Direct proportionality

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 $B$?

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?

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 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 $20\%$.

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.

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?

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.

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 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 $k = \frac{Y}{X}$.