Exercises: Compound proportion

Three packaging machines produce $500$ parts in $4$ hours of continuous operation. How many parts will $5$ identical machines produce if they operate for $6$ hours?

A team of $4$ painters manages to paint a wall of $100 \text{ m}^2$ in $5$ days. If $2$ more painters are hired to work at the same rate, how many days in total will the $6$ painters need to paint a wall of $150 \text{ m}^2$?

Five identical taps, open at the same time, fill a $2000$-litre tank in $8$ hours. How many hours will $4$ of these taps take to fill a $3000$-litre tank?

On a farm, $10$ cows consume $300 \text{ kg}$ of grass in $6$ days. Assuming all cows eat the same daily amount, how many kilograms of grass will $15$ cows consume in $10$ days?

In an office, $6$ printers working simultaneously print $1200$ pages in $2$ hours. How many printers with the same specifications are needed to print $2400$ pages in $3$ hours?

A hotel charges $600$ € to accommodate $2$ people for $5$ nights. Keeping the same rate per person per night, how much will the hotel charge to accommodate $4$ people for $3$ nights?

A group of $8$ workers builds a perimeter fence of length $40 \text{ m}$ in $5$ days. How many workers will be needed to build a similar fence of length $60 \text{ m}$ in just $4$ days?

At a camp, $12$ students consume $40$ loaves of bread in $5$ days. How many loaves of bread will $15$ students consume over $8$ days if they maintain the same level of consumption?

To prepare for the local patron saint festivities of a municipality, $4$ machines in a local printing house print a run of posters in $6$ days operating $8$ hours a day. Due to a breakdown, only $3$ machines remain available, and it is decided to extend the working day to $10$ hours a day to try to finish on time.

Lucas tries to calculate how many days they will take and writes in his notebook:

"Printing presses and days is an inverse relationship, so I write $\frac{3}{4}$. Hours per day and days is direct, because more working hours mean more days, so I write $\frac{10}{8}$.

Therefore: $\frac{x}{6} = \frac{3}{4} \cdot \frac{10}{8} \Rightarrow x = 6 \cdot \frac{30}{32} \approx 5,6$ days".

1. a) Identify and explain with reasoning the two mistakes made by Lucas (one when classifying the proportionality relationship of the daily hours and another when setting up the ratio of the machines with the unknown in the numerator).
2. b) Set up the correct proportion, calculate the exact number of days that the $3$ machines will take, and reason whether it makes sense that the final result is greater than the initial $6$ days.

During the olive harvest campaign in an olive-growing cooperative in Jaén, a crew of $6$ day laborers harvests $12\,000$ kg of olives in $4$ full workdays. In the next grove, the estimated harvest is $18\,000$ kg and the crew is reinforced with $2$ more people (making a total of $8$ day laborers with the same work rate).

1. a) Without carrying out operations with formulas yet, reasonably estimate whether they will take more, fewer, or exactly the same $4$ days as in the first grove. (Hint: compare the proportional increase in harvest to the increase in staff).
2. b) Set up the compound proportion with the corresponding ratios, calculate the exact number of days they will take, and check whether your initial estimate was correct.

A financial investment of $5000$ € at a given simple interest rate generates a profit of $400$ € in $2$ years. How much interest will a capital of $7500$ € generate when invested for $3,5$ years at exactly the same interest rate?

A data centre has $12$ servers that, working in parallel, process $4,5 \text{ TB}$ of information in $6$ hours. If $3$ servers break down and stop working, how long will it take the remaining servers to process a $6,75 \text{ TB}$ data package?

To harvest a cereal field of $12 \text{ ha}$, $5$ combine harvesters working $8$ hours a day need $3$ full days. How many days will $4$ combine harvesters working $10$ hours a day need to harvest a field of $15 \text{ ha}$?

A logistics company charges $1250$ € to transport $8$ tonnes of goods over a distance of $300 \text{ km}$. If the rates remain proportional to weight and distance, how much will the company charge to transport $10,5$ tonnes over a distance of $450 \text{ km}$?

A printing house uses $4$ rotary presses working $10$ hours a day and takes $15$ days to print the first edition of a novel. If the publisher demands that a second identical edition be finished in $12$ days and the printing house has $5$ machines, how many hours a day must the machines work to meet the deadline?

A team of $6$ cyclists consumes $150$ litres of water during $4$ stages of a Grand Tour. Assuming the average daily consumption rate and physical demands remain constant, how many litres of water will a team composed of $9$ cyclists consume over $7$ stages?

A group of $15$ workers manages to resurface $2,4 \text{ km}$ of road in $8$ days working $6$-hour daily shifts. How many kilometres of road can $20$ workers resurface in $12$ days if the company decides to increase the shifts to $8$ hours a day?

In a craft workshop, a team of $4$ sculptors needs $24$ days working $6$ hours a day to carve $5$ marble statues. How many sculptors will be needed to carve $8$ statues with the same characteristics in $16$ days working $9$ hours a day?

In the forest fire fighting system (helitack crew operations in Spain), a crew of $12$ specialists can manually clear a firebreak $600$ meters long in $5$ hours of continuous work in scrubland. In the face of a declared emergency in the region of Las Hurdes, the coordination center urgently needs to clear a firebreak of the same characteristics measuring $1\,500$ meters within an unextendable maximum deadline of $8$ hours before the wind direction shifts.

1. a) Classify the proportionality relationship of each magnitude (length of the firebreak and available time) with respect to the unknown (number of specialists required), justifying each case by keeping the other magnitude constant.
2. b) Set up the compound proportion and calculate how many specialists would theoretically be needed.
3. c) Knowing that the specialists travel in helicopters in indivisible crews of $7$ members, how many full crews must compulsorily be mobilized to guarantee that the firebreak is completed on time? Reason why standard mathematical rounding cannot be applied in this safety situation.

In a youth hostel in the Pyrenean region of Sobrarbe (Huesca), a group of $25$ hikers consumed $150$ liters of milk during a stay of $6$ days. For next week, the arrival of a new group of $40$ hikers staying for $8$ days is expected. To place the order with the regional distributor, the two warehouse managers propose different methods:

• Method A (Unitary method): first calculate the liters of milk that one hiker consumes in a single day and then multiply by the new conditions.

• Method B (Compound proportion with ratios): classify the magnitudes and set up the equation $\frac{150}{x}$ equal to the product of the corresponding ratios.

1. a) Solve the problem step by step applying both methods and check whether they lead to the same amount of liters.
2. b) If the hostel continuously receives bookings throughout the entire summer from groups of very different sizes and lengths of stay, which of the two methods is faster and more practical for the manager to calculate supplies for each group without having to reset the entire setup? Justify your answer.

A factory has $N$ machines producing $P$ items in $D$ days working $H$ hours a day. Determine the simplified algebraic expression representing the number of days $x$ required for twice as many machines ($2N$) to produce $50\%$ more items ($\frac{3}{2}P$) working one-third of the usual working day ($\frac{H}{3}$ hours a day).

A gang of $12$ workers, with an estimated efficiency of $80\%$, builds a wall of $600 \text{ m}$ in $15$ days. How many highly qualified workers, with an efficiency of $120\%$, will be needed to build a wall of $800 \text{ m}$ in a maximum of $10$ days?

Four type A water extraction pumps, each with a constant flow rate of $C$ litres per minute, are capable of filling $3$ identical swimming pools in $12$ hours. How many type B pumps, given that their flow rate is $1,5C$ litres per minute, are needed to fill $6$ of those same swimming pools in $8$ hours?

A school canteen has sufficient provisions to feed $120$ students for $20$ days, giving each student $3$ daily rations. After $5$ days have passed, $30$ students leave the school to go on a permanent field trip and the administration decides to reduce the daily rations per student to $2$. How many more days will the remaining provisions last for the students who stay at the school?

A manufacturing company produces $5000$ units of a product in $10$ days employing $20$ workers, with a total wage cost of $C$ euros. If market demand leads to increasing the workforce by $25\%$ and reducing the working days by $20\%$, how many units will be produced? Furthermore, if the daily wage per worker increases by $10\%$, what will be the new total wage cost expressed in terms of $C$?

To paint a spherical tank of radius $R$, a team of $5$ painters needs $8$ days working $6$ hours a day. How many days will $4$ painters, working $8$ hours a day, need to paint a spherical tank of radius $2R$? (Reminder: the surface area to paint on a sphere is proportional to the square of its radius).

Eight tractors are able to plough a rectangular plot of land of $400 \text{ m} \times 300 \text{ m}$ in $6$ days. How many tractors will be needed to plough a square plot of land with side length $600 \text{ m}$ in $9$ days, knowing that the soil hardness of the second plot is twice that of the first (that is, it requires twice as much effort and time per square metre)?

Ten weavers, working $8$ hours a day, produce a piece of cloth $40 \text{ m}$ long and $2 \text{ m}$ wide in $5$ days. How many weavers, working $10$ hours a day, will be required to produce a piece of cloth $60 \text{ m}$ long and $3 \text{ m}$ wide in $6$ days? (Note that the amount of work done is directly proportional to the total surface area of the cloth).

An e-commerce logistics platform located in the Corredor del Henares (Madrid) operates with a homogeneous fleet of delivery vans. Under standard conditions, $8$ vans deliver $3\,840$ packages in $6$ days working a shift of $h$ hours per day per vehicle.

During the sales season, the company expanded its fleet to $12$ vans to deliver a total volume of $9\,600$ packages. At the end of the sales period, records confirmed that it took exactly $8$ full days working at the maximum shift allowed by collective agreement, set at $9$ hours per day per van.

1. a) From the final confirmed results in the sales campaign ($8$ days at $9$ hours per day), reconstruct the value of the standard shift $h$ (in hours and minutes) that the vans worked during the standard period.
2. b) Deduce a simplified formula that allows you to calculate the number of days $D$ that those same $12$ vans would take to deliver the $9\,600$ packages as a function of the chosen daily shift $h'$. From your formula, determine how many full days would be needed if the delivery drivers worked a reduced shift of $6$ hours per day.

In an irrigation community in Campo de Cartagena (Murcia), a solar pumping station uses square photovoltaic panels to lift water from a well into a storage reservoir. The following data from the original installation are known:

• It has $60$ square panels with sides of $1,2$ meters.

• With an average solar irradiance of $8$ peak sun hours per day, they manage to pump $432\text{ m}^3$ of water in $5$ days.

To cope with the summer season, the community upgrades the station by replacing all the panels with $80$ units of a new model, also square, but with sides of $1,8$ meters. The power and flow rate generated by each panel are directly proportional to its surface area. In the month of July, there are $10$ peak sun hours per day and a total volume of $2\,160\text{ m}^3$ needs to be pumped.

1. a) Justify why the power depends on the square of the panel side length and calculate the numerical ratio of change between the surface area of the new panel and that of the old one.
2. b) Set up the complete compound proportion and calculate the exact number of days that the new installation will take to pump the $2\,160\text{ m}^3$.
3. c) If the reservoir has reserves to guarantee crop irrigation for a maximum of $12$ days without receiving new water inputs, will the new installation complete the filling on time before the reserve runs out? Interpret the available safety margin.