Word Problems and Applied Math

A merchant combines $10$ kilograms of Brand A tea costing $\$20$ per kilogram with $30$ kilograms of Brand B tea costing $\$40$ per kilogram. What is the average cost per kilogram of the combined mixture?

A commercial bakery prepares a specialty grain blend using Organic Wheat, Spelt, and Rye flour. Initially, the ratio of Organic Wheat to Spelt to Rye flour in the batch is $4 : 3 : 2$ by weight. To meet specific recipe requirements, $30$ kilograms of Organic Wheat and $15$ kilograms of Spelt are added to the batch, while $10$ kilograms of Rye are removed. After these adjustments, the ratio of Organic Wheat to Rye flour in the batch becomes $3 : 1$. What was the total weight, in kilograms, of the initial batch of flour before any adjustments were made?

At a technology company, a team of $60$ software engineers works on product development. Of these engineers, $35$ work on the mobile application, $28$ work on the web platform, and $10$ work on neither project. How many engineers work on both the mobile application and the web platform?

A boutique perfumery blends three essential oils—Jasmine, Sandalwood, and Bergamot—in an initial volume ratio of $4 : 3 : 2$, respectively. To modify the fragrance profile, the perfumer adds $20\text{ milliliters}$ of Sandalwood and $40\text{ milliliters}$ of Bergamot to the mixture, leaving the amount of Jasmine unchanged. If the resulting volume ratio of Sandalwood to Bergamot is $5 : 6$, what was the total initial volume, in milliliters, of the fragrance batch?

A pharmaceutical laboratory prepares a liquid vaccine solution by blending three ingredients: Active Compound X, Active Compound Y, and Distilled Water in the volume ratio of $3 : 4 : 8$, respectively. To meet updated formulation guidelines, a chemist adds $12$ liters of Active Compound Y and $36$ liters of Distilled Water to the solution, leaving the quantity of Active Compound X unchanged. If the new ratio of Active Compound X to Distilled Water in the resulting solution is $1 : 4$, what was the total volume, in liters, of the solution before any ingredients were added?

A container initially holds $100$ liters of an acid solution that is $40\%$ acid by volume. First, $x$ liters of the solution are drained and replaced with an equal volume of pure water. After the mixture is thoroughly stirred, $x$ liters of the new solution are drained and replaced with an equal volume of pure acid. If the final solution is $45.6\%$ acid by volume, what is the value of $x$?

A oceanographic research institute uses three autonomous submersibles���Submersible A, Submersible B, and Submersible C—to perform high-resolution seabed mapping. Working together at their respective constant rates, Submersible A and Submersible B can complete a full mapping mission in $15$ hours. Working together at their respective constant rates, Submersible B and Submersible C can complete the same mapping mission in $24$ hours. During a specialized operation, Submersible A works alone for $6$ hours, after which it is recalled. Submersible B then works alone for $10$ hours, after which Submersible C joins Submersible B, and both work together for an additional $12$ hours to complete the remaining portion of the mission. How many hours would it take Submersible A to complete the entire seabed mapping mission working alone at its constant rate?

A barista has $40$ ounces of a flavored drink mixture that is $30\%$ syrup by volume. How many ounces of pure water must be added to reduce the syrup concentration to $24\%$ by volume?

In a corporate workshop of $80$ executives, $52$ executives attended the Leadership module and $40$ executives attended the Negotiation module. If $18$ executives attended neither module, how many executives attended both modules?

At a software firm, an audit of 200 developers evaluated proficiency in three programming languages: Python, Java, and C++.

- 120 developers are proficient in Python.
- 105 developers are proficient in Java.
- 95 developers are proficient in C++.
- 15 developers are proficient in all three languages.
- 10 developers are proficient in none of the three languages.
- The ratio of the number of developers proficient in BOTH Python and Java ONLY to the number of developers proficient in BOTH Python and C++ ONLY is $3 : 2$.
- The number of developers proficient in ONLY Java is equal to the number of developers proficient in ONLY C++.

How many developers are proficient in ONLY Python?

Vessel A contains $40$ kilograms of an organic fertilizer mixture that is $60\%$ Nitrogen by weight. Vessel B contains $60$ kilograms of a fertilizer mixture that is $20\%$ Nitrogen by weight. First, $x$ kilograms of the mixture are removed from Vessel A and transferred into Vessel B, where the contents are thoroughly mixed. Then, $x$ kilograms of the newly formed mixture in Vessel B are transferred back into Vessel A. If the final mixture in Vessel A is $50\%$ Nitrogen by weight, what is the value of $x$?

A regional logistics company maintains a fleet consisting of delivery vans, medium trucks, and heavy cargo trucks in the initial ratio of $3 : 4 : 5$, respectively. After the company retires 4 delivery vans, purchases 12 medium trucks, and purchases 6 heavy cargo trucks, the ratio of delivery vans to heavy cargo trucks becomes $1 : 2$. What was the total number of vehicles in the fleet initially?

A retailer purchases a jacket for $80. The retailer marks up the purchase price by 25% to establish a list price. During a clearance sale, the jacket is sold at a 10% discount off the list price. What is the retailer's net profit on the sale of the jacket?

At a research foundation, annual grant funding was initially allocated among three divisions—Artificial Intelligence, Biotechnology, and Renewable Energy—in the ratio $7 : 4 : 3$, respectively. At mid-year, the foundation redistributed the funding by transferring $\$10$ million from the Artificial Intelligence division to the Renewable Energy division, while the Biotechnology division's funding remained unchanged. After this transfer, the ratio of Artificial Intelligence funding to Renewable Energy funding became $3 : 2$. What was the total annual grant funding, in millions of dollars, allocated across all three divisions?

An agricultural processing plant produces a livestock feed blend consisting of corn, soybeans, and oats in an initial weight ratio of $5 : 3 : 2$, respectively. To increase the protein content of a $1,200\text{ kg}$ batch of this blend, a technician removes a certain quantity of oats and replaces it with an equal weight of soybeans. If the resulting weight ratio of corn to soybeans to oats becomes $10 : 9 : 1$, how many kilograms of oats were replaced by soybeans?

A storage tank is completely filled with $150$ liters of a solution containing Compounds X, Y, and Water in a ratio of $2:3:5$ by volume. First, $30$ liters of the mixture are drained and replaced with $30$ liters of a liquid mixture that is $20\%$ Compound Y by volume. Next, $50$ liters of the resulting mixture are drained and replaced with $50$ liters of another liquid mixture that is $70\%$ Compound Y by volume. What is the percentage of Compound Y, by volume, in the final mixture?

A textile manufacturing plant uses two automated weaving looms, Loom $P$ and Loom $Q$, to produce standardized fabric orders. Working alone at its constant rate, Loom $P$ can complete a full fabric order in $8$ hours, while Loom $Q$ working alone at its constant rate can complete the same order in $12$ hours. Loom $P$ begins working on a full order alone. After $3$ hours, Loom $Q$ joins Loom $P$, and both looms work together at their respective constant rates until the order is completed. How many total hours does it take, from the moment Loom $P$ begins, to complete the entire fabric order?