Exercises: Direct proportion

State which of the following pairs of quantities are directly proportional, justifying your answer in each case:

• The time a tap runs at a constant flow rate and the litres of water it dispenses.

• A person's age and their height in centimetres.

• The number of workers painting a fence and the time they take to finish it.

• The kilograms of oranges bought at the greengrocer's and the total price to pay.

The quantities $A$ and $B$ are directly proportional. Knowing that when $A = 5$, the value of $B = 15$:

• Calculate the constant of direct proportionality $k = \frac{B}{A}$.

• If $A = 2$, what is the value of $B$?

• If $B = 24$, what is the value of $A$?

• If $A = 12$, what is the value of $B$?

To make a sponge cake recipe for 4 people, 200 grams of flour, 150 grams of sugar and 2 eggs are needed. Calculate the exact quantities needed of each of these ingredients if we want to make the cake for 10 people.

In a stationer's, a sealed pack with 8 identical notebooks costs 14.40 €. Use the unitary method to calculate how much it would cost to buy exactly 5 of those notebooks.

Calculate the value of the unknown $x$ in the following proportion so that the fractions are equivalent:

$\frac{18}{x} = \frac{45}{20}$

Two friends, Alba and Bruno, decide to buy a lottery ticket that costs 20 €. Alba contributes 8 € and Bruno contributes 12 €. If the ticket wins a prize of 50 000 €, how should they share out the prize so that the amounts received are directly proportional to the amount each contributed?

On a map drawn to a scale of 1:50,000, the straight-line distance between two towns is 8 cm. What is the actual distance between both towns expressed in kilometres?

At an airport bureau de change, we are informed that 50 euros gives us 55 US dollars. How many dollars will we get if we decide to exchange 320 euros?

A pair of trainers costing $60\ \text{€}$ has a discount of $25\%$ and, on top of the already discounted price, an additional discount coupon of $10\%$ is applied. Calculate the final price paid for the trainers.

The monthly membership fee for a gym was $40\ \text{€}$. In January it went up by $10\%$ and in June it went up again by another $10\%$ on the current fee. Calculate the current fee and reason whether the two increases are equivalent to a single increase of $20\%$.

During the Black Friday campaign, a tech store announces that a laptop that cost 800 € will first undergo a 10 % price increase due to component shortages and, a week later, a 10 % discount will be applied as a promotional offer. Lucas claims: “Since it first goes up by 10 % and then goes down by 10 %, the final price will be exactly 800 € again, as $+10\, - 10\, = 0\,\%$”. Identify in a reasoned way the conceptual error in Lucas's statement based on variation multipliers, calculate the actual final price of the laptop, and indicate whether this sequence of changes benefits or harms the customer compared to the initial price.

An electric vehicle consumes energy directly proportionally to the distance travelled. On a test circuit, it has consumed 12,6 kWh after travelling 70 km. You must calculate the consumption it will record on three different journeys: Journey A (45 km), Journey B (110 km), and Journey C (230 km).

Two procedures are suggested:

• Method 1: Set up three independent direct rules of three (one for each journey).

• Method 2: Reduce to the unit by calculating the constant of proportionality $k$ (consumption per kilometre) and apply the function $y = k \cdot x$.

1. a) Compare both methods and justify which one is more efficient in this situation.
2. b) Calculate the exact consumption on each of the three journeys using the method you selected.

Three partners decide to open a catering business and contribute an initial capital of 15 000 €, 25 000 € and 40 000 € respectively. After one year, the business has generated a net profit of 12 800 €. If they agree to share the profits in direct proportion to the capital contributed by each, how much money does each partner receive?

The following graph shows the relationship between the distance travelled by a car at a constant speed and the litres of fuel consumed.

From the graph, determine the constant of proportionality representing the vehicle's fuel consumption (litres consumed per 100 km). Then, calculate mathematically how many litres the car will consume on a long trip of 425 km.

An industrial tap discharges a constant flow of water. It has been timed that in 15 minutes it pours exactly 45 litres. How long will it take to completely fill an empty swimming pool with a capacity of 13.5 m³? Express the final result in hours and minutes. (Given: 1 m³ is equivalent to 1 000 litres).

On an architect's floor plan, drawn to a scale of 1:150, the living room of a house is rectangular and measures 4 cm wide by 6 cm long. First, calculate the area of the living room on the plan and then deduce its actual area expressed in square metres.

A mechanical system is made up of three connected cogwheels: A, B and C. It is known that the number of turns they make is in direct proportion. For every 4 full turns wheel A makes, wheel B makes 3 turns. Furthermore, for every 5 turns of wheel B, wheel C makes 2 turns. If the motor turns wheel A 600 times, how many turns will wheel C have made?

A grandfather decides to divide 25 200 € among his three grandchildren in direct proportion to their ages, which are 12, 15 and 18 years old. However, before making the official division, he calculates how much each would get and notices that the middle grandchild would receive a specific amount. What is that amount? Also calculate the money received by the other two grandchildren.

A goldsmith melts two metals to create a special alloy. The ratio of silver to copper in the alloy is 9 to 2. If he used exactly 45 grams of copper for a piece, how many grams of silver did he need to melt? What is the total weight of the alloy created?

At a certain time in the morning, the sun's rays hit at the same angle, creating shadows proportional to the heights of the objects. A vertical post 3.2 metres tall casts a shadow of 2.4 metres. At that same moment, a nearby building casts a shadow measuring 18 metres. Using direct proportion, what is the actual height of the building?

The price of a mobile phone excluding VAT is $350\ \text{€}$. First, VAT of $21\%$ is applied and, on the resulting price, a discount of $15\%$ for cash payment.

• Calculate the final price paid for the mobile phone.

• What percentage increase does the final price represent compared to the price excluding VAT?

State whether the following statement is true or false, giving reasons for your answer: "if the rent of a flat increases by $5\%$ one year and decreases by $5\%$ the following year, the rent goes back to costing exactly the same as at the beginning". Check this with an initial rent of $800\ \text{€}$ and justify it also using multipliers.

An irrigation community in the Murcian fertile plain must divide a shared irrigation ditch maintenance expense of 15.600 € among three plots (A, B, and C) directly proportionally to their irrigated area. Plot A has 4 hectares, Plot B has 9 hectares, and Plot C has 13 hectares.

1. a) Before calculating the exact amounts, make a reasoned estimate: will Plot C pay more than half of the total bill? Approximately what fraction of the total should Plot B take on? Justify your estimates by analysing the proportion of hectares relative to the total.
2. b) Calculate the distribution constant $k$ and determine the exact amount in euros that each of the three plots must pay.
3. c) Compare the calculated amounts with your previous estimate and explain how this prior analysis helps detect possible errors when solving proportional distribution problems.

According to technical data on photovoltaic solar energy in Spain, a domestic installation generates an amount of electrical energy directly proportional to the Peak Sun Hours (PSH) recorded. A house with 6 identical solar panels produces $13,5\text{ kWh}$ on a clear spring day with 5 PSH.

1. a) Model the situation using a linear function $E(h)$ that expresses the energy produced $E$ (in kWh) as a function of the peak sun hours $h$. State the constant of proportionality with its units and interpret its practical meaning.
2. b) On a day in July, 7,2 PSH are reached. If the expected electricity consumption of the house for that day is $18\text{ kWh}$, use your model to determine whether production will cover demand, and calculate the resulting energy surplus or deficit.
3. c) If the family decides to expand the installation by adding 2 more panels with identical characteristics, what will be the new constant of proportionality of the function? Justify your answer without needing to take new field measurements.

Three friends decide to invest their savings in a joint fund for a project. The first contributes $\frac{1}{4}$ of the total capital, the second contributes $\frac{2}{5}$ of the total and the third contributes the rest of the required money. At the end of the investment, they obtain a net profit of 8 400 €. If the distribution must be in direct proportion to the fraction of capital contributed, how much money in euros does each of the friends receive?

A company manufactures model sports cars to a scale of 1:24. The boot of the model has an internal volume of 15 cm³. Calculate the actual volume of the car's boot expressed in litres. Furthermore, if the model weighs 1.2 kg and is manufactured using exactly the same materials and densities as the real car, what would be the weight of the real car expressed in tonnes?

Solve the following equation based on direct proportion, finding the value of $x$ that makes the two ratios equivalent:

$\frac{3x - 1}{4} = \frac{2x + 5}{3}$

The following graph represents the distance travelled by two trains, A and B, over time, travelling at a constant speed from the same departure station.

Looking at the graph, determine the constant speed of each train. If both maintain their speed, at what exact distance will train A be from its origin when train B has travelled 400 km?

In a local sports competition, a cash prize is distributed directly proportional to the points obtained by the top three finishers in the final. The points were $180$, $150$, and $120$, respectively. Knowing that the first-place finisher received €$900$ more than the third-place finisher, what was the total amount of money distributed in prizes?

To prepare a high-strength cement and sand mix, the technical ratio must be 2 parts cement to 5 parts sand. A builder needs to prepare 350 kg of mix in total. Calculate how many kilograms of cement and how many of sand he must use initially. If by a calculation error he puts 120 kg of cement into the cement mixer, how much sand in total should there be in the final mix to maintain the correct ratio without wasting the cement?

The quantities $M$ and $N$ are directly proportional, with constant of proportionality $k$. Prove mathematically that if the value of quantity $M$ increases by $25\%$, the corresponding value of quantity $N$ will also increase by exactly $25\%$.

The variables $X$ and $Y$ are directly proportional. In an experiment, it was recorded that when $X = m + 2$, the value of the dependent variable is $Y = 15$. On the other hand, under the same conditions, when $X = m - 2$, the value is $Y = 9$. Set up an algebraic proportion to calculate the numerical value of parameter $m$ and then determine the constant of proportionality $k = \frac{Y}{X}$.

A bicycle shop applies a discount of $20\%$ to a model and then adds VAT of $21\%$. The final price at the till is $484\ \text{€}$. What was the price of the bicycle before the discount, excluding VAT?

Two shops sell the same console for $300\ \text{€}$. Shop A reduces it by $30\%$ and offers an additional $10\%$ off the reduced price if paid with their store card. Shop B reduces it directly by $40\%$.

• Calculate the final price in each shop.

• Calculate the single percentage discount equivalent to the two successive discounts in shop A and explain why it does not equal the $40\%$ in shop B.

Three university research centres receive an extraordinary budget allocation of 240.000 € that is distributed directly proportionally to the number of full-time researchers employed at each centre.

For data privacy reasons, the staff size of each centre is not initially disclosed, but the following conditions of the distribution are known:

• Centre 2 has exactly twice as many researchers as Centre 1.

• Centre 3 receives 40.000 € more than Centre 2.

• Centre 1 is allocated exactly 40.000 €.

1. a) Using the properties of direct proportionality, deduce the exact amount of money allocated to each of the three centres.
2. b) Knowing that the three centres have a total of 60 researchers combined, calculate the value of the constant of proportionality of the distribution (specifying its units) and reconstruct the exact number of researchers working at each of the centres. Explain the reasoning that allows you to deduce these initial values from the final data.

In the water bill of a Spanish municipality, the bi-monthly amount consists of a fixed service fee of 15 € plus a variable rate of $1,40\text{ \euro{}/m}^3$ of water consumed.

Alba claims: "The cost of water is directly proportional to the volume consumed, because the more volume you use, the more you pay on the bill".

For next year, the city council approves two changes to the rates: the fixed fee increases by 10 %, while the price per cubic metre decreases by 5 %.

1. a) Apply the algebraic definition of directly proportional quantities rigorously (by analysing the quotient between the total cost and the volume consumed) to refute or confirm Alba's claim. Use a numerical counterexample comparing the average cost per cubic metre if $10\text{ m}^3$ are consumed versus if $20\text{ m}^3$ are consumed.
2. b) A family that typically consumes $30\text{ m}^3$ per two-month period plans to keep exactly that consumption next year. Formally demonstrate why it is not correct to calculate the percentage change of their bill by adding or subtracting the percentage changes of the rates ($+10\, % - 5\, % = +5\, %$), and determine the actual percentage change that the total cost of their bill will undergo.