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 $\text{RD\$}\ 630$.

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?

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.

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 $x^2 = 9 \times 16$.

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.