Exercises: Simple proportion

At a greengrocer's, $5$ kg of apples cost $8,50$ €. How much will $3$ kg of the same apples cost?

A team of $4$ painters takes $6$ days to paint the facade of a building. If the company only has $3$ painters available for the next identical job, how many days will they take to finish it?

A cake recipe for $6$ people requires $150$ grams of sugar. If we want to adapt the recipe to serve $10$ people, how many grams of sugar will we need?

A car travelling at a constant speed of $90$ km/h takes $4$ hours to complete a journey between two cities. At what constant speed should it travel to complete the same journey in $3$ hours?

On a road map, a distance of $5$ cm represents $600$ km in reality. If two cities are separated by $12$ cm on the map, what is the actual distance between them in kilometres?

To fill a water tank, $8$ identical taps pouring water at the same rate are used, taking $12$ hours to fill it completely. If we want the tank to be filled in $16$ hours, how many taps identical to the previous ones must be left open?

A machine in a factory produces $450$ parts in $3$ hours of continuous operation. How many parts will it produce if it is kept running for $8$ hours at the same rate?

The prize of a raffle is divided equally among the winners. If there are $5$ winners, each receives $120$ €. How much money would each receive if there had been $8$ winners?

According to technical data from the IDAE (Institute for the Diversification and Saving of Energy), an efficient petrol car consumes an average of $5,4$ litres for every $100$ km travelled on a dual carriageway at a constant legal speed. A family is going to make a one-way trip from Madrid to Valencia along the A-3 dual carriageway, covering a distance of $355$ km.

1. a) Briefly explain why the total petrol consumption and the kilometres travelled are directly proportional quantities on this journey.
2. b) Using a direct rule of three or by reduction to the unit, calculate how many litres of petrol the vehicle will consume during the entire one-way journey.
3. c) If the price of 95-octane petrol at the service station is $1,62$ € per litre, how much money will they spend on fuel during that trip? Round the result to the nearest euro cent.

To prepare and pack $300$ food packages for a school charity drive, $3$ volunteers take $40$ minutes working at the same constant pace. For the next batch of identical packages, $2$ more people join, so there are now $5$ volunteers working at the same pace.

Lucía tries to calculate how long they will take and writes in her notebook:

"The more people, the more time it will take:

Therefore, they will take almost $67$ minutes."

1. a) Explain in your own words, without performing calculations, why Lucía's conclusion makes no sense in real life.
2. b) Point out the mathematical error Lucía made in her setup and indicate which numerical value remains constant when the number of people changes (that is, what the constant of the product $x \cdot y$ represents in this context).
3. c) Correct the setup and calculate the exact time the $5$ volunteers will take.

With a stable internet connection, downloading a $3,2$ GB file takes exactly $15$ minutes. If you need to download a $8,4$ GB update, how long will it take? Express the final result in minutes and seconds.

A gear transmission system consists of a driving gear with $45$ teeth rotating at $120$ revolutions per minute (rpm). This gear is meshed with a second driven gear with $36$ teeth. At what speed (in rpm) will the second gear rotate?

At a currency exchange bureau, a British tourist found that changing $125$ £ (pounds sterling) gave her $145$ €. The next day, at the same exchange rate, a Spanish tourist wants to obtain $250$ £. How many euros will he have to give?

A water extraction pump operates at a flow rate of $2,5$ litres per second and takes $4$ hours and $30$ minutes to empty a swimming pool. If it is replaced by a more powerful pump with a flow rate of $3,0$ litres per second, how long will it take to empty the same pool?

In a chemistry experiment, it is known that the mass of a compound is directly proportional to the amount of substance measured in moles. If $3,5$ moles of this compound have a mass of $154$ grams, what will be the exact mass of a sample containing $6,2$ moles of the same compound?

A plot of agricultural land is rectangular with a length of $120$ m and a width of $85$ m. The local council expropriates the land, but offers in exchange another rectangular plot with the exact same area. If the new plot has a length of $150$ m, what must its width be?

The following graph shows the relationship between the weight of a metal alloy and its market cost. Observing point $A$ marked on the graph, determine the constant of proportionality and calculate how much a $45$ kg piece of this alloy will cost.

According to Boyle's Law, at constant temperature, the volume occupied by a gas is inversely proportional to the pressure to which it is subjected. A gas occupies a volume of $15,6$ litres when the pressure is $1,2$ atmospheres. What pressure in atmospheres will need to be applied for the gas to be compressed until it occupies exactly $10,4$ litres?

A regular intercity bus covers a $108$ km stretch along the AP-7 motorway. At an average speed of $120$ km/h, it takes exactly $54$ minutes to complete the journey. On a day with heavy rain and dense traffic regulated by the DGT, the average speed of the bus on that same stretch drops to $80$ km/h.

1. a) Before doing any exact numerical calculation, make a reasoned estimate: will the new travel time be greater or less than one hour? Will it be closer to $60$ minutes or to $90$ minutes? Justify your mental estimate.
2. b) Identify the type of proportionality that exists between the average speed and the time spent travelling a fixed distance. What is the value and what are the units of the constant of proportionality in this case?
3. c) Set up the corresponding proportion, calculate the exact time (in minutes) that the bus will take at $80$ km/h, and check whether your initial estimate in part a) was accurate.

In a municipal market butcher's shop, a sign states that a $350$ g tray of turkey breast fillets costs $4,20$ €. Two customers, Sara and Mario, want to find out the price of a family-sized tray of $1,4$ kg of the same product.

• Sara says: "I prefer to reduce to the unit: I calculate how much $1$ kg costs by dividing $4,20$ by $0,35$ kg, and then I multiply that price by $1,4$."

• Mario says: "It is easier to observe the relationship between the data: $1,4$ kg is $1\,400$ g, and since $1\,400$ is exactly four times $350$ g, I only have to multiply $4,20$ by $4$."

1. a) Carry out the calculations following Sara's method and Mario's method. Check whether both obtain the same final result.
2. b) Justify which of the two methods seems more efficient and convenient to you for solving this specific case without needing a calculator.
3. c) If they later want to know the price of a small tray weighing $210$ g, is Mario's direct reasoning still as advantageous compared to Sara's? Justify which of the two paths it would be better to take in this second case and calculate the price for those $210$ g.

It is known that a quantity $Y$ is directly proportional to another quantity $X$. When $X = a$, the value of quantity $Y$ is $b + 3$. If the value of $X$ quadruples (that is, $X = 4a$), determine the algebraic expression that defines the new value of $Y$ in terms of $b$.

Two physical quantities $P$ and $Q$ are inversely proportional. If in a given process quantity $P$ undergoes an increase of $25\%$, by what exact percentage will quantity $Q$ decrease? Prove your answer using the constant of proportionality.

The following graph shows the relationship between two inversely proportional quantities $X$ and $Y$. Two points $P(x_1, y_1)$ and $Q(x_2, y_2)$ are marked on the curve (hyperbola), and the rectangles they form with the coordinate axes have been drawn. Prove algebraically, using the definition of inverse proportion, why the area of the blue rectangle is exactly equal to the area of the red rectangle, regardless of the points chosen on the curve.

According to Hooke's Law, the extension of a spring is directly proportional to the mass attached to it. When attaching a mass $M$, the spring extends by $L$ centimetres. If we add an additional mass of $m$ kilograms, the total extension becomes $L + \Delta L$. Express the initial mass $M$ in terms of $m$, $L$ and $\Delta L$ only.

A team of $N$ workers agrees to finish an engineering project in $D$ days working at a constant rate. After exactly a quarter of the scheduled days ($D/4$) have elapsed, a third of the workers leave the project due to an unforeseen circumstance. The rest of the team continues working at the same rate. Express, in terms of $D$, the total number of days the entire project will take from start to finish.

A 3D printer takes $45$ seconds to print a plastic layer $0,2$ mm thick. The aim is to print a solid object of height $H$ mm. If the operator changes the setting so that each layer has a thickness of $0,15$ mm (which keeps exactly the same printing time per layer of $45$ seconds, but requires more layers), printing the complete object will take $2$ hours and $5$ minutes longer than with the original setting. Set up the necessary proportional relationships and determine the height $H$ of the object.

In a long-distance race, the average speed of a runner and the time taken to reach the finish line are inversely proportional quantities. Runner A finishes the course in $t$ minutes. Runner B runs at an average speed that is $20\%$ higher than that of Runner A. Express the time taken by Runner B in terms of $t$.

It is known that the mass of a solid steel sphere is directly proportional to its volume. A student claims that if we double the radius of the sphere, its mass will also double by applying direct proportion with respect to the radius. Is this statement correct? Knowing that a sphere with a radius of $2$ cm has a mass of $260$ g, calculate the actual mass of a sphere made of the same material with a radius of $3$ cm, demonstrating why applying direct proportion with respect to the radius is a conceptual error in this case.

You are designing a written Mathematics test on proportionality for your classmates, and you want to create a problem about the electricity cost of running a water drainage pump.

The conditions that your exercise must meet are:

1. The cost in euros (€) is directly proportional to the operating time in hours (h).

2. The final solution that your classmates must obtain must be exactly a cost of $17,50$ € for an operating time of $3,5$ hours.

3. In the problem statement that you write, as the initial given data, the time must be an integer number of hours and the cost must be an integer number of euros strictly less than $15$ €.

1. a) Find the value of the constant of direct proportionality $k$ (in €/h) required by condition 2.
2. b) Deduce which pairs of given data $(t\text{ hours}, C\text{ euros})$ simultaneously satisfy condition 3 and the calculated constant. Choose one of those pairs for your problem.
3. c) Write the complete, self-contained statement of the problem you invented, using the chosen pair of data. Then, solve it step by step by reduction to the unit to show that the solution is exactly $17,50$ €.

In road safety studies by the Dirección General de Tráfico (DGT), stopping a vehicle is analysed by dividing the total distance into two distinct phases:

• Reaction distance: distance travelled at a constant speed during the time it takes the driver to perceive the hazard and press the brake pedal (estimated at $1$ second on average).

• Braking distance: distance covered by the car from the moment the brake is applied until it comes to a complete stop due to friction.

According to DGT tests, at $50$ km/h a passenger car covers a reaction distance of approximately $13,9$ metres.

1. a) Justify why the reaction distance and the driving speed are directly proportional quantities. What physical quantity is represented in this case by the constant of proportionality $\dfrac{\text{distance}}{\text{speed}}$?
2. b) Using a direct rule of three, calculate the reaction distance of that same driver if travelling on a dual carriageway at the maximum legal speed of $120$ km/h. Round the result to the nearest tenth of a metre.
3. c) An inexperienced driver claims: "If at $50$ km/h the braking distance on dry asphalt is $12$ metres, at $120$ km/h it will be $28,8$ metres, since I apply a direct rule of three by multiplying by $\frac{120}{50} = 2,4$." However, actual DGT measurements show that at $120$ km/h the braking distance is around $69$ metres. Reason mathematically why the braking distance is NOT proportional to the speed and explain the serious road safety danger of applying simple proportionality to quantities that do not satisfy it.