Algebraic Word Problems and Modeling

A commercial coffee roaster creates a custom blend by mixing two existing bean blends: Blend X and Blend Y. Blend X consists of 60% Arabica beans and 40% Robusta beans by weight, whereas Blend Y consists of 30% Arabica beans and 70% Robusta beans by weight. The roaster mixes a quantity of Blend X with a quantity of Blend Y to produce a total of 50 pounds of a new mixture that is 42% Arabica beans by weight. How many pounds of Blend X are in the final mixture?

A manufacturing company produces two models of office chairs, Model P and Model Q. Producing each Model P chair requires $3$ hours of assembly and $1$ hour of finishing. Producing each Model Q chair requires $2$ hours of assembly and $2$ hours of finishing. On a given day, the assembly department was scheduled for $96$ total hours of work and the finishing department was scheduled for $48$ total hours of work. If both departments operated at full capacity and used all scheduled hours, how many Model P chairs were produced on that day?

A hotel renovated its guest rooms and purchased a total of 100 lighting fixtures, consisting of wall sconces costing $\$45$ each and ceiling pendants costing $\$70$ each. If the average (arithmetic mean) cost per fixture for the entire purchase was $\$52$, how many wall sconces were purchased?

A food processing plant packages grain using two automated lines, Line A and Line B. Line A operates at a constant rate of $40$ bags per minute, and Line B operates at a constant rate of $60$ bags per minute. Line A begins filling a batch of $4,200$ bags alone. After $15$ minutes, Line A's rate decreases by $25\%$, and Line B is turned on to work alongside Line A until the batch is complete. How many total minutes does Line A operate to complete the batch?

A university archive uses two high-speed scanners, Machine A and Machine B, to digitize manuscript pages. Machine A operates at a constant rate of $60$ pages per hour, and Machine B operates at a constant rate of $80$ pages per hour. On Monday, Machine A starts scanning at 8:00 AM. At 9:30 AM, Machine B is brought online to work alongside Machine A. If both machines continue scanning together at their respective constant rates until a total of $510$ pages have been scanned, at what time will the scanning project be completed?

A IT consultant charges a flat setup fee of $\$150$ plus a standard hourly rate of $\$80$ for regular hours worked on a project. For any rush hours worked, the hourly rate increases by $50\%$. On a recently completed project, the consultant worked a total of $t$ hours, of which $r$ hours were rush hours (where $0 \leq r \leq t$). Which of the following statements must be true regarding the total charge $C$, in dollars, for this project? Select all such statements.

A boutique perfume workshop creates a signature fragrance blend by combining two fragrance oils, Oil A and Oil B. Oil A costs $12 per ounce and contains 40$ 20 per ounce and contains 80% pure essential oil by volume. The perfumer creates a 30-ounce batch of the signature blend at a total cost that averages exactly $15 per ounce. What is the total volume, in ounces, of pure essential oil contained in this 30-ounce blend?

A logistics company calculates the shipping cost $C$, in dollars, for a parcel of weight $w$ pounds shipped over a distance of $d$ miles using the following piecewise pricing model:
- A fixed base fee of $\$10$.
- An additional weight surcharge of $\$1.50$ per pound for every pound (or fraction thereof) exceeding $5$ pounds.
- An additional distance surcharge of $\$0.05$ per mile for every mile exceeding $100$ miles.
- A flat combined heavy-distance surcharge of $\$25$ applied if and only if $w > 10$ AND $d > 500$.

A customer ships two parcels, Parcel 1 and Parcel 2:
- Parcel 1 weighs $w_1$ pounds (where $w_1$ is an integer greater than 10) and is shipped a distance of $d_1$ miles (where $d_1$ is an integer greater than 500).
- Parcel 2 weighs $w_2 = w_1 - 4$ pounds and is shipped a distance of $d_2 = d_1 + 200$ miles.

Let $C_1$ be the shipping cost of Parcel 1 and $C_2$ be the shipping cost of Parcel 2.

Which of the following statements MUST be true? Select all such statements.

A reservoir initially contains $4,800$ gallons of water. A drainage pump operates at a constant rate of $120$ gallons per minute to empty the reservoir. How many minutes of continuous pumping will it take for the amount of water remaining in the reservoir to be reduced to $1,200$ gallons?

A train travels from Station A to Station B, a distance of 180 miles, at a constant speed of $v$ miles per hour. On the return trip from Station B to Station A, the train travels the first half of the distance at a constant speed that is $20\%$ less than $v$, and the remaining half of the distance at a constant speed that is $25\%$ greater than $v$. If the total time for the return trip is 6 minutes longer than the total time for the trip from Station A to Station B, what is the value of $v$?

Pipeline A operating alone can fill a storage tank in $x$ hours, whereas Pipeline B operating alone takes $50\%$ longer than Pipeline A to fill the same tank. Drainage Pipe C operating alone can empty a full tank in $2x$ hours. If all three pipes are opened simultaneously when the tank is empty, the tank becomes completely full in $12$ hours. What is the value of $x$?

A pharmaceutical laboratory produces a custom solution in a mixing tank using two automated pumps, Pump A and Pump B. Initially, the tank contains $600$ liters of a solution that is $20\%$ active reagent by volume. Pump A adds pure active reagent to the tank at a constant rate of $15$ liters per minute, while Pump B simultaneously adds distilled water ($0\%$ active reagent) to the tank at a constant rate of $25$ liters per minute. At the exact instant when the mixture in the tank reaches $35\%$ active reagent by volume, Pump B is turned off while Pump A continues to add pure active reagent at $15$ liters per minute. How many total minutes, from the moment both pumps were initially started, does it take for the solution in the tank to reach $50\%$ active reagent by volume?

A coffee shop owner creates a $30$-pound blend of coffee by mixing Bean X, which costs $\$8$ per pound, with Bean Y, which costs $\$14$ per pound. If the final mixture costs $\$10$ per pound, how many pounds of Bean X are in the mixture?

Working alone at a constant rate, Pump A can drain a full water reservoir in $6$ hours, while Pump B working alone at a constant rate can drain the same full reservoir in $4$ hours. Which of the following statements must be true? Select all such statements.

A car rental agency charges a flat daily rate of $45 plus$ 0.20 per mile driven. If a customer rented a car for one day and the total rental cost before taxes was $75, how many miles did the customer drive?

Working independently at their respective constant rates, Alex and Blair can complete a certain job together. Alex works alone for $4$ hours, after which Blair joins Alex, and together they work for an additional $6$ hours to finish the entire job. If Alex takes strictly less time to complete the job working alone than Blair takes working alone, which of the following statements must be true? Select all that apply.

Train $X$ departs from a station traveling due east at a constant speed of $50$ miles per hour. Exactly $1$ hour later, Express Train $Y$ departs from the same station along the same track, traveling due east at a constant speed of $75$ miles per hour. How many hours after Express Train $Y$ departs will it catch up to Train $X$?