Fractions and Rational Numbers

A water storage tank is initially filled to $\frac{4}{5}$ of its total capacity with water. First, $\frac{1}{4}$ of the water in the tank is drained for irrigation. Next, an amount of water equal to $\frac{1}{3}$ of the remaining water in the tank is added back. Finally, $\frac{3}{8}$ of the water currently in the tank is removed for domestic use. Which of the following statements about the volume of water in the tank must be true? Select all such statements.

In a manufacturing facility, three automated machines—Machine A, Machine B, and Machine C—produce components at constant individual rates. Machine A working alone completes $\frac{3}{10}$ of a standard daily order in $3$ hours. Machine B working alone completes $\frac{2}{5}$ of the order in $6$ hours. Machine C working alone completes $\frac{1}{4}$ of the order in $5$ hours.

Initially, all three machines work together for $3$ hours. At the end of $3$ hours, Machine A malfunctions and stops, while Machines B and C continue working together without interruption until the daily order is finished. How many total hours does it take from the start to complete the entire daily order?

A technology company allocates its quarterly research budget among three key projects: Project Alpha, Project Beta, and Project Gamma. Initially, Project Alpha receives $\frac{2}{5}$ of the total budget. Of the remaining budget, Project Beta is allocated $\frac{4}{7}$, and Project Gamma receives the rest. During a mid-quarter review, $\frac{1}{6}$ of Project Alpha's allocated budget and $\frac{1}{4}$ of Project Beta's allocated budget are transferred to Project Gamma. After these transfers, Project Gamma's final budget is what fraction of the total initial quarterly research budget?

A charity foundation allocates its annual grant budget among three sectors: Healthcare, Education, and Environmental Conservation. First, $\frac{2}{5}$ of the total annual budget is allocated to Healthcare. Next, $\frac{5}{9}$ of the remaining budget is allocated to Education. The final remainder of $\$120,000$ is allocated to Environmental Conservation. What was the foundation's total annual grant budget?

In a community library, $\frac{4}{15}$ of the total book collection consists of fiction books, $\frac{2}{5}$ consists of non-fiction books, and the remaining 240 books are children's books. If $\frac{3}{8}$ of the fiction books are hardcovers and the rest are softcovers, how many softcover fiction books are in the library?

An agricultural research station divides an experimental plot of land among three crops: Wheat, Rye, and Barley. Initially, Wheat occupies $\frac{4}{9}$ of the total plot area, Rye occupies $\frac{1}{3}$ of the total area, and Barley occupies the remaining area. First, $\frac{1}{3}$ of the Wheat area is converted to Rye. Next, $\frac{2}{5}$ of the new total Rye area is converted to Barley. Finally, $\frac{1}{4}$ of the new total Barley area is converted back to Wheat. If the final area occupied by Wheat is $126$ hectares, what was the total area, in hectares, of the experimental plot?

At a financial trading firm, incoming trade orders pass through a three-stage automated verification pipeline:

1. In Stage 1, $\frac{1}{6}$ of all incoming orders are rejected as invalid, and $\frac{1}{5}$ of the remaining valid orders are flagged for manual review.
2. In Stage 2, of the orders that were cleared directly without manual review, $\frac{3}{8}$ are executed immediately, while the rest are placed in a waiting queue.
3. In Stage 3, $\frac{1}{3}$ of the queued orders are processed, leaving the remaining queued orders unprocessed.

If exactly $250$ queued orders remain unprocessed at the end of Stage 3, how many incoming orders were flagged for manual review in Stage 1?

On Monday, a community kitchen starts with a bulk container of rice. First, $\frac{1}{3}$ of the initial supply of rice is used for lunch. Next, $\frac{1}{4}$ of the remaining rice in the container is used for dinner. After preparing both meals, exactly $60$ kilograms of rice remain in the container. How many kilograms of rice were in the container initially?

A specialty baking recipe consists of three ingredients: flour, sugar, and butter. By weight, $\frac{1}{2}$ of the recipe is flour and $\frac{1}{3}$ of the recipe is sugar, with the remaining portion consisting entirely of butter. Which of the following statements about the ingredients in the recipe must be true? Select all such statements.

At the beginning of a dry season, a municipal water reservoir contains an initial volume of $V$ gallons of water. During the first month, $\frac{1}{4}$ of the initial volume is released for agricultural irrigation. During the second month, $\frac{1}{5}$ of the remaining volume evaporates, after which $30,000$ gallons of water are pumped into the reservoir from an underground aquifer. If the final volume of water in the reservoir is equal to $\frac{3}{4}$ of the initial volume $V$, what was the initial volume $V$, in gallons?

At a research institute, an annual equipment budget is distributed among three departments. First, $\frac{2}{5}$ of the total budget is allocated to the Biology department. Next, $\frac{3}{8}$ of the remaining budget is allocated to the Chemistry department. The Physics department receives the remaining balance of $\$45,000$. What was the institute's total annual equipment budget?

Let $r$ and $s$ be rational numbers such that $0 < r < \frac{1}{2}$ and $1 < s < 2$. Which of the following inequalities MUST be true? Select all that apply.