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, $25\ \text{L}$.

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?

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?

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?

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 $\frac{5}{8}$

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.

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?

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