Exercises: Ratios and proportions

Calculate the value of the ratio of each pair of quantities and interpret the result:

1. a) $3\ \text{kg}$ of cherries cost $10,50\ \text{€}$.

1. b) A gravel cyclist covers $95\ \text{km}$ in $4\ \text{h}$.

1. c) A bottle contains $2\ \text{L}$ and a drum, $25\ \text{L}$.

Check, by calculating the product of the extremes and the product of the means, whether the following pairs of ratios form a proportion:

1. a) $\frac{6}{4}$ and $\frac{15}{10}$

1. b) $\frac{5}{8}$ and $\frac{7}{12}$

1. c) $\frac{2,5}{4}$ and $\frac{5}{8}$

Find the fourth proportional in each case:

1. a) $\frac{x}{4} = \frac{9}{12}$

1. b) $\frac{5}{x} = \frac{15}{21}$

1. c) $\frac{8}{6} = \frac{20}{x}$

Two metal rods measure $8\ \text{dm}$ and $12\ \text{dm}$.

1. a) Calculate the ratio of the length of the first to the second, and simplify it.

1. b) Calculate the ratio of the second to the first.

1. c) How are the two ratios obtained related to each other? Does this always happen?

Using the numbers $3$, $6$, $4$ and $8$, form:

1. a) A proportion in which $3$ and $8$ are the extremes.

1. b) A proportion in which $3$ and $8$ are the means.

1. c) Check in both cases that the product of the extremes is equal to the product of the means.

A road map is drawn to a scale of $1 : 300.000$.

1. a) Explain the meaning of this ratio.

1. b) What real-life distance is represented by $5\ \text{cm}$ on the map?

1. c) Two towns are $21\ \text{km}$ apart in reality. How far apart are they on the map?

Lucía and her brother prepare a soft drink by mixing orange juice and sparkling water in a ratio of $3 : 2$ (that is, for every $3$ glasses of orange juice, they add $2$ glasses of water). Since friends arrive home and they need a larger quantity, Lucía suggests: "To make more soft drink while keeping exactly the same taste, let's add $2$ more glasses of orange juice and $2$ more glasses of water; since we are adding the same quantity to both ingredients, the proportion doesn't change."\na) Calculate the numerical value of the original ratio and the value of the new ratio after applying Lucía's suggestion. Do both soft drinks have the same taste? Justify your answer by comparing the decimal values of both ratios.\nb) Identify and explain, with reasoning, the conceptual error that Lucía has made regarding the meaning of ratio and proportion.\nc) Correct Lucía's recipe: if she decides to add $3$ more glasses of sparkling water, how many glasses of orange juice must she add to maintain exactly the initial proportion?

In an animal shelter, the ratio of the number of cats to the number of dogs housed is $4 : 7$.

1. a) Knowing that there are more than $30$ animals and fewer than $40$ in total, deduce how many cats and how many dogs there are in the shelter without resorting to blind trial and error. Clearly explain the numerical property on which you base your conclusion.
2. b) Write the complete mathematical proportion that relates the number of cats and dogs to the given simplified ratio, explicitly indicating which are the extremes and which are the means.
3. c) If $2$ cats and $2$ dogs are adopted tomorrow, check using the cross-multiplication rule whether the proportion between both species remains the same or changes.

Find the value of $x$ in these proportions:

1. a) $\frac{x}{2,4} = \frac{7}{4,8}$

1. b) $\frac{3}{x} = \frac{\frac{1}{2}}{\frac{3}{4}}$

1. c) $\frac{1,5}{4,5} = \frac{x}{13,5}$

Find the mean proportional in each case, that is, the positive value of $x$ that satisfies the continuous proportion:

1. a) $\frac{x}{4} = \frac{25}{x}$

1. b) $\frac{3}{x} = \frac{x}{27}$

1. c) $\frac{x}{2} = \frac{50}{x}$

A carpenter makes picture frames whose central decorative section is a square. So that the frame maintains the same proportion in all sizes, the side $x$ of the square must be the mean proportional between the inner width and the outer width of the frame.

1. a) For a frame with an inner width of $9\ \text{cm}$ and an outer width of $25\ \text{cm}$, set up the continuous proportion $\frac{9}{x} = \frac{x}{25}$ and find the side length of the square.

1. b) Check that the two ratios have the same value.

1. c) If the inner width is $8\ \text{cm}$ and the decorative square has side length $12\ \text{cm}$, what is the outer width of the frame?

Two wine regions compare their production. Region A harvests $4.500\ \text{t}$ of grapes from $1.800\ \text{ha}$ of vineyard; Region B harvests $6.240\ \text{t}$ from $2.400\ \text{ha}$.

1. a) Calculate the ratio of production to surface area in each region, including its units.

1. b) Which of the two obtains a higher yield per hectare?

1. c) If Region A were to increase its area to $2.000\ \text{ha}$ while maintaining the yield, what production would be expected?

A javelin thrower trains with a mark set at $60\ \text{m}$ and takes $4,8\ \text{s}$ to cover that distance in her run-up.

1. a) Calculate the ratio of distance to time, in $\text{m/s}$.

1. b) Express it in $\text{km/h}$.

1. c) Is it correct to calculate the ratio between $60\ \text{m}$ and $4,8\ \text{s}$ without considering the units? Why must units always be specified when interpreting a ratio?

The Parthenon in Athens was designed seeking the golden ratio: the ratio between the longer side and the shorter side of its rectangles is approximately $\frac{8}{5} = 1,6$.

1. a) A window is $1,92\ \text{m}$ high and $1,20\ \text{m}$ wide. Calculate the ratio between its sides. Is it in the golden ratio?

1. b) Another window is $2\ \text{m}$ high. How wide should it be so that the ratio of height to width is $1,6$?

1. c) Measure three everyday rectangular objects at home (a door, a book, a screen) and check whether any of them is close to the golden ratio.

To organize a hiking trip in the Sierra de Guadarrama, you use an official map from the Instituto Geográfico Nacional (IGN) at a scale of $1 : 25.000$. On the map, the length of the route you want to take measures exactly $18,4\text{ cm}$.\na) Before calculating the exact value, make an estimate: reason how many actual meters $1\text{ cm}$ on the map represents and mentally estimate between which two integer values of kilometers the actual distance of the route will be.\nb) Set up the proportion that defines the map scale, calculate the actual distance in kilometers using the equality of cross-products, and check whether your initial estimate was correct.\nc) If the group walks at a constant rate of $4\text{ km/h}$, calculate how long it will take to complete the route (express the result in hours and minutes) and assess whether it is reasonable to complete the hike in a single morning.

According to the official report from Spain's Dirección General del Agua, the Guadalquivir river basin has a total capacity of $8.030\text{ hm}^3$ and contains $2.409\text{ hm}^3$ in dammed water, while the Eastern Cantabrian basin has a total capacity of $70\text{ hm}^3$ and contains $56\text{ hm}^3$ in dammed water.\na) Model the situation by calculating the ratio between the dammed volume and the total capacity for each of the two basins. Interpret what information each result provides beyond the raw water figure.\nb) A newspaper headline states: "The Guadalquivir basin enjoys a much better water situation because it accumulates more than $40$ times the amount of water of the Eastern Cantabrian basin." Critically argue why this statement is misleading, supporting your reasoning with the calculated ratios.\nc) Set up a proportion to determine how many cubic hectometers of water the Guadalquivir basin would need to have dammed to be at the same relative fill proportion as that of the Eastern Cantabrian basin, and calculate this fourth proportional.

A student states the following:

"Since $\frac{6}{9} = \frac{10}{15}$, then $\frac{6}{10} = \frac{15}{9}$ also holds, because I have only changed the order of the numbers."

1. a) Check whether the first equality is a true proportion.

1. b) Check whether the second equality is also true. Is she correct?

1. c) The correct rule states that in a proportion you can swap the means or swap the extremes. Apply this to $\frac{6}{9} = \frac{10}{15}$ and write down two equivalent true proportions.

State whether each statement is true or false, assuming that $\frac{a}{b} = \frac{c}{d}$ is a true proportion. Justify your answer using the product of the extremes and means:

1. a) "$\frac{a}{c} = \frac{b}{d}$"

1. b) "$\frac{d}{b} = \frac{c}{a}$"

1. c) "$\frac{a + b}{b} = \frac{c + d}{d}$"

1. d) "$\frac{a}{b} = \frac{a + c}{b + d}$"

A square is to be constructed that has exactly the same area as a rectangle measuring $9\ \text{cm}$ by $16\ \text{cm}$, as shown in the figure.

1. a) If the side length of the square is $x$, set up the equation $x^2 = 9 \cdot 16$.

1. b) Rewrite the equation as a continuous proportion $\frac{9}{x} = \frac{x}{16}$. What is $x$ relative to $9$ and $16$?

1. c) Calculate $x$ and check that the areas match.

1. d) What would be the side length of the square equivalent to a rectangle measuring $5\ \text{cm}$ by $45\ \text{cm}$?

Decision task. An electronics shop advertises three monitors as "16:9 widescreen", meaning that the ratio between width and height is $\frac{16}{9} \approx 1,78$. The actual measurements of each model's screen are:

• Model 1: $53\ \text{cm}$ wide and $30\ \text{cm}$ high.

• Model 2: $62\ \text{cm}$ wide and $34\ \text{cm}$ high.

• Model 3: $71\ \text{cm}$ wide and $40\ \text{cm}$ high.

1. a) Calculate the width/height ratio of each model to two decimal places.

1. b) Which models truly satisfy the 16:9 proportion (allow an error of $0,02$)?

1. c) A customer wants to place the monitor in a space $65\ \text{cm}$ wide and $38\ \text{cm}$ high. Which model, among those satisfying the proportion, fits in the space? Justify your decision with calculations.

An architecture studio designs two rectangular windows. Window A measures $60\text{ cm}$ wide by $90\text{ cm}$ high, and window B measures $80\text{ cm}$ wide by $120\text{ cm}$ high. To verify whether both windows are proportional, three methods are considered:\n- Method 1: Calculate the internal ratio $\dfrac{\text{height}}{\text{width}}$ of each window and compare whether the results match.\n- Method 2: Calculate the external ratios between corresponding measurements ($\dfrac{\text{width B}}{\text{width A}}$ and $\dfrac{\text{height B}}{\text{height A}}$) and check whether they are equal.\n- Method 3: Set up the proportion $\dfrac{60}{90} = \dfrac{80}{120}$ and check whether the equality of cross-products holds: $60 \cdot 120 = 90 \cdot 80$.\na) Work out all three methods numerically and formally prove why the equality $\dfrac{a}{b} = \dfrac{c}{d}$ mathematically guarantees that $\dfrac{b}{a} = \dfrac{d}{c}$ and that $\dfrac{c}{a} = \dfrac{d}{b}$.\nb) Now you are asked to analyze a window C of $75\text{ cm} \times 110\text{ cm}$. Reasonably choose which of the three methods is most direct to check whether it is proportional to window A, and apply it justifying your conclusion.\nc) If a window D is built whose dimensions multiply the perimeter of window A by $3$ while maintaining the proportion between its sides, by what ratio do its width, its height, and the value of its aspect ratio change?

In a continuous proportion, the means are equal: $\dfrac{a}{x} = \dfrac{x}{b}$, so that the term $x = \sqrt{a \cdot b}$ is called the mean proportional (geometric mean) between the extremes $a$ and $b$.\na) Find all pairs of positive integers $(a, b)$ with $a < b$ such that their mean proportional is exactly $x = 12$. Show the deductive procedure based on the prime factorization of $12^2$, without resorting to unsystematic trial and error.\nb) Out of all the pairs obtained in the previous section, identify which one has the smallest possible difference between its extremes ($b - a$) and write the corresponding continuous proportion, verifying that the cross-products match.\nc) Prove algebraically and in general that if $x$ is the mean proportional between any two positive numbers $a$ and $b$, then the mean proportional between $k \cdot a$ and $k \cdot b$ (with $k > 0$) is exactly equal to $k \cdot x$.