Ratios, Rates, and Proportions

A beverage mixture is prepared by combining apple juice, orange juice, and grape juice in a ratio of $3 : 4 : 5$ by volume. If a pitcher contains a total of $48$ fluid ounces of this mixture, how many fluid ounces of orange juice are in the pitcher?

A renewable energy facility distributes stored electricity among three battery banks: Bank 1, Bank 2, and Bank 3. Initially, the ratio of the energy stored in Bank 1 to Bank 2 to Bank 3 is $3 : 4 : 5$.

To balance the system, two sequential transfers of energy are performed without any energy loss:
1. Energy is transferred from Bank 3 to Bank 1 such that the ratio of the energy in Bank 1 to Bank 2 becomes $5 : 4$.
2. Energy is then transferred from Bank 2 to Bank 3 such that the ratio of the energy in Bank 2 to Bank 3 becomes $1 : 4$.

If Bank 3 contains $60\text{ MWh}$ more energy after the second transfer than it did initially, what was the total amount of energy, in MWh, stored across all three battery banks?

In a school club of $40$ students, the ratio of the number of boys to the number of girls is $3 : 5$. Which of the following statements must be true? Select all such statements.

A pharmaceutical facility uses two liquid compounding lines, Line $A$ and Line $B$, to produce a standardized saline solution. Line $A$ operates at a constant flow rate of $120$ liters per hour, producing a solution with active compound and purified water in a ratio of $1 : 3$ by volume. Line $B$ operates at a constant flow rate of $180$ liters per hour, producing a solution with active compound and purified water in a ratio of $2 : 3$ by volume. If Line $A$ runs for $4$ hours and Line $B$ runs for $5$ hours, and all output is collected into a single storage tank, what is the ratio of active compound to purified water in the storage tank?

An artisan bakery produces a signature flour blend by combining wheat, rye, and oat flour. Initially, the ratio of wheat flour to rye flour by weight is $5:3$, and the ratio of rye flour to oat flour by weight is $4:1$. The baker prepares an initial batch of this mixture weighing exactly $140$ pounds. To adjust the recipe for a special order, pure oat flour is added to the batch until oat flour accounts for exactly $20\%$ of the total weight of the new mixture. How many pounds of pure oat flour must the baker add?

In an industrial mechanical system, Gear $A$ has $16$ teeth, Gear $B$ has $24$ teeth, and Gear $C$ has $40$ teeth. Gear $A$ is meshed directly with Gear $B$, and Gear $B$ is meshed directly with Gear $C$. If Gear $A$ rotates at a constant speed of $150$ revolutions per minute ($\text{rpm}$), what is the rotational speed, in revolutions per minute ($\text{rpm}$), of Gear $C$?

A bakery uses flour and sugar in a ratio of $5:2$ by weight to make a cake batter. If a baker needs to prepare a batch using a total of $28$ pounds of flour and sugar combined, how many pounds of flour are needed?

Printer X prints at a constant rate of $60$ pages per minute, and Printer Y prints at a constant rate of $40$ pages per minute. If both printers operate simultaneously at their respective constant rates, which of the following statements must be true? Select all that apply.

A laboratory technician prepares a cleaning solution by mixing a concentrated formula and distilled water in a ratio of $2 : 7$ by volume. If the technician uses $14$ liters of the concentrated formula, how many liters of distilled water are required?

A car travels a total distance of $120$ miles. For the first $60$ miles, the car travels at an average speed of $30$ miles per hour, and for the remaining $60$ miles, the car travels at an average speed of $60$ miles per hour. What is the average speed of the car, in miles per hour, for the entire $120$-mile trip?

A contractor prepares a concrete mix using gravel and sand in a ratio of $4 : 7$ by weight. If the contractor uses $28$ tons of sand for a project, what is the total weight, in tons, of the concrete mix?

Pipes $A$ and $B$, operating together at their respective constant rates, can fill an empty storage tank in $6$ hours. Pipes $B$ and $C$, operating together at their constant rates, can fill the same tank in $8$ hours. The rate at which Pipe $A$ fills the tank is $1.5$ times the rate at which Pipe $C$ fills the tank. If all three pipes operate together for $3$ hours to fill the empty tank, after which Pipe $B$ is closed, how many additional hours will it take for Pipes $A$ and $C$ together to finish filling the tank?

Three industrial machines, $X$, $Y$, and $Z$, continuously produce liquid chemical mixtures composed solely of Compounds $A$ and $B$.

- Machine $X$ produces the mixture at a constant rate of $100$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $3:2$ by volume.
- Machine $Y$ produces the mixture at a constant rate of $150$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $1:4$ by volume.
- Machine $Z$ produces the mixture at a constant rate of $R$ liters per hour, with Compound $A$ and Compound $B$ in a ratio of $7:3$ by volume.

If all three machines operate simultaneously for $4$ hours, the resulting combined mixture contains equal total volumes of Compound $A$ and Compound $B$.

Which of the following statements must be true? Select all that apply.

An industrial laboratory prepares a batch of Alloy $Z$ weighing $260$ pounds, which consists solely of copper and zinc in a ratio of $7 : 6$ by weight. A technician then adds $40$ pounds of pure copper to this batch to create Alloy $W$. What is the ratio of copper to zinc in Alloy $W$?

Three water pumps, $P_1$, $P_2$, and $P_3$, operate at constant individual rates to fill a large reservoir. The ratio of the pumping rate of $P_1$ to that of $P_2$ is $3 : 4$, and the ratio of the pumping rate of $P_2$ to that of $P_3$ is $3 : 5$.

At 8:00 AM, all three pumps begin filling an empty reservoir together. At 10:00 AM, pump $P_1$ shuts down, while $P_2$ and $P_3$ continue operating at their original rates. At 11:00 AM, the operating rate of $P_2$ is decreased by $25\%$, and the operating rate of $P_3$ is increased by $25\%$. The two remaining pumps continue at these adjusted rates until the reservoir is completely full at 1:00 PM.

If pump $P_3$ were to fill the empty reservoir working alone at its original constant rate, how many hours would it take?

A renewable energy facility stores electricity in three separate battery modules: Module $X$, Module $Y$, and Module $Z$. The ratio of the initial energy stored in Module $X$ to Module $Y$ is $2 : 3$, and the ratio of the initial energy stored in Module $Y$ to Module $Z$ is $5 : 8$. Each module discharges its stored energy at a constant individual rate. Operating alone, Module $Y$ can completely discharge its initial stored energy in $5$ hours. Module $X$ discharges energy at a rate $33\frac{1}{3}\%$ greater than Module $Y$, and Module $Z$ discharges energy at a rate $25\%$ less than the combined discharge rate of Modules $X$ and $Y$. If all three modules begin discharging simultaneously, how many hours will it take to completely discharge the total initial energy stored across all three modules?