Direct Proportionality

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

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

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

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

• The kilograms of oranges bought at the grocery store 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 288 pesos. 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:

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

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 kilometers?

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

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

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

An industrial faucet discharges a constant flow of water. It has been timed that in 15 minutes it pours exactly 45 liters. 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 liters).

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 meters.

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 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 meters tall casts a shadow of 2.4 meters. At that same moment, a nearby building casts a shadow measuring 18 meters. Using direct proportion, what is the actual height of the building?

A company manufactures model sports cars to a scale of 1:24. The trunk of the model has an internal volume of 15 cm³. Calculate the actual volume of the car's trunk expressed in liters. 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 metric tons?

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?

A pair of sneakers costing $\$1\,200$ 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 sneakers.

The monthly membership fee for a gym was $\$800$. 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 $20\%$.

The price of a cell phone excluding VAT is $\$7\,000$. First, a VAT of $21\%$ is applied and, on the resulting price, a discount of $15\%$ for cash payment.

• Calculate the final price paid for the cell 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 an apartment 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 $\$16\,000$ and justify it also using multipliers.

A bicycle shop applies a discount of $20\%$ to a model and then adds sales tax of $21\%$. The final price at the register is $\$9\,680$. What was the price of the bicycle before the discount, excluding sales tax?

Two shops sell the same console for $\$6\,000$. 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.

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

Three partners decide to open a catering business and contribute an initial capital of 15 000 pesos, 25 000 pesos and 40 000 pesos respectively. After one year, the business has generated a net profit of 12 800 pesos. 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 pesos 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.

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 pesos. If the distribution must be in direct proportion to the fraction of capital contributed, how much money in pesos does each of the friends receive?

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

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

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 pesos$900$ 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 $k = \frac{Y}{X}$.