All practice questions

For all real numbers $x$ and $y$, the custom operator $\Delta$ is defined by $x \Delta y = x^2 - 2y$. If $k$ is a real number such that $(3 \Delta k) \Delta 4 = 17$, what is the sum of all possible values of $k$?

Printer Alpha can fabricate a set of architectural prototypes in $4$ hours operating alone at a constant rate. Printer Beta can fabricate the exact same set of prototypes in $12$ hours operating alone at a constant rate. If both printers operate simultaneously at their respective constant rates, how many hours will it take them to fabricate one set of architectural prototypes?

For all real numbers $p$ and $q$, the operation $\nabla$ is defined by $p \nabla q = 2p + 3q - pq$. What is the value of $3 \nabla 4$?

In a technology firm of $100$ software engineers, $65$ engineers know Python and $45$ know Java. If $20$ engineers know both Python and Java, how many of the engineers know neither Python nor Java?

Let $S$ be the set of all real solutions to the polynomial equation $(x^2 - 5x + 6)(x^2 - 7x + 12) = (x - 3)^2(x - 2)$. What is the sum of all elements in set $S$?

Data Pipeline A can process a standard batch of records in $12$ hours, while Data Pipeline B can process the same batch of records in $18$ hours. Data Pipeline A begins processing a batch alone and operates for $4$ hours before Data Pipeline B is brought online to assist. Both pipelines then continue processing together at their respective constant rates until the batch is completely processed. What is the total time, in hours, required to process the entire batch of records from start to finish?

At a software company, the ratio of senior developers to junior developers was initially $3 : 7$. During a quarterly hiring push, $12$ senior developers and $18$ junior developers were hired, but $6$ junior developers resigned shortly thereafter. As a result, the ratio of senior developers to junior developers became $4 : 9$. If a certain number $N$ of senior developers were subsequently promoted out of the developer pool such that the ratio of senior developers to junior developers became $5 : 12$, what is the value of $N$?

What is the sum of all real solutions to the equation $x(x - 3) = 10$?

Scanner X can scan a batch of archival documents in $6$ hours operating alone at a constant rate. Scanner Y can scan the exact same batch of documents in $3$ hours operating alone at a constant rate. Working together simultaneously at their respective constant rates, how many hours will it take Scanner X and Scanner Y to scan the entire batch of documents?

Three automated data-processing pipelines, $A$, $B$, and $C$, process incoming dataset files at constant individual rates. Pipeline $A$ operating alone can complete a standard processing job in $10$ hours, while Pipeline $B$ operating alone can complete the same job in $15$ hours.

The processing begins with Pipelines $A$ and $B$ working together at their standard rates for $2$ hours. At that point, Pipeline $A$ encounters a network slowdown that reduces its processing rate by $50\%$ for the remainder of the job. Pipelines $A$ and $B$ continue working at these modified rates for another $2$ hours, after which Pipeline $C$ is brought online to assist them. If Pipeline $C$ operating alone at its constant rate could finish the entire job in $10$ hours, how many total hours from the start of the processing will it take for the job to be completed?

Pumps X, Y, and Z, operating simultaneously at their respective constant rates, can fill an empty reservoir in $4$ hours. Operating alone at its constant rate, Pump X can fill $\frac{1}{5}$ of the reservoir in $2$ hours. The rate at which Pump Y fills the reservoir is $50\%$ greater than the rate at which Pump Z fills the reservoir. If Pump X and Pump Z work together for $3$ hours, and then Pump Y is turned on so that all three pumps work together until the reservoir is full, how many additional hours will it take to finish filling the remainder of the reservoir?

A beverage recipe requires mixing orange juice and cranberry juice in a ratio of $4 : 1$. If a caterer uses $12$ liters of orange juice, how many liters of cranberry juice must be added to maintain this ratio?

A corporate marketing team initially allocates its quarterly budget among Digital, Print, and Event advertising in the ratio $5 : 3 : 2$, respectively. Midway through the quarter, $\$12,000$ is reallocated from Digital advertising to Event advertising, while the Print advertising budget remains unchanged. As a result of this transfer, the ratio of the Digital advertising budget to the Event advertising budget becomes $4 : 3$. What was the total initial quarterly budget, in dollars, allocated across all three advertising categories?

For all real numbers $x$ and $y$, the custom operation $\diamondsuit$ is defined by $x \diamondsuit y = 3x - 2y + 1$. What is the value of $5 \diamondsuit 4$?

To reduce mortality rates from aggressive skin cancers, a network of dermatology clinics plans to deploy a new automated scanning device that detects subtle tissue irregularities undetectable by visual examination alone. Clinic administrators claim that because the device identified early-stage tissue irregularities in 30 percent more patients during trials than visual exams did, deploying this technology across all clinics will significantly decrease overall skin cancer mortality among their patients. Which of the following, if true, most seriously undermines the clinic administrators' argument?

An executive board must arrange 7 books side-by-side on a display shelf: 3 identical copies of a Finance textbook, 2 identical copies of an Economics textbook, and 2 distinct novels (Novel A and Novel B). How many distinct linear arrangements of these 7 books are possible such that Novel A and Novel B are not placed adjacent to each other?

For all non-zero real numbers $x$, the function $f$ is defined by $f(x) = \frac{x^2 - 4}{2x}$, and the function $g$ is defined by $g(x) = 3x - 1$. What is the positive value of $a$ for which $f(g(a)) = 0$?

At a bio-tech research facility, the ratio of male to female scientists is initially $5 : 3$. After $4$ female scientists leave and $12$ male scientists join the facility, the ratio of male to female scientists becomes $4 : 1$. If a group of $N$ additional female scientists subsequently joins the facility, changing the ratio of male to female scientists to $8 : 5$, what is the value of $N$?