Word Problems and Applied Math

An express delivery truck travels from Warehouse A to Warehouse B along a $180$-mile route. For the first $60$ miles, the truck maintains a constant speed of $v$ miles per hour. For the remaining $120$ miles, heavy traffic reduces its constant speed to $\frac{v}{2}$ miles per hour. If the average speed of the truck for the entire $180$-mile trip was $36$ miles per hour, what was the truck's speed $v$, in miles per hour, during the first $60$ miles?

A commuter drives from home to work along a straight route at a constant speed of $v$ miles per hour. On the return trip along the exact same route, heavy traffic causes the drive to take $50\%$ longer than the outbound trip. If the commuter's average speed for the entire round trip was $48$ miles per hour, what was the value of $v$, in miles per hour?

An artisanal coffee roasting company purchases unroasted green coffee beans at a cost of $\$12.00$ per kilogram. During the roasting process, moisture loss reduces the total weight of the coffee beans by $20\%$. To set the regular retail price per kilogram of roasted beans, the company marks up the effective cost price per kilogram of roasted beans by $50\%$. If the company sells the roasted beans at a promotional discount of $10\%$ off the regular retail price, what is the net profit percentage earned on the total cost of the unroasted green coffee beans?

An jeweler creates a custom silver alloy by melting together two available alloys. Alloy X is $40\%$ silver by weight, and Alloy Y is $70\%$ silver by weight. If the jeweler uses $15$ grams more of Alloy Y than Alloy X to produce a final alloy mixture that is $60\%$ silver by weight, what is the total weight, in grams, of the final alloy mixture?

Car 1 departs from City A at 8:00 AM and travels along a straight highway toward City B at a constant speed of $50$ miles per hour. At 8:30 AM, Car 2 departs from City B and travels along the same highway toward City A at a constant speed of $60$ miles per hour. At 9:15 AM, Car 1 encounters heavy traffic and reduces its speed by $20$ percent, maintaining this reduced speed for the rest of its journey, while Car 2 continues at its original speed. If the two cars pass each other at exactly 10:00 AM, what is the distance, in miles, between City A and City B?

A market research firm surveyed $200$ smartphone owners regarding two security features enabled on their devices: Facial Recognition and Fingerprint Scanning. Exactly $140$ owners enabled Facial Recognition, $110$ owners enabled Fingerprint Scanning, and $30$ owners enabled neither feature. How many of the surveyed smartphone owners enabled Facial Recognition ONLY?

A passenger train and a cargo train both travel along parallel tracks on a $120$-mile route between Station P and Station Q. The passenger train travels at a constant speed of $50$ miles per hour for the entire trip. The cargo train completes the first $60\%$ of the total distance at a constant speed of $30$ miles per hour and the remaining $40\%$ of the distance at a constant speed of $60$ miles per hour. By what percentage is the average speed of the passenger train for the entire trip greater than the average speed of the cargo train for the entire trip?

A commercial retailer purchased 100 identical smartwatches for a total cost of $\$10,000$. Before any sales were made, 10 smartwatches were damaged and could not be sold. The retailer marked up the unit cost of each remaining smartwatch by $P$ percent to determine its list price. During a promotional period, 50 smartwatches were sold at the full list price. The remaining 40 undamaged smartwatches were subsequently sold during a clearance sale at a 20 percent discount off the list price. If the retailer earned a total net profit of 23 percent on the initial investment of $\$10,000$, what is the value of $P$?

Container X initially contains $30$ liters of a solution that is $20\%$ acid by volume. Container Y contains $V$ liters of a solution that is $50\%$ acid by volume. If $10$ liters of pure water are added to Container X and the entire contents of Container X are then mixed with Container Y, the resulting mixture is $30\%$ acid by volume. What is the value of $V$, in liters?

A publishing firm evaluated $160$ manuscript submissions based on three quality criteria: Plot Originality ($P$), Character Development ($C$), and Dialogue Quality ($D$). Each manuscript was evaluated against all three criteria.

- $75$ manuscripts met the criteria for Plot Originality ($P$).
- $80$ manuscripts met the criteria for Character Development ($C$).
- $70$ manuscripts met the criteria for Dialogue Quality ($D$).
- $30$ manuscripts met the criteria for both Plot Originality and Character Development ($P \cap C$).
- $25$ manuscripts met the criteria for both Character Development and Dialogue Quality ($C \cap D$).
- $20$ manuscripts met the criteria for both Plot Originality and Dialogue Quality ($P \cap D$).
- $10$ manuscripts met all three criteria ($P \cap C \cap D$).

How many manuscripts met exactly one of the three criteria?

An athlete completes a three-leg training course consisting of cycling, running, and swimming, covering a total distance of $60$ miles in a total time of $4$ hours. The athlete cycles at a constant speed of $24$ miles per hour, runs at a constant speed of $10$ miles per hour, and swims at a constant speed of $2$ miles per hour. If the time spent cycling is equal to the combined time spent running and swimming, how many miles did the athlete run?

A pharmaceutical research laboratory tested $120$ chemical compounds for the presence of three distinct bio-reactivity markers: Enzyme Inhibition ($E$), Protein Binding ($P$), and Receptor Affinity ($R$). The test results revealed the following:

- $58$ compounds exhibited Enzyme Inhibition ($E$).
- $52$ compounds exhibited Protein Binding ($P$).
- $46$ compounds exhibited Receptor Affinity ($R$).
- $8$ compounds exhibited all three markers.
- $15$ compounds exhibited none of the three markers.

How many of the tested compounds exhibited exactly two of the three markers?

An automated delivery van and an electric cargo scooter both travel along a straight route from Warehouse X to Warehouse Y, which are $D$ miles apart.

The van travels at a constant speed of $30$ miles per hour for the first half of the total distance, and at a constant speed of $60$ miles per hour for the second half of the total distance.

The scooter travels at a constant speed of $30$ miles per hour for the first half of its total travel time, and at a constant speed of $60$ miles per hour for the second half of its total travel time.

If the scooter completes the entire journey in $15$ minutes less time than the van, what is the value of $D$, in miles?

At a technology firm, an evaluation was conducted for 150 software developers to test their proficiency in three programming paradigms: Object-Oriented ($O$), Functional ($F$), and Reactive ($R$). The evaluation revealed that 85 developers are proficient in $O$, 60 are proficient in $F$, and 45 are proficient in $R$. Furthermore, 25 developers are proficient in both $O$ and $F$, 18 are proficient in both $F$ and $R$, 22 are proficient in both $O$ and $R$, and 10 developers are proficient in all three paradigms. How many of the software developers surveyed are proficient in exactly two of these three programming paradigms?

Vessel A contains $60$ liters of a beverage mixture that is $75\%$ fruit concentrate by volume, and Vessel B contains $90$ liters of a beverage mixture that is $25\%$ fruit concentrate by volume. First, $10$ liters of pure fruit concentrate are added to Vessel B. Next, $x$ liters of the mixture in Vessel A are removed and replaced with $x$ liters of pure water. If the concentration of fruit concentrate in Vessel A is now equal to the concentration of fruit concentrate in Vessel B, what is the value of $x$?

A management consulting firm charges clients based on three staff tiers: Junior Consultants, Senior Consultants, and Directors. Junior Consultants are billed at a rate of $\$120$ per hour, Senior Consultants at $\$210$ per hour, and Directors at $\$390$ per hour. On a strategic assignment, the ratio of hours worked by Junior Consultants to Senior Consultants to Directors was $5 : 3 : 2$, respectively. What was the overall average hourly billing rate per consultant hour for this assignment?

Two cyclists, Cyclist A and Cyclist B, depart simultaneously from Town X and Town Y, respectively, traveling toward each other along a straight 150-mile path. Cyclist A travels at a constant speed of 18 miles per hour throughout the journey. Cyclist B initially travels at a constant speed of 30 miles per hour. After traveling for 2 hours, Cyclist B encounters a steep incline and decreases speed by 40 percent, maintaining this reduced speed for the remainder of the trip. How many hours after their departure will Cyclist A and Cyclist B meet?

A pharmacist has $80$ milliliters of a topical liquid formulation that is $5\%$ active ingredient by volume. How many milliliters of pure active ingredient must the pharmacist add to this formulation so that the resulting mixture is $24\%$ active ingredient by volume?