Direct proportionality

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 centimeters.
4. d) The time you leave your cell 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 $22.50$ pesos. Complete the following table and calculate the constant of proportionality $k$ (price per ticket):

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 $3.75$ pesos. Copia la siguiente tabla en tu cuaderno y complétala calculando la razón de proporcionalidad $k$. En la gráfica se ha marcado el dato conocido.

A radio-controlled car covers $12$ meters 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 $30$ meters?

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 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 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.

To organise a school event, it has been decided to prepare a recipe for homemade lemonade. The original recipe states that for $6$ people, $1.5$ liters of water and $450$ g of sugar are needed. If $42$ people have ultimately signed up, how many liters of water and how many kilograms of sugar should be bought to maintain the exact proportion?

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?

A hiking map of the Cordillera Central 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 kilometers 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?

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).

Your cell 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 store. Its original price is $2\,700$ pesos, but today it has a $15\%$ discount. How many pesos do you save thanks to the discount? (It is not necessary to calculate the final price).

A recycling center pays users based on the weight of the plastic delivered. On Monday, a school delivered $120$ kg and received $1\,080$ pesos. On Tuesday, another school delivered an unknown quantity and received $1\,485$ pesos.

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

2. How many kilograms did the second school deliver?

Three friends, Alba, Bruno and Carla, bought a pack of $50$ trading card booster packs together for $75$ pesos. Alba contributed $15$ pesos, Bruno $27$ pesos 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?

At the same time of day, the shadow of a road sign $2.2$ meters high is $1.4$ meters long. If right next to it there is an electricity pylon whose shadow is $10.5$ meters 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.

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 $12.5\text{ meters}$ 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 $50\text{-meter}$ 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 ($1\text{ GB} = 1024\text{ MB}$).

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).