbf{(D) } 258 \qquad \textbf{(E) } 264$

2020 AMC 10, 4
Tags: AMC , AMC 10 , AMC 12 , 2020 AMC , 2020 AMC 12A , 2020 AMC 10A , rate problems
A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $\$0.50$ per mile, and her only expense is gasoline at $\$2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense? $\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$