Quiz: Linear Equations and Matrix Algebra — 64 questions

1. What distinguishes an algebraic expression from an equation?

A. An algebraic expression contains variables, while an equation contains numerical constants without variables.

B. An algebraic expression describes a graph, while an equation describes only a numerical calculation.

C. An algebraic expression links quantities through operations, while an equation asserts equality between two expressions.

D. An algebraic expression asserts equality between quantities, while an equation links quantities through operations.

Answer: (C)

Explanation

An algebraic expression combines quantities using mathematical operations without necessarily asserting equality. An equation, by contrast, states that two expressions have equal values.

2. Which statement best defines a linear equation?

A. It contains variables raised to the first power along with constants or constants multiplied by those variables.

B. It contains at least one variable raised to a power greater than one and may include several constants.

C. It relates two expressions through equality but permits variables with any whole-number exponent.

D. It contains a variable in a denominator and combines it with constants through multiplication.

Answer: (A)

Explanation

A linear equation has variables of degree one and includes constants or constants multiplied by first-power variables. Equations with higher powers or variables in denominators do not meet this definition.

3. What does the slope of a line measure?

A. The change in the value of $X$ for a one-unit change in $Y$, indicating horizontal position.

B. The value of $Y$ where the line crosses the vertical axis, indicating its starting height.

C. The change in the value of $Y$ for a one-unit change in $X$, indicating the line’s steepness.

D. The total distance between two points on the line, indicating the length of the graph.

Answer: (C)

Explanation

Slope measures the change in $Y$ relative to a one-unit change in $X$ and therefore describes how steeply the line rises or falls. The vertical-axis crossing is the intercept, not the slope.

4. In the linear form $Y=mX+b$, what do $m$ and $b$ represent?

A. $m$ is the slope and $b$ is the $Y$-intercept.

B. $m$ is the fixed cost and $b$ is the variable cost.

C. $m$ is the $Y$-intercept and $b$ is the slope.

D. $m$ is the dependent variable and $b$ is the independent variable.

Answer: (A)

Explanation

In $Y=mX+b$, $m$ determines the rate of change and therefore is the slope, while $b$ is the value of $Y$ when $X=0$. The variables $X$ and $Y$ represent the independent and dependent variables, respectively.

5. Which set of information can be used with the slope-intercept form to develop a line equation?

A. The line’s slope and its vertical change.

B. One coordinate and the line’s slope.

C. The line’s slope and its $Y$-intercept.

D. Two coordinates located on the line.

Answer: (C)

Explanation

The slope-intercept form is built from a slope and a $Y$-intercept. Two coordinates instead provide the starting information for the two-point approach, while one point and a slope support the slope-point form.

6. Which equation represents a line with slope $m$ and $Y$-intercept $b$?

A. $Y=mX+b$

B. $m=\frac{Y_2-Y_1}{X_2-X_1}$

C. $Y-Y_1=m(X-X_1)$

D. $X=\text{constant}$

Answer: (A)

Explanation

The slope-intercept equation is $Y=mX+b$, where $m$ is the slope and $b$ is the $Y$-intercept. The other expressions represent slope-point form, the slope calculated from two points, and a vertical line.

7. A non-vertical line has slope $3$ and passes through $(2,5)$. Which equation is its slope-point form?

A. $Y=3X+5$

B. $Y-5=3(X-2)$

C. $Y+5=3(X+2)$

D. $Y-2=3(X-5)$

Answer: (B)

Explanation

Substituting $m=3$, $X_1=2$, and $Y_1=5$ into $Y-Y_1=m(X-X_1)$ gives $Y-5=3(X-2)$. The expression $Y=3X+5$ incorrectly treats the point’s $Y$-coordinate as the intercept.

8. How can a line be identified as horizontal rather than vertical?

A. Its slope is negative, so its equation has a changing $X$-value.

B. Its slope is positive, so its equation has a changing $Y$-value.

C. Its points have the same $X$-coordinate, so its equation is $X=\text{constant}$.

D. Its points have the same $Y$-coordinate, so its equation is $Y=\text{constant}$.

Answer: (D)

Explanation

A horizontal line keeps $Y$ constant while $X$ changes, so it has the form $Y=\text{constant}$. Equal $X$-coordinates instead characterize a vertical line.

9. Which expression correctly represents total cost in terms of total variable cost and total fixed cost?

A. $TC=VC\times FC$

B. $TC=VC+FC$

C. $TC=FC-VC$

D. $TC=VC-FC$

Answer: (B)

Explanation

Total cost is the sum of variable and fixed costs, so $TC=VC+FC$. Fixed cost does not get subtracted from variable cost or multiplied by it in this definition.

10. A business sells each unit for $P$ and sells $Q$ units. Which pair of formulas gives total revenue and profit?

A. $TR=\frac{P}{Q}$ and $\Pi=TR+TC$

B. $TR=P+Q$ and $\Pi=TC-TR$

C. $TR=PQ$ and $\Pi=TR-TC$

D. $TR=TC-PQ$ and $\Pi=TR+TC$

Answer: (C)

Explanation

Sales revenue equals unit price multiplied by quantity, $TR=PQ$, and profit equals revenue minus total cost, $\Pi=TR-TC$. Fixed or total costs are not added to revenue to calculate profit.

11. In a linear cost model, what happens to average fixed cost as production quantity increases?

A. It decreases because the fixed cost is spread across more units.

B. It becomes equal to marginal cost because variable cost is constant per unit.

C. It increases because producing more units raises total fixed cost.

D. It remains constant because fixed cost does not vary with output.

Answer: (A)

Explanation

Average fixed cost falls as quantity increases because the same fixed cost is allocated across more units. Fixed cost itself remains constant, but its per-unit average does not.

12. A company reaches its break-even point when which condition holds?

A. Total variable cost equals total fixed cost, so production becomes cost-free.

B. Unit selling price equals total revenue, so the company covers each expense.

C. Total revenue equals total cost, so the company makes neither profit nor loss.

D. Total revenue exceeds total cost, so the company earns its maximum profit.

Answer: (C)

Explanation

At break-even, total revenue and total cost are equal, producing zero profit and zero loss. A revenue amount greater than total cost would instead indicate a profit.

13. What distinguishes a matrix from a scalar?

A. A matrix is a rectangular array of numbers, whereas a scalar is one real number

B. A matrix is one real number, whereas a scalar is a rectangular array of numbers

C. A matrix is a square array, whereas a scalar is any rectangular array

D. A matrix is a list of variables, whereas a scalar is a geometric vector

Answer: (A)

Explanation

A matrix contains numbers arranged in rows and columns, while a scalar is a single real number. Confusing a matrix with a scalar reverses these two definitions.

14. In a matrix, what does the element $a_{ij}$ identify?

A. The entry on diagonal position $i+j$

B. The entry in row $j$ and column $i$

C. The entry in row $i$ and column $j$

D. The entry in the matrix's $i$th column and $j$th row

Answer: (C)

Explanation

The first subscript gives the row and the second gives the column, so $a_{ij}$ is in row $i$ and column $j$. Reversing the subscripts changes the element's location.

15. Which statement correctly describes an identity matrix?

A. It is square, has ones on the main diagonal, and leaves conformable matrices unchanged under multiplication

B. It is rectangular, has ones in every position, and leaves matrices unchanged under addition

C. It is diagonal, has arbitrary diagonal entries, and produces a zero matrix under multiplication

D. It is square, has zeros on the main diagonal, and changes matrix dimensions under multiplication

Answer: (A)

Explanation

An identity matrix is square with ones on its primary diagonal and zeros elsewhere, and it acts as a multiplicative identity. A rectangular matrix with ones throughout does not have this property.

16. Two matrices can be added or subtracted when which condition holds?

A. They have the same number of rows and the same number of columns

B. They are both square, even if their dimensions differ

C. Their corresponding entries are equal before the operation

D. The first matrix has as many columns as the second matrix has rows

Answer: (A)

Explanation

Addition and subtraction require equal dimensions, after which corresponding entries are combined. Matching the first matrix's columns to the second matrix's rows is the condition for multiplication instead.

17. If a scalar $X$ multiplies a matrix $A$, which equality expresses commutativity of scalar multiplication?

A. $XA=X+A$

B. $XA=A-X$

C. $XA=A+X$

D. $XA=AX$

Answer: (D)

Explanation

A real scalar can be placed on either side of a matrix without changing the scalar-matrix product, so $XA=AX$. The additive expressions do not represent scalar multiplication.

18. Which equation correctly applies distributivity to a scalar multiplying a matrix sum?

A. $(X+Y)A=XYA$

B. $X(A+B)=XA+XB$

C. $X(A+B)=X+A+B$

D. $X(A+B)=AB+X$

Answer: (B)

Explanation

Scalar multiplication distributes across matrix addition, giving $X(A+B)=XA+XB$. Combining the scalars as $XYA$ applies to a different expression involving scalar multiplication.

19. If $A$ has dimension $4\times 3$ and $B$ has dimension $3\times 2$, what is the dimension of $AB$?

A. $3\times 3$

B. $2\times 4$

C. $4\times 3$

D. $4\times 2$

Answer: (D)

Explanation

The inner dimensions, $3$ and $3$, match, so the product is conformable and has the outer dimensions $4\times 2$. The other dimensions either preserve an input size or reverse the outer dimensions.

20. Which statement correctly characterizes matrix multiplication?

A. It is associative and distributive, but it is not generally commutative

B. It is commutative and associative, but it does not distribute over addition

C. It is generally neither associative nor distributive

D. It is distributive and commutative, but it is not associative

Answer: (A)

Explanation

Matrix multiplication satisfies associativity and distributivity, but products may depend on order, so $AB$ does not generally equal $BA$. Treating matrix multiplication like ordinary scalar multiplication creates the commutativity error.

21. For a square matrix $A$, what condition defines $A^{-1}$ as its inverse?

A. $AA^{-1}=I$ while $A^{-1}A=0$

B. $AA^{-1}=A^{-1}A=I$

C. $AA^{-1}=A$ and $A^{-1}A=A^{-1}$

D. $AA^{-1}=A^{-1}A=0$

Answer: (B)

Explanation

An inverse is defined by producing the identity matrix on both sides, so $AA^{-1}=A^{-1}A=I$. A zero product does not establish that a matrix is invertible.

22. How does an invertible matrix differ from a singular matrix?

A. An invertible matrix has no inverse, whereas a singular matrix has a unique inverse

B. An invertible matrix has determinant zero, whereas a singular matrix has an identity inverse

C. An invertible matrix must be rectangular, whereas a singular matrix must be square

D. An invertible matrix has an inverse, whereas a singular matrix has no inverse

Answer: (D)

Explanation

An invertible, or nonsingular, matrix possesses an inverse, while a singular matrix does not. The determinant and shape claims in the other choices do not state this distinction correctly.

23. Which statement about matrix inverses is correct?

A. Only diagonal matrices can have inverses, and their inverses need not be unique

B. Any matrix with a zero entry has an inverse, and that inverse is unique

C. Every rectangular matrix has an inverse, and it may have several inverses

D. Only square matrices can have inverses, and an existing inverse is unique

Answer: (D)

Explanation

Inverses are defined for square matrices, and whenever an inverse exists it is unique. Rectangular shape and diagonal structure are not sufficient conditions for invertibility.

24. What is the standard row-operation procedure for finding the inverse of a square matrix $A$?

A. Augment $A$ with $I$, row-reduce the left side to $I$, and read the inverse on the right

B. Augment $A$ with a zero matrix, row-reduce the right side to $I$, and read the inverse on the left

C. Add the rows of $A$ together and place the resulting vector beside the original matrix

D. Multiply $A$ by itself until the left side becomes zero, then use the resulting matrix

Answer: (A)

Explanation

The augmented matrix $[A\mid I]$ is transformed through elementary row operations into $[I\mid A^{-1}]$. Reducing the right side or using a zero matrix does not implement the inverse-finding method.

25. Which sequence correctly describes the inverse method for solving $AX=B$?

A. Substitute each constant into the equations, eliminate variables, and verify $A=B$

B. Augment $A$ with $B$, row-reduce to identity form, and read off $X$

C. Convert to matrix form, find $A^{-1}$, and compute $X=A^{-1}B$

D. Find the determinant of $B$, invert $B$, and compute $X=B^{-1}A$

Answer: (C)

Explanation

The inverse method solves the system by obtaining the inverse of the coefficient matrix and multiplying it by the constant vector. Row reduction describes Gaussian elimination rather than the inverse method.

26. For which coefficient matrix can the inverse method be applied to a system $AX=B$?

A. A square matrix that is invertible

B. A square matrix whose determinant equals zero

C. A rectangular matrix with dependent columns

D. A coefficient matrix from any consistent system

Answer: (A)

Explanation

The inverse method requires a square coefficient matrix with an inverse. A zero determinant means the matrix is not invertible, and consistency alone does not guarantee that the inverse method applies.

27. What does Gaussian elimination do after adjoining the constant vector to a coefficient matrix?

A. It applies elementary row operations to reveal the solution type

B. It multiplies the coefficient matrix by its own transpose

C. It replaces every equation with a probability transition rule

D. It calculates the inverse of the constant vector before solving

Answer: (A)

Explanation

Gaussian elimination augments the coefficient matrix with the constants and uses elementary row operations to reach a revealing form. Computing an inverse is associated with the inverse method, not the defining step of Gaussian elimination.

28. A reduced augmented matrix contains a row whose coefficient entries are all zero while its constant entry is nonzero; what does this show?

A. The system requires an inverse coefficient matrix

B. The system has infinitely many solutions

C. The system has exactly one solution

D. The system has no solution

Answer: (D)

Explanation

A zero coefficient row paired with a nonzero constant represents a contradiction, so the system has no solution. Free variables in a consistent system, rather than a contradiction, indicate infinitely many solutions.

29. Which procedure best represents a word problem as a solvable algebraic model?

A. Assign every quantity a different variable, substitute randomly, and simplify

B. Choose numerical values, form a matrix, invert it, and accept the result

C. Solve each sentence separately, ignore units, and compare the final numbers

D. Define an unknown, express related quantities, form equations, solve, and check

Answer: (D)

Explanation

A reliable solution defines an unknown, relates other quantities to it, translates the information into equations, solves, and checks the result against the statement. Omitting the check can preserve an algebraic result that does not fit the original conditions.

30. What does a Markov chain model?

A. How a system evolves through time using transition probabilities between states

B. How a fixed equation produces a single output from a selected input

C. How a database stores unrelated records without temporal relationships

D. How a geometric figure changes shape under a sequence of reflections

Answer: (A)

Explanation

A Markov chain is a probabilistic model of repeated changes between states, described by transition probabilities. A fixed input-output equation does not capture the state-to-state evolution that defines a Markov chain.

31. What information determines the state of a Markov system during a given period?

A. The number of periods elapsed without any state information

B. The long-run equilibrium distribution without the preceding state

C. The preceding state together with the transition probabilities

D. The initial state alone, regardless of transition probabilities

Answer: (C)

Explanation

The Markov rule uses the preceding state and the transition probabilities to determine the next period's state distribution. The long-run distribution describes eventual behavior but does not replace the preceding state in the step-by-step update.

32. What is the transition probability from state $i$ to state $j$?

A. The probability that the system began in state $i$ at the first period

B. The fraction of time eventually spent in state $j$ after equilibrium

C. The total number of transitions observed between all pairs of states

D. The likelihood of moving from state $i$ to state $j$ in the next period

Answer: (D)

Explanation

A transition probability measures the likelihood of moving from the current state $i$ to state $j$ during the next period. A long-run fraction describes a steady-state distribution rather than a single-period transition.

33. If a row state vector is used, how is the next state vector calculated from the current vector?

A. $V(n)=PV(n-1)$

B. $V(n)=V(n-1)+P$

C. $V(n)=V(n-1)P$

D. $V(n)=P^{-1}V(n-1)$

Answer: (C)

Explanation

With row state vectors, the next vector is obtained by multiplying the previous vector on the right by the transition matrix, giving $V(n)=V(n-1)P$. Left multiplication corresponds to a different vector convention and is not the stated rule here.

34. What is the long-run distribution in the two-store example?

A. One-third of customers are at store 1 and two-thirds at store 2

B. Half of the customers are at each store after many periods

C. Two-thirds of customers are at store 1 and one-third at store 2

D. All customers eventually concentrate at store 1

Answer: (C)

Explanation

The stated steady-state distribution places two-thirds of customers at store 1 and one-third at store 2. Equal shares and complete concentration at one store are different long-run distributions.

35. What do decision variables represent in a linear programming model?

A. Unknown controllable quantities that the decision maker must determine

B. Fixed numerical effects assigned to resources and production activities

C. Restrictions that limit which proposed actions are feasible

D. Outcomes showing the unused portion of each scarce resource

Answer: (A)

Explanation

Decision variables are the unknown controllable quantities whose values the decision maker selects. Fixed numerical effects are parameters, while restrictions are constraints and unused resources are slack.

36. Which statement best describes the objective of a linear programming model?

A. It balances two unrelated quantities by optimizing both at once

B. It lists the controllable quantities that management must choose

C. It identifies every legal and contractual restriction on decisions

D. It optimizes one measurable quantity, such as profit or cost

Answer: (D)

Explanation

A linear programming model has one objective, which may involve maximizing profit or minimizing cost. Constraints describe restrictions, and decision variables identify the quantities being chosen.

37. Which set contains the main assumptions of linear programming models?

A. Linearity, divisibility, certainty, and non-negativity

B. Scarcity, competition, forecasting, and market equilibrium

C. Continuity, randomness, integrality, and unrestricted values

D. Profitability, flexibility, uncertainty, and proportionality

Answer: (A)

Explanation

The standard assumptions are linearity, divisibility, certainty, and non-negativity. The other sets include concepts that are not the model's four main assumptions.

38. A factory cannot exceed its available machine hours or violate a production policy; how are these limits represented in a linear programming model?

A. As decision variables selected by the production manager

B. As parameters measuring the objective's contribution

C. As objective terms measuring the value of production

D. As constraints restricting feasible courses of action

Answer: (D)

Explanation

Constraints represent restrictions such as scarce machine hours and company policies, thereby limiting feasible decisions. Decision variables are the quantities chosen, while objective terms measure what is optimized.

39. Which sequence correctly describes the main steps in formulating a linear programming model?

A. Specify constraints, calculate slack, choose corners, and then define the problem

B. Define the problem, identify variables, state the objective, and specify constraints

C. Choose production quantities, maximize profit, plot results, and formalize the problem

D. Identify the objective, solve graphically, define variables, and estimate parameters

Answer: (B)

Explanation

Formulation begins by defining the problem, then identifies decision variables, specifies the objective function, and identifies constraints. Solving and evaluating corner points occur after formulation.

40. In the microcomputer model, what does the objective function $Z=60X_1+50X_2$ represent?

A. The available capacity of the two production resources

B. The maximum number of computers demanded each week

C. Machine hours required to produce the two computer types

D. Weekly profit from producing the two computer types

Answer: (D)

Explanation

The objective coefficients represent weekly profit contributions, so the expression measures total weekly profit from the two computer types. The inequalities describe resource limits rather than the objective.

41. Why must the microcomputer model include $X_1,X_2\ge0$?

A. Negative production quantities are not meaningful decisions

B. Both computer types must be produced in equal quantities

C. The objective function must contain two positive coefficients

D. The production resources must remain unused after production

Answer: (A)

Explanation

Non-negativity requires each decision variable to be zero or positive because negative production quantities are unrealistic. It does not require equal production or unused resources.

42. What distinguishes the graphical method from the algebraic simplex method in terms of problem size?

A. Simplex handles one objective, while the graphical method handles two

B. Both methods require exactly two variables to produce a solution

C. The graphical method handles many variables, while simplex handles two

D. The graphical method handles two variables, while simplex handles more than two

Answer: (D)

Explanation

Graphical analysis is limited to linear programs with two decision variables, whereas the simplex method can address problems with more variables. Both methods can optimize a single objective.

43. Which procedure correctly identifies an optimum using the graphical method?

A. Find every point on the graph, compare variable values, and select the most balanced point

B. Plot the objective alone, select its highest intercept, and ignore constraint intersections

C. List resource amounts, calculate averages, and choose the largest production quantity

D. Plot constraints, find the feasible region, evaluate its corner points, and choose the best value

Answer: (D)

Explanation

The graphical method plots the constraints, identifies their common feasible region, and evaluates corner points using the objective function. The best maximum or minimum value determines the optimum.

44. Why can an optimal solution to a linear programming problem be found at a corner point of its feasible region?

A. At least one optimum occurs at an extreme point when an optimum exists

B. Corner points are the only points that satisfy every linear constraint

C. The objective function has a different value at every feasible point

D. Interior points cannot represent nonnegative decision variables

Answer: (A)

Explanation

For a linear programming problem with an optimum, at least one optimal solution occurs at an extreme or corner point. Feasible interior points can also satisfy the constraints, and objective values need not differ at every point.

45. What characterizes a binding constraint in a linear programming solution?

A. It has zero slack and limits the solution

B. It has positive slack and exceeds the requirement

C. It measures profit above the objective target

D. It applies to a greater-than-or-equal-to requirement

Answer: (A)

Explanation

A binding constraint is exactly satisfied, so its slack is zero and it restricts the feasible solution. A nonbinding constraint instead has positive slack, meaning some resource or allowance remains unused.

46. A production plan exceeds a required minimum of 120 units by 15 units under a greater-than-or-equal-to constraint. What is the surplus?

A. 120 units

B. 105 units

C. 135 units

D. 15 units

Answer: (D)

Explanation

Surplus measures how much the achieved amount exceeds the required minimum, so it equals $135-120=15$ units. Slack is associated with a less-than-or-equal-to constraint, not this greater-than-or-equal-to requirement.

47. Which description best defines the simplex method?

A. An iterative procedure that improves a feasible solution until no further improvement is possible

B. A graphical procedure that evaluates every point in the feasible region before choosing one

C. A calculation that begins with an arbitrary solution and corrects violations afterward

D. A substitution method that solves the objective equation without checking constraints

Answer: (A)

Explanation

The simplex method starts from a feasible solution and repeatedly improves it until the stopping condition shows that no further improvement is available. An arbitrary starting point can violate constraints, so it does not meet the method's required starting condition.

48. What distinguishes a basic feasible solution from a basic solution?

A. A basic solution must satisfy non-negativity, while feasibility ignores the remaining constraints

B. A basic feasible solution is obtained from objective coefficients without using constraints

C. A basic feasible solution also satisfies all constraints, including non-negativity

D. A basic solution always gives the best objective value, while feasibility concerns resource use

Answer: (C)

Explanation

A basic feasible solution is a basic solution that satisfies every constraint, including non-negativity conditions. A basic solution may come from the constraint equations without meeting all those restrictions.

49. In a maximization simplex problem with less-than-or-equal-to constraints, what sequence correctly describes the main procedure?

A. Graph the constraints, discard the tableau, and compare only the intercepts of the feasible region

B. Replace the objective function with inequalities, select the smallest coefficient, and stop after one pivot

C. Choose the final variables, eliminate all constraints, and calculate profit from the original equations

D. Convert to standard form, build a tableau, pivot through selected variables, and recompute $C_j-Z_j$

Answer: (D)

Explanation

The simplex procedure standardizes the model, creates an initial tableau, selects entering and leaving variables, performs pivot row operations, and recalculates $C_j-Z_j$. It repeats these steps rather than stopping after one pivot or relying on graphical intercepts.

50. When is a maximization simplex tableau optimal?

A. When every entry in the $C_j-Z_j$ row is zero or negative

B. When each basic variable has a positive reduced cost

C. When the objective value becomes zero after the first pivot

D. When every entry in the $C_j-Z_j$ row is positive or zero

Answer: (A)

Explanation

For a maximization problem, the tableau is optimal when the $C_j-Z_j$ row contains no positive values. A positive entry would indicate that an improvement is still possible through another entering variable.

51. What is the optimal production plan in the lawn-mower application?

A. 24 push-type mowers and 42 self-propelled mowers, earning Br. 3600

B. 42 push-type mowers and 24 self-propelled mowers, earning Br. 4020

C. 18 push-type mowers and 48 self-propelled mowers, earning Br. 4020

D. 24 push-type mowers and 42 self-propelled mowers, earning Br. 4020

Answer: (D)

Explanation

The optimal lawn-mower plan produces 24 push-type and 42 self-propelled mowers for a profit of Br. 4020. It leaves 9 engines unused, while assembly and packing resources are fully used.

52. What happens to woodworking capacity in the optimal bentwood furniture plan?

A. Eight woodworking hours remain unused after producing 13 rocking chairs and 4 coffee tables

B. All woodworking hours are used after producing 13 rocking chairs and 4 coffee tables

C. Eight woodworking hours remain unused after producing 13 tables and 4 rocking chairs

D. Eight woodworking hours are added after producing 4 rocking chairs and 13 coffee tables

Answer: (A)

Explanation

The optimal plan produces 13 rocking chairs and 4 coffee tables, achieves a maximum profit of Br. 648, and leaves 8 woodworking hours unused. The unused woodworking capacity shows that other resources determine the limiting combination.

53. What production plan minimizes operating cost in the tire-machine application?

A. 40 tires on machine I and 20 tires on machine II, at a cost of Br. 4020

B. 40 tires on machine I and 20 tires on machine II, at a cost of Br. 3600

C. 20 tires on machine I and 40 tires on machine II, at a cost of Br. 3600

D. 30 tires on machine I and 30 tires on machine II, at a cost of Br. 3600

Answer: (B)

Explanation

The minimum operating cost of Br. 3600 is obtained by producing 40 tires on machine I and 20 tires on machine II. Reversing the machine assignments changes the production plan and is not the stated cost-minimizing solution.

54. What does interest represent in a lending or investment transaction?

A. The total amount repaid before subtracting the principal

B. The price paid for using money over a period of time

C. The number of periods required to repay the borrowed amount

D. The original amount lent, invested, or borrowed

Answer: (B)

Explanation

Interest is the cost of using money over time and is generally expressed as a percentage of the principal. The principal is the original amount, so it is not the payment for using the money.

55. What is the key difference between simple interest and compound interest?

A. Simple interest uses accrued interest, while compound interest uses the initial principal

B. Simple interest uses periodic rates, while compound interest uses annual rates

C. Simple interest applies to investments, while compound interest applies to loans

D. Simple interest uses the initial principal, while compound interest also uses accrued interest

Answer: (D)

Explanation

Simple interest is calculated on the original principal, whereas compound interest is calculated on the principal plus interest accumulated in earlier periods. The distinction concerns the interest base, not whether the transaction is a loan or investment.

56. An account earns simple interest at an annual rate of $6\%$ for $3$ years on a principal of $\mathrm{Br.}\ 2{,}000$. What maturity amount does it produce?

A. $\mathrm{Br.}\ 2{,}600$

B. $\mathrm{Br.}\ 2{,}360$

C. $\mathrm{Br.}\ 2{,}120$

D. $\mathrm{Br.}\ 2{,}380$

Answer: (B)

Explanation

Using $A=P(1+rt)$ gives $A=2{,}000(1+0.06\times3)=\mathrm{Br.}\ 2{,}360$. The interest is based on the original principal rather than on previously earned interest.

57. Which expression gives the present value of a compound amount due after $n$ conversion periods?

A. $P=A\frac{(1+i)^n-1}{i}$

B. $P=A(1+i)^n$

C. $P=A(1+i)^{-n}$

D. $P=A\frac{i}{(1+i)^n-1}$

Answer: (C)

Explanation

Present value discounts the future amount back to the present, so the compound growth factor is raised to a negative exponent. The positive exponent expression calculates a future amount from a present principal.

58. When are payments made in an ordinary annuity?

A. At irregular dates throughout the payment term

B. At the end of each payment period

C. At the midpoint of each payment period

D. At the beginning of each payment period

Answer: (B)

Explanation

An ordinary annuity consists of equal payments made at the end of each period. Payments at the beginning describe an annuity due, not an ordinary annuity.

59. An ordinary annuity pays $\mathrm{Br.}\ 500$ per period for $n$ periods at a periodic interest rate of $i$. Which formula gives its future value?

A. $A=500(1+i)^{-n}$

B. $A=500\frac{(1+i)^n-1}{i}$

C. $A=500\frac{1-(1+i)^{-n}}{i}$

D. $A=500\frac{i}{(1+i)^n-1}$

Answer: (B)

Explanation

The future value of an ordinary annuity is found by accumulating each equal payment to the end of the term, giving $A=R\frac{(1+i)^n-1}{i}$. The expression involving $1-(1+i)^{-n}$ calculates an annuity's present value.

60. Depositing $\mathrm{Br.}\ 100$ at the end of each quarter for one year at $4\%$ compounded quarterly produces approximately what future value?

A. $\mathrm{Br.}\ 400.00$

B. $\mathrm{Br.}\ 406.04$

C. $\mathrm{Br.}\ 416.24$

D. $\mathrm{Br.}\ 404.00$

Answer: (B)

Explanation

The four end-of-quarter deposits earn interest for different lengths of time, producing a future value of approximately $\mathrm{Br.}\ 406.04$. Treating the deposits as earning no interest would give $\mathrm{Br.}\ 400.00$.

61. What is the purpose of a sinking fund?

A. To retire an existing debt through equal periodic payments

B. To accumulate a definite future amount through equal periodic deposits

C. To calculate interest on an original principal during a loan

D. To discount future payments to their value at the beginning

Answer: (B)

Explanation

A sinking fund receives equal deposits to build a specified amount by a specified date. Retiring an existing debt through regular payments is amortization, which has a different purpose.

62. A sinking fund must accumulate a target amount $A$ in $n$ periods at rate $i$ per period. Which formula gives the required periodic deposit?

A. $R=A\frac{1-(1+i)^{-n}}{i}$

B. $R=A\frac{(1+i)^n-1}{i}$

C. $R=A\frac{i}{1-(1+i)^{-n}}$

D. $R=A\frac{i}{(1+i)^n-1}$

Answer: (D)

Explanation

The sinking-fund deposit is obtained by dividing the target amount by the future-value factor for an ordinary annuity, yielding $R=A\frac{i}{(1+i)^n-1}$. The annuity present-value factor applies when valuing a stream of payments at the beginning of the term.

63. What does amortization describe?

A. Accumulating equal deposits to a specified date without reducing debt

B. Finding the current value of payments scheduled for future periods

C. Retiring a debt over time through equal payments that include compound interest

D. Building a future fund through deposits toward a specified target

Answer: (C)

Explanation

Amortization retires a debt through equal periodic payments containing both interest and repayment of principal. A sinking fund builds a future balance, whereas amortization reduces an existing loan balance.

64. A loan has present value $P$, periodic rate $i$, and $n$ equal payments. Which formula gives the amortization payment?

A. $R=P\frac{i}{1-(1+i)^{-n}}$

B. $R=P(1+i)^{-n}$

C. $R=P\frac{1-(1+i)^{-n}}{i}$

D. $R=P\frac{i}{(1+i)^n-1}$

Answer: (A)

Explanation

The amortization payment is the payment that makes the present value of the payment stream equal to the loan amount, so $R=P\frac{i}{1-(1+i)^{-n}}$. The sinking-fund formula uses the future accumulation factor and applies to building a target fund.