Finite Mathematics Quiz: Translating To Mathematical Expressions

A investment portfolio consists of stocks, bonds, and cash. The value of the stock portion changes by $r\%$ annually, bonds change by $s\%$ annually, and cash remains constant. Initially, stocks represent 60% of the portfolio, bonds represent 30%, and cash represents 10%. After one year, the portfolio is rebalanced so that stocks again represent 60% of the total value. If $V$ is the initial portfolio value, which expression represents the dollar amount moved from stocks to achieve this rebalancing?

A subscription service charges a monthly fee plus an additional cost per hour of usage. In January, Sarah used the service for 12 hours and paid $47. In February, she used it for 8 hours and paid $35. If $$x$ $represents the monthly fee and$ $y$ $ represents the cost per hour, which system of equations correctly models this situation?

A parking garage charges $3 for the first hour, $2 for each additional hour up to 5 hours total, and a flat rate of $12 for any stay exceeding 5 hours. Which piecewise function correctly represents the cost $$C(h)$ $for parking$ $h$ $ hours?

A telecommunications company offers a data plan where customers pay a base fee plus charges based on data usage. The plan includes 2 GB free, then $5 per GB for the next 3 GB, and $8 per GB for any usage beyond 5 GB total. If the base fee is $25 and a customer uses $$d$ $GB of data, which expression correctly represents their bill when$ $d > 5$ $?

A city's population growth follows this pattern: it increases by 2% annually, but every 5 years, an additional 1000 people move to the city due to a recurring economic development program. If the initial population is $P_0$ and $t$ represents years, which expression best models the population after $t$ years (where $t$ is a multiple of 5)?

A delivery service uses the following pricing structure:

• Base delivery fee: $8 • Distance charge: $0.75 per mile for the first 10 miles • Distance charge: $0.50 per mile for miles 11-20 • Distance charge: $0.25 per mile for any miles beyond 20 • Rush delivery surcharge: additional 25% of total delivery cost • Weekend surcharge: additional $3 (applied before rush surcharge if both apply)

Based on the pricing structure above, which expression correctly represents the total cost for a rush weekend delivery that travels 25 miles?

A chemist has two acid solutions. Solution A is 15% acid by volume, and Solution B is 40% acid by volume. The chemist first mixes $x$ liters of Solution A with $y$ liters of Solution B. Then, the chemist adds $z$ liters of pure water (0% acid) to this new mixture. Which expression represents the final concentration of acid in the mixture, as a percentage?

A company manufactures two types of bicycles: standard and deluxe. Let $s$ be the number of standard models and $d$ be the number of deluxe models produced each week. Each standard model requires 2 hours of assembly time, and each deluxe model requires 5 hours. The company has a maximum of 400 assembly hours available per week. For finishing, each standard model takes 1 hour and each deluxe model takes 1.5 hours; a minimum of 95 finishing hours must be used. Due to market demand, the number of standard models produced must be at least half the number of deluxe models produced but no more than three times the number of deluxe models. Which of the following is NOT a valid constraint for this production plan?

An employee contributes $P$ dollars at the end of each year for 10 years to a retirement account earning an annual interest rate of $r$, compounded annually. After 10 years, the employee increases the annual contribution to $1.5P$ dollars for the next 15 years. Let $i=r$ be the interest rate per period. Which expression represents the total value of the account after the full 25-year period?

An organization consists of $M$ managers and $E$ engineers. Of these, $M_f$ are female managers and $E_f$ are female engineers. A committee of $N$ people is to be randomly selected from all employees. Which expression represents the probability that the selected committee consists entirely of female engineers, assuming $N \le E_f$?

A manufacturer produces an item at a cost of $C$. The manufacturer sells it to a wholesaler at a 20% markup over its cost. The wholesaler then sells it to a retailer at a 25% markup over the price they paid. Finally, the retailer sells the item to a customer at a 40% markup over the price they paid. Which expression represents the final price paid by the customer in terms of $C$?

A university is organizing its course schedule. Let $m$ be the number of mathematics courses and $p$ be the number of physics courses offered. The total number of courses must be between 40 and 50, inclusive. The number of mathematics courses must be at least 15. The number of physics courses offered must be no less than one-third of the number of mathematics courses. Which of the following systems of inequalities correctly represents all of these constraints?

A bakery sells two types of cakes, chocolate and vanilla. Let $C$ be the event that a randomly selected customer buys a chocolate cake, and $V$ be the event that they buy a vanilla cake. The bakery knows that $P(C) = 0.6$ and $P(V) = 0.5$. The probability that a customer buys both types of cake is $P(C \cap V) = 0.2$. Which expression represents the probability that a randomly selected customer buys a vanilla cake but not a chocolate cake?

A rental car company charges a daily fee plus a fee per mile driven. For a one-day rental, driving 100 miles costs $75, and driving 160 miles costs $93. Let $D$be the fixed daily fee and$M$ be the fee per mile. Which system of linear equations correctly models this situation?

A manufacturing process has a setup cost of $500 per batch regardless of batch size. The variable cost per unit is $12, but due to economies of scale, this cost decreases by $0.10 for every 100 units produced in a batch (applied to all units in the batch). If a batch contains $$n$ $units where$ $n > 100$ $, which expression correctly represents the total cost per unit?

A manufacturer produces widgets at a cost of $8 each. The fixed monthly costs are $2000, and each widget sells for $15. However, for every 50 widgets produced beyond 200 widgets, the production cost per widget increases by $0.50 due to overtime labor. If $$x$ $represents the number of widgets produced (where$ $x > 200$ $), which expression represents the monthly profit?

A company manufactures smart thermostats. The monthly cost of production includes a fixed cost of $15,000. The production cost per unit is $80 for the first 1,000 units produced in a month. For any additional units produced beyond 1,000, the cost per unit is reduced by 25%. Let $x$be the total number of thermostats produced in a month, where$x > 1000$. Which expression represents the total monthly cost,$C(x)$?

The population of a city is $P_0$. Each year, the population increases by a factor of $g$. However, at the end of each year, after the growth has occurred, a fixed number of people, $K$, emigrates from the city. Which recurrence relation describes the population $P_n$ after $n$ years?

A company's revenue $R(x)$ from selling $x$ units is given by $R(x) = px$, where the price $p$ is not constant but depends on the number of units sold according to the linear demand equation $p = a - bx$, where $a$ and $b$ are positive constants. The cost $C(x)$ of producing $x$ units consists of a fixed cost $F$ and a variable cost of $v$ per unit. Which expression represents the company's profit function, $\pi(x)$?

A survey of students revealed their usage of three social media platforms. Let $T$, $F$, and $C$ be the sets of students who use each platform. The number of students in various categories is given by $|T|$, $|F|$, $|C|$, $|T \cap F|$, $|T \cap C|$, $|F \cap C|$, and $|T \cap F \cap C|$. Which expression correctly calculates the number of students who use exactly one of these three platforms?