Poklicna matura (Vocational Matura Slovenia) · Applied Mathematics

Commercial Arithmetic, Applied Algebra, Geometry and Statistics

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A wholesale product has a base price of $200 EUR$ before tax. With standard Slovenian VAT at $22%$ , what is the final gross price?
A retailer buys an electric drill for $80 EUR$ and applies a markup of $35%$ on the cost price. What is the selling price before tax?
During an autumn sale, a laptop originally priced at $650 EUR$ is discounted by $20%$ . What is the discounted sales price?
A gross invoice including $22%$ VAT totals $366 EUR$ . What is the net price of the goods before tax?
A machine purchased for $12000 EUR$ depreciates by $15%$ of its value each year under the declining-balance method. What is its book value after one year?
A deposit of $4000 EUR$ is placed in a bank account offering simple annual interest of $3.5%$ . How much interest is earned after $3$ years?
A business takes out a short-term bank credit of $10000 EUR$ for $90$ days at a simple annual interest rate of $6%$ , using an ordinary $360$ -day business year. What is the total interest owed?
An investment of $2500 EUR$ is deposited at a compound interest rate of $5%$ compounded annually. What is the total accumulated sum at the end of $2$ years?
A technical workshop produces two models of valves. A bulk order of $500$ units contains $320$ standard valves and the rest are precision valves. What percentage of the order is precision valves?
A supplier offers trade discounts of $10%$ followed by an additional cash discount of $5%$ on the remaining balance. What single net multiplier applies to the original list price?
A firm purchases raw materials for $8000 EUR$ plus $22%$ VAT and sells the processed goods for $14000 EUR$ plus $22%$ VAT. What net VAT amount must the firm pay to the tax authority?
A delivery vehicle valued at $20000 EUR$ depreciates by straight-line depreciation over $5$ years with zero salvage value. What is the annual depreciation expense?
An employee in Ljubljana has a gross monthly salary of $1800 EUR$ . Social security contributions deducted at source total $22.1%$ . What is the amount deducted for social security?
A commercial loan of $15000 EUR$ is repaid in $12$ equal monthly principal installments plus interest. What is the monthly principal portion of each installment?
A manufacturing plant produces $480$ parts per hour at $80%$ capacity. How many parts per hour does the plant produce at $100%$ capacity?
A retail item is priced at $122 EUR$ including $22%$ VAT. If the government temporarily reduces the VAT rate to $9.5%$ and the base net price remains unchanged, what is the new gross price?
A capital fund grows from $8000 EUR$ to $10648 EUR$ over $3$ years with annual compounding. What is the constant annual compound growth rate $r$ ?
An annuity pays $500 EUR$ at the end of each year for $3$ years. With an annual discount rate of $4%$ , what is the present value of this ordinary annuity?
A piece of industrial equipment costing $30000 EUR$ is depreciated using the sum-of-the-years-digits method over $4$ years with zero residual value. What is the depreciation in the first year?
A company offers credit terms of 2/10 net 30. What effective annual interest rate does a customer roughly forfeit by foregoing the $2%$ cash discount and paying on day $30$ instead of day $10$ ?
A merchant calculates the gross margin on a product sold for $150 EUR$ with a cost of goods sold of $90 EUR$ . What is the gross margin percentage?
A company issues bonds with a par value of $1000 EUR$ paying an annual coupon of $50 EUR$ . If the current market trading price of the bond is $950 EUR$ , what is its current yield?
A company projects net cash inflows of $4000 EUR$ in year 1 and $6000 EUR$ in year 2. With an initial investment of $8500 EUR$ and a discount rate of $10%$ , what is the project n…
A commercial real estate lease costs $1200 EUR$ per month. If the rent increases by $3%$ annually, what will the monthly rent be during the third year?
A bank requires a debt service coverage ratio of at least $1.25$ . If a company generates net operating income of $50000 EUR$ , what is the maximum annual debt service allowed?
A technical alloy requires copper and zinc in a mass ratio of $7 : 3$ . To produce $50 kg$ of this alloy, how many kilograms of copper are needed?
On an architectural blueprint with a scale of $1 : 50$ , a living room wall measures $12 cm$ in length. What is the actual length of the wall in meters?
Four synchronized automated milling machines produce $120$ precision parts in $3$ hours. Working at the same rate, how many parts will $6$ machines produce in $5$ hours?
A commercial transport company charges a base fee of $15 EUR$ plus $1.80 EUR$ per kilometer driven. If a delivery invoice totals $69 EUR$ , what distance was driven?
A contractor purchases $5$ packs of structural screws and $3$ tubes of adhesive for $54 EUR$ . Another contractor buys $2$ packs of screws and $3$ tubes of adhesive for $36 EUR$ . What is…
A total of $60$ liters of a $10%$ chemical cleaning solution is mixed with $40$ liters of a $25%$ solution. What is the concentration of the resulting mixture?
A manufacturing line has fixed monthly costs of $8000 EUR$ and variable production costs of $12 EUR$ per unit. If each unit sells for $20 EUR$ , what is the break-even volume in un…
A technician tests an electrical circuit where total current $I = 15 A$ splits between two parallel resistors in inverse proportion to their resistances $R_{1} = 4 Ω$ and $R_{2} = 6 Ω$ .…
Two water inlet pipes can fill a municipal storage tank. Pipe A fills the tank alone in $6$ hours, while Pipe B fills it in $3$ hours. How many hours will it take to fill the tank if both pipes operat…
A rectangular parking area has a perimeter of $140 m$ , and its length is $10 m$ longer than its width. What is the width of the parking area?
A warehouse inventory contains $240$ items consisting of standard crates and oversized crates. Standard crates weigh $15 kg$ each and oversized crates weigh $25 kg$ each, giving a total…
A steel cable expands linearly with temperature according to $Δ L = L_{0} α Δ T$ . A $50 m$ bridge cable with coefficient $α = 1.2 \times 1 0^{- 5} K^{- 1}$ undergoes a te…
A production line has a daily cost function $C (x) = 2 x^{2} - 40 x + 500$ , where $x$ is the number of batches produced. How many batches minimize the daily cost?
A small workshop produces tables and chairs. Each table requires $3 hours$ of assembly and each chair requires $1 hour$ . If assembly labor is constrained to at most $60 hours$ per…
A vehicle accelerates uniformly from rest along a test track so that distance covered in meters is $s (t) = 1.5 t^{2}$ . How many seconds does it take to travel $96 meters$ ?
An enterprise model has a total revenue function $R (x) = 120 x - x^{2}$ and total cost function $C (x) = 20 x + 1500$ . What production level $x$ maximizes total profit?
A courier service charges $10 EUR$ for parcels up to $2 kg$ and $3 EUR$ for each additional kilogram or fraction thereof. What is the fee for a package weighing $7.5 kg$ ?
In a commercial bakery, mixer A kneads a batch of dough in $40 minutes$ and mixer B kneads it in $60 minutes$ . Working together, how many minutes does it take to knead the batch?
A technical drafting system maps GPS coordinates using a linear transformation where $x^{'} = 2 x - 3 y$ and $y^{'} = x + 2 y$ . What is the transformed coordinate of the point $(4, 1)$ ?
A chemical storage tank is drained by two valves. Valve 1 empties the full tank in $15 minutes$ . When both valves are open, the tank is emptied in $10 minutes$ . How many minutes would Va…
An investment analysis establishes that project cash flow $F (t)$ follows an exponential decay model $F (t) = 5000 e^{- 0.2 t}$ . At what value of $t$ in years does the cash flow fall to approximately…
A manufacturing plant produces widgets using two inputs: labor $L$ and capital $K$ . The production function is $Q = 10 L^{0.5} K^{0.5}$ . If $16$ units of labor and $25$ units of capital are employed…
A linear programming problem seeks to maximize profit $Z = 5 x + 4 y$ subject to $x + 2 y \leq 14$ , $3 x + y \leq 18$ , $x \geq 0$ , and $y \geq 0$ . At which corner point is profit maximized?
A hydraulic press operates on Pascal's principle where $\frac{F _{1}}{A _{1}} = \frac{F _{2}}{A _{2}}$ . A small piston with area $A_{1} = 5 cm^{2}$ is acted upon by an effort force of $250 N$ . What forc…
A technical firm models the learning curve of its assembly technicians as $T (n) = 80 n^{- 0.322}$ , where $T (n)$ is the hours needed to assemble the $n$ -th unit. If unit $1$ takes $80 hours$ , app…
A rectangular warehouse floor measures $24 m$ in length and $15 m$ in width. What is the total floor area?
A cylindrical water storage tank has an inner diameter of $4 m$ and a height of $5 m$ . What is the total volume of the tank in cubic meters?
A surveying transit measures an angle of elevation of $3 0^{\circ}$ to the top of an antenna mast from a point on level ground $60 m$ from its base. What is the height of the antenna mast?
A right-angled triangular gable roof has a horizontal run of $8 m$ and a vertical rise of $6 m$ . What is the slope length of the rafter?
A steel pipe has an outer radius of $10 cm$ and an inner radius of $8 cm$ . What is the cross-sectional area of the steel ring?
A cone-shaped hopper has a base radius of $3 m$ and a height of $4 m$ . What is the volume of the hopper?
A technician designs a ramp with a horizontal length of $12 m$ and a rise of $1 m$ . What is the gradient of the ramp expressed as a percentage?
A circular gear has a radius of $14 cm$ . What distance along its circumference does a point on the rim travel during a rotation of $9 0^{\circ}$ ?
A concrete pillar has the shape of a regular hexagonal prism with base edge $2 m$ and height $6 m$ . What is the total volume of the pillar?
A ladder of length $5 m$ leans against a vertical wall, making an angle of $6 0^{\circ}$ with the horizontal ground. How high up the wall does the top of the ladder reach?
A land plot has the shape of a trapezoid with parallel sides measuring $30 m$ and $50 m$ , and a perpendicular distance between them of $20 m$ . What is the area of the plot?
A spherical metallic float has a diameter of $60 cm$ . What is the volume of the float in liters, using $π \approx 3.1416$ ?
A road curve forms a circular arc with a central angle of $4 5^{\circ}$ and a curve radius of $200 m$ . What is the length of the curved roadway?
In a triangle, side lengths are $a = 7 m$ , $b = 8 m$ , and $c = 9 m$ . What is the cosine of the angle $γ$ opposite side $c$ ?
A closed rectangular ventilation duct has a cross-section of $0.6 m \times 0.4 m$ and a length of $10 m$ . What is the total external surface area of sheet metal required for its f…
A land plot survey gives triangular vertices at coordinates $(0, 0)$ , $(80, 0)$ , and $(40, 60)$ in meters. What is the area of this parcel of land?
A conical roof has a base diameter of $12 m$ and a slant height of $10 m$ . What is the lateral surface area of the roof covering?
A conveyor belt inclined at an angle $θ$ rises $5 m$ vertically over a belt travel distance of $13 m$ . What is the cosine of the angle of inclination $θ$ ?
A technician cuts a $6 0^{\circ}$ sector from a circular sheet metal disk of radius $12 cm$ . What is the area of this cut sector?
A hydraulic cylinder operates with a bore diameter of $80 mm$ and a piston stroke of $250 mm$ . What is the displacement volume of the cylinder in cubic centimeters?
A solar collector panel of area $2.5 m^{2}$ is tilted at an angle of $3 0^{\circ}$ to the horizontal. What is the area of its horizontal ground shadow when the sun is directly overhead at solar noon…
A hollow concrete retaining wall has a cross-section formed by a trapezoid with base $1.2 m$ , top width $0.4 m$ , and height $3.0 m$ . What volume of concrete is needed for a…
A surveying team measures two points $A$ and $B$ separated by a deep ravine. From an observation post $C$ , distances are measured as $C A = 100 m$ , $C B = 150 m$ , and angle…
A regular pyramid has a square base with edge length $10 m$ and height $12 m$ . What is the total surface area of the four triangular lateral faces?
A parabolic reflector has the equation $y = 0.05 x^{2}$ , where measurements are in centimeters. What is the focal distance $f$ of this reflector from its vertex, given the standard form $x^{2} = 4 f y$ ?
A quality inspector records the following diameter measurements in millimeters: $20.1, 20.3, 20.0, 20.4, 20.2$ . What is the sample mean?
In a shipment of $200$ electrical fuses, $8$ fuses are found to be defective. What is the probability that a randomly chosen fuse from this shipment is defective?
A production supervisor records weekly machine downtime in hours: $4, 7, 2, 9, 5, 8, 3$ . What is the median downtime?
A batch of $500$ printed circuit boards undergoes inspection. If historical quality data shows a defect rate of $1.6%$ , how many defective boards are expected in the batch?
A machine produces steel pins whose lengths have a variance of $0.09 mm^{2}$ . What is the standard deviation of the pin lengths?
In a dual-component safety system, Component A functions with probability $0.95$ and Component B functions independently with probability $0.90$ . What is the probability that both components function?
A technician measures the voltage output of five power supplies: $12.0 V, 12.2 V, 11.8 V, 12.1 V, 11.9 V$ . What is the range of these voltage readings?
An assembly line produces parts with two independent stages. The probability of a defect at Stage 1 is $0.03$ and at Stage 2 is $0.02$ . What is the probability that a finished part has no defects from…
A survey of $120$ technical apprentices reveals that $72$ prefer computer-aided design (CAD), $36$ prefer computerized machining (CNC), and $12$ prefer manual drafting. What is the relative frequency…
A technician takes $6$ repeated temperature readings in Celsius: $45, 47, 50, 52, 53, 55$ . What is the interquartile range (IQR) of these readings, taking $Q_{1} = 47$ and $Q_{3} = 53$ ?
A batch of $10$ electric motors contains $2$ defective units. If a technician randomly selects $2$ motors without replacement, what is the probability that neither motor is defective?
A machine produces bolts whose diameters are normally distributed with mean $μ = 10.00 mm$ and standard deviation $σ = 0.05 mm$ . What percentage of bolts fall between…
A testing laboratory evaluates the compressive strength of $4$ concrete cubes: $28, 30, 32, 34 MPa$ . What is the sample standard deviation $s$ of these measurements?
A delivery vehicle makes $5$ independent deliveries each day. The probability of being delayed at any delivery is $0.2$ . What is the probability that the vehicle is delayed at exactly $1$ delivery on…
In an electronic testing setup, test equipment correctly identifies a defective circuit with probability $0.98$ (sensitivity). If a batch has a true defect prevalence of $2%$ , what is the probability…
A regression line between advertising expenditure $x$ in thousands of euros and sales revenue $y$ in thousands of euros is estimated as $y = 15 + 4.5 x$ . If $x = 8 thousand EUR$ , what is the pr…
A manufacturing line produces steel rods with a target mean length of $200 mm$ . A sample of $25$ rods has mean $\overset{x}{ˉ} = 202 mm$ and standard error $S E = 1.0 mm$ . What is the $t$ …
A factory has two machines producing screws: Machine 1 produces $60%$ of all screws and has a defect rate of $2%$ , while Machine 2 produces $40%$ of all screws and has a defect rate of $4%$ . What…
Using Bayes' theorem for the screws in the previous problem, if a randomly chosen screw is found to be defective, what is the probability that it was produced by Machine 1?
A continuous random variable $X$ has probability density function $f (x) = 2 x$ for $0 \leq x \leq 1$ and $0$ elsewhere. What is the expected value $E [X]$ ?
A quality control team plots sample means on an $\overset{ˉ}{X}$ -chart where individual part weights have mean $μ = 50 g$ and standard deviation $σ = 2 g$ . For samples of size $n = 16$ …
A technician tests a sample of $100$ components and finds $5$ failures. What is the upper bound of the $95%$ confidence interval for the proportion of failures, using $z = 1.96$ and…
An automotive battery manufacturer claims that average battery life is $48 months$ with standard deviation $6 months$ . Assuming a normal distribution, what fraction of batteries fail bef…
The correlation coefficient between machine operating hours $x$ and maintenance cost $y$ is calculated as $r = 0.8$ . What percentage of the variation in maintenance cost is explained by machine operat…
A failure-rate model gives the time to failure $T$ of a cutting tool as an exponential distribution with rate parameter $λ = 0.02 per hour$ . What is the mean time to failure (MTTF)?

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