Changing-resource problems: growing grass, leaking ships, pumps, escalators, mixed consumers, queues, renewable resources, and changing work capacity.
13 questions · 120 points · Manual assessment
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Paired trials are separate: each starts with the same quantity unless different pasture areas are stated. “Feeds for,” “lasts,” and “empties in” mean the resource first runs out at the stated time. A new herd or team starts again with that original quantity unless a mid-trial change is specified. Workers, pumps, gates, and animals of the same type have equal constant rates within each problem. Inflow continues while the resource is used unless stated otherwise. Read each situation carefully: some questions use a fixed stock.
Questions 1–10 · 6 points each · 60 points
A ship strikes a reef and a hole is made in its hull. Seawater enters the ship at a constant rate. By the time the leak is discovered, some water has already entered. Twelve people can bail all the water out in 3 hours, while five people can bail it out in 10 hours. How many people are needed to bail it out in 2 hours?
Twenty-four cows can eat all the grass in a pasture in 6 days, while 21 cows can eat all the grass in the same pasture in 8 days. The grass grows by the same amount each day. What is the greatest number of cows that can graze there so that the grass will never be completely eaten?
An escalator moves upward at a constant speed. A boy and a girl think it is too slow, so they walk up the moving escalator. Both travel the full length; walking rates count steps taken relative to the moving escalator. The boy walks 30 steps per minute and reaches the top in 3 minutes. The girl walks 20 steps per minute and reaches the top in 4 minutes. How many steps does the escalator have when it is stationary?
Canteens A, B, and C share the pork from seven equal-weight pigs, with each canteen receiving one-third of the total. Canteen A supplies four pigs, Canteen B supplies three pigs, and Canteen C pays A and B a total of 840 yuan for its share. How much more money should Canteen A receive than Canteen B?
No grass is added to a fixed pile. It can feed 3 cows and 5 sheep for 15 days, or 5 cows and 6 sheep for 10 days. Starting with an identical full pile, for how many days can 8 cows and 11 sheep eat?
Water continuously flows into a pool from a spring at the bottom. Ten pumps can empty the pool in 8 hours, while eight pumps can empty it in 12 hours. How many hours will six pumps need?
The grass in a pasture grows by the same amount every day. The pasture can feed 10 cows for 20 days, or 15 cows for 10 days. For how many days can it feed 30 cows?
A pasture is initially full of grass. It can feed 10 cows for 3 days, or 5 cows for 8 days. If the grass grows at a constant daily rate, how many cows can it feed for 2 days?
In a hypothetical mathematical model, renewable resources grow at a constant rate and each person consumes the same constant amount. This is not an estimate of Earth’s actual carrying capacity. At that rate, the resources could support 11 billion people for 90 years, or 9 billion people for 210 years. What is the greatest population Earth could support indefinitely?
Eight sheep eat 168 kilograms of feed in one week. Over the same length of time, one cow eats 2.8 times as much as one sheep. How many kilograms of feed will 200 sheep and 180 cows need for one 30-day month?
Questions 11–13 · 20 points each · 60 points
Twenty-two cows can eat all the grass on 33 qing of pasture in 54 days. Seventeen cows can eat all the grass on 28 qing of the same kind of pasture in 84 days. Assume that every qing initially has the same amount of grass and that the grass on every qing grows at the same daily rate. How many cows can 40 qing of the same pasture feed for 24 days?
A qing is the area unit used in the practice problem. Treat it simply as a consistent unit of pasture area.
An art exhibition opens at 9:00, but people have already formed a queue. The queue is initially empty before the first visitor arrives. From that time onward, the same number of visitors arrive every minute, including after opening; nobody leaves or is admitted before 9:00. All gates begin serving at 9:00 and work at the same constant rate. With three gates, the queue first becomes empty exactly at 9:09. With five gates, it first becomes empty exactly at 9:05. At what time after 8:00 did the first visitor arrive?
Warehouses A, B, and C each contain the same initial amount of flour, and no new flour arrives while it is being removed. At Warehouse A, one conveyor belt and 12 workers can move all the flour out in 5 hours. At Warehouse B, one conveyor belt and 28 workers can move all the flour out in 3 hours. Warehouse C has two conveyor belts. Every worker has the same hourly work rate, and every conveyor belt has the same hourly work rate. If the flour in Warehouse C must be moved out within 2 hours, what is the least number of workers needed alongside the two conveyor belts?