Exercises: Direct proportion

State whether the following pairs of quantities are directly proportional. Briefly justify your answer:

1. a) The number of followers of an influencer and the number of likes on their latest photo.
2. b) The weight of apples in kilograms that you buy and the total price to pay.
3. c) A person's age and their height in centimetres.
4. d) The time you leave your phone charging and the percentage of battery restored (assuming it charges at a constant rate).

A group of friends wants to go to the cinema. We know that $3$ tickets cost $22,50$ €. Complete the following table and calculate the constant of proportionality $k$ (price per ticket):

To prepare a fruit smoothie for $4$ people, $600$ ml of milk is needed. If next Saturday you are going to invite $10$ friends to your house, how much milk will you need to buy so that everyone gets a smoothie? Solve using direct proportion.

A radio-controlled car covers $12$ metres in just $8$ seconds, maintaining a constant speed. If it continues moving in the same way:

1. a) What distance will it cover in one minute ($60$ seconds)?
2. b) How long will it take to cover a track of $30$ metres?

Your mobile phone has a total storage capacity of $128$ GB. If the operating system and installed apps already occupy $25\%$ of the space, how many gigabytes are currently occupied?

You have seen a hoodie that you love in an online shop. Its original price is $45$ €, but today it has a $15\%$ discount. How many euros do you save thanks to the discount? (It is not necessary to calculate the final price).

A 3D printer takes $45$ minutes to manufacture a part weighing $60$ grams. If we want to print a figure of the same material weighing $100$ grams, how long will the printer take? Express the result in hours and minutes.

Marcos accompanies his grandmother to a greengrocer's shop in their neighborhood market in Valencia. They see a sign indicating that a $3\text{ kg}$ bag of oranges costs $4,20\text{ €}$. Marcos wants to calculate how much a $5\text{ kg}$ bag of the same variety will cost and performs the calculation $3 : 4,20 \approx 0,71$ on his mobile phone. He then tells his grandmother: "A kilo costs $0,71\text{ €}$, so the $5\text{ kg}$ bag will cost us $5 \cdot 0,71 = 3,55\text{ €}$".

1. a) Explain in a reasoned way why Marcos's final result is clearly absurd without needing to do complex calculations.
2. b) Identify the conceptual error that Marcos made when calculating the constant of direct proportionality. What magnitude does the number $0,71$ that he obtained really represent, and what are its units of measurement?
3. c) Correct the procedure: calculate the appropriate constant of proportionality (price per kilogram), write the formula for the direct proportionality function $y = k \cdot x$, and determine the exact price they will pay for the $5\text{ kg}$ bag.

At an AEMET weather station located in Grazalema (Cádiz), a digital rain gauge records constant precipitation during an autumn storm. The device indicates that exactly $18\text{ liters per square meter}$ ($18\text{ mm}$) have fallen over a continuous period of $40\text{ minutes}$.

1. a) Before performing any exact mathematical calculation on paper, estimate mentally how many liters per square meter will have been collected if the storm continues at the same constant intensity for a total of $2\text{ hours and }15\text{ minutes}$. Explain in writing what approximations or quick deductions you used to obtain your initial estimate.
2. b) Check your estimate by calculating the exact value using direct proportionality: find the constant of proportionality (in liters per square meter and minute) and obtain the final accumulated amount for the $135\text{ minutes}$.
3. c) Compare your estimate from section a) with the exact result from section b) and assess whether your initial approximation was reasonable, explaining why this check is useful in real-life situations.

Un creador de contenido en una plataforma de vídeo recibe ingresos directamente proporcionales al número de visualizaciones de sus vídeos. Por cada $1250$ visualizaciones, el sistema le genera un beneficio de $3,75$ €. Copia la siguiente tabla en tu cuaderno y complétala calculando la razón de proporcionalidad $k$. En la gráfica se ha marcado el dato conocido.

An electric scooter's battery is consumed in proportion to the distance travelled. If completing a journey of $4,5$ km used up $12 \%$ of the total battery capacity, calculate using the unitary method how much battery will be used on a journey of $15$ km over the same type of terrain.

To organise a school event, it has been decided to prepare a recipe for homemade lemonade. The original recipe states that for $6$ people, $1,5$ litres of water and $450$ g of sugar are needed. If $42$ people have ultimately signed up, how many litres of water and how many kilograms of sugar should be bought to maintain the exact proportion?

Three friends, Alba, Bruno and Carla, bought a pack of $50$ trading card booster packs together for $75$ €. Alba contributed $15$ €, Bruno $27$ € and Carla the rest. If they decide to share the packs in direct proportion to the money each person contributed, how many packs does each person get?

A hiking map of the Sierra de Guadarrama is drawn to a scale of $1:25000$. A group of students is planning a route that measures $14,4$ cm on the map.

1. How many actual kilometres will they cover in total?

2. If they walk at a constant speed and take $45$ minutes to cover the first $3$ km, how long will it take them in total to complete the route?

A recycling centre pays users based on the weight of the plastic delivered. On Monday, a school delivered $120$ kg and received $18$ €. On Tuesday, another school delivered an unknown quantity and received $24,75$ €.

1. What is the price per kilogram of recycled plastic?

2. How many kilograms did the second school deliver?

At the same time of day, the shadow of a road sign $2,2$ metres high is $1,4$ metres long. If right next to it there is an electricity pylon whose shadow is $10,5$ metres long, how tall is the pylon? Draw a sketch of the situation before solving the problem.

You can do it on paper and send a photo with your answer.

A smartphone component manufacturing company uses a machine that produces $150$ microchips every $20$ minutes.

1. How many microchips will it produce in a $8$-hour working day without stopping?

2. If due to a technical fault the machine halves its production speed, how many hours will it take to manufacture an urgent order of $2250$ microchips?

Laura is the kitchen manager of a state secondary school with a school canteen. She knows from official records that to serve $180$ daily diners, exactly $27\text{ kg}$ of rice are cooked. Next week, orientation days will be held and the number of diners will vary: on Tuesday, $210$ students will eat; on Thursday, $195$; and on Friday, $240$.

To calculate the amount of rice needed for each of those three days, Laura is hesitating between two strategies:

• Method 1 (Rule of three): Set up three independent direct rules of three using cross-multiplication, one for each day.

• Method 2 (Reduction to the unit): First calculate the constant of direct proportionality (the portion of rice per diner) and then apply it to the diners for each day.

1. a) Apply both methods to calculate the kilograms of rice for Tuesday ($210$ diners) and verify that both procedures lead to the same result.
2. b) Use Method 2 to calculate the kilograms needed for Thursday ($195$ diners) and Friday ($240$ diners).
3. c) Justify with arguments which of the two methods is more advantageous and efficient in this particular case. Would your recommendation change if Laura only had to calculate the food for a single isolated day? Reason your answer.

A charging station from the municipal public network supplies electrical energy at a constant net power of $22\text{ kW}$ (that is, it transfers exactly $22\text{ kWh}$ for each full hour of continuous connection). A user plugs in their electric car, whose battery has a total capacity of $50\text{ kWh}$, at the moment when the battery is at $10\,\%$ of its charge. The municipal rate sets a price of $0,38\text{ €}$ for each $\text{kWh}$ recharged.

1. a) Calculate how many $\text{kWh}$ of energy the vehicle needs to receive to reach $80\,\%$ of its capacity (the level usually recommended by manufacturers to extend battery life).
2. b) Model the relationship between the elapsed time $t$ (in hours) and the transferred energy $E$ (in $\text{kWh}$) using a direct proportionality function. Exactly how much time (expressed in hours, minutes, and seconds) must the car remain plugged in to reach that $80\,\%$?
3. c) Model the function that expresses the total economic cost $C$ (in euros) as a function of the charging time $t$ (in hours). Find the constant of direct proportionality of this relationship, state its units, and interpret what it means in practice. How much will the user pay in total for that recharge?

An architecture firm is designing a scale model of a skyscraper using a high-precision 3D printer. They have verified that printing a section of the structure with a volume of $450\text{ cm}^3$ requires $12,5\text{ metres}$ of plastic filament and the printer takes $3\text{ hours and }45\text{ minutes}$. If the complete skyscraper in the scale model requires a total plastic volume of $2,16\text{ dm}^3$:

1. a) How many $50\text{-metre}$ filament spools must they buy at a minimum to complete the scale model?
2. b) If the printer works without interruption, how many days and hours will it take to finish the job?

A video streaming server adjusts file sizes proportionally to the duration and resolution (total number of pixels). A video lasting $10\text{ minutes}$ at a resolution of $1280 \times 720$ pixels takes up $450\text{ MB}$.

Calculate how much space a movie lasting $1\text{ hour and }45\text{ minutes}$ will take up if it is uploaded at a resolution of $1920 \times 1080$ pixels, knowing that the relationship between file size and total pixels is directly proportional. Express the result in Gigabytes ($1\text{ GB} = 1024\text{ MB}$).

On a star map, the distance between two stars is $18\text{ cm}$. It is known that this map uses a scale where $1\text{ cm}$ corresponds to $4,5$ light years. If a spacecraft travels at a constant speed and takes $240\text{ years}$ to cover $0,8$ light years:

1. a) What is the actual distance in light years between the two stars?
2. b) How many centuries would it take the spacecraft to travel from one star to the other? (Assume that the relationship between time and distance covered is directly proportional).

A desalination plant produces drinking water in direct proportion to the operating time of its turbines. Over a period of $8\text{ hours}$, the plant produces enough water to supply a population of $3.500\text{ people}$, consuming $1.400\text{ kWh}$ of energy.

Due to a heatwave, water consumption per person increases by $25\%$. If the plant must now supply a population of $5.000\text{ people}$ under these new conditions, for how many hours will it need to operate and what will be the total expected energy consumption? (Assume that energy consumption is also directly proportional to operating time).

In the 3rd year ESO Physics laboratory, two elastic springs, $A$ and $B$, are analyzed, which obey Hooke's law: the applied force $F$ (in Newtons, $\text{N}$) is directly proportional to the elongation experienced $x$ (in centimeters, $\text{cm}$), satisfying $F = k \cdot x$.

The students find an old laboratory worksheet in which several data have been erased, but it keeps three notes intact:

1. When subjecting spring $A$ to a force of $10\text{ N}$, its recorded elongation is $8\text{ cm}$.

2. Under the same unknown force $F_1$, spring $A$ stretches exactly $3\text{ cm}$ more than spring $B$.

3. If that unknown force is doubled (applying $2 \cdot F_1$), the total elongation experienced by spring $B$ is equal to the elongation that spring $A$ had under the original force $F_1$.

1. a) Calculate the elastic constant of proportionality $k_A$ of spring $A$ and interpret its physical meaning, stating its units.
2. b) Exclusively from the theoretical properties of direct proportionality and conditions 2 and 3, deduce the elastic constant of proportionality $k_B$ of spring $B$ without previously calculating the numerical value of the force $F_1$.
3. c) Reconstruct the lost test data: find the exact numerical value of the force $F_1$ (in $\text{N}$) and the elongations that both springs had under that force.

The Empresa Municipal de Transportes (EMT) monitors the energy consumption of its urban electric buses. Under normal driving conditions, energy consumption is directly proportional to the distance traveled, with a proportionality constant of $1,20\text{ kWh/km}$. However, during the summer months, the uninterrupted use of the air conditioning system increases consumption per kilometer by $25\,\%$.

1. a) Write the direct proportionality equation $E = k \cdot d$ that relates the energy consumed $E$ (in $\text{kWh}$) to the distance traveled $d$ (in $\text{km}$) in each of the two scenarios: standard regime ($k_1$) and summer regime ($k_2$).
2. b) A bus starts its day with the battery fully charged ($320\text{ kWh}$). Due to safety regulations, the depot requires the vehicle to return before the battery drops to $15\,\%$ of its capacity. Calculate the maximum range (in kilometers) that the bus can travel in the summer regime before having to return to the depot.
3. c) During a special test day, a bus travels a total route of $180\text{ km}$: the first $120\text{ km}$ with the air conditioning running at full blast and the last $60\text{ km}$ in standard mode. Calculate the overall proportionality constant for the day (the actual average consumption per kilometer). A draft technical report proposed calculating that constant by taking the simple arithmetic mean of $k_1$ and $k_2$: $\frac{1,20 + 1,50}{2} = 1,35\text{ kWh/km}$. Rigorously justify why that simple mean is conceptually wrong and how the weighting of distances is involved in the correct calculation.