Ratios and Proportions
A "16:9" screen, a "1:300,000" map, a recipe with "2 of flour for every 1 of water"… These are all ratios: ways of comparing two quantities by dividing. They are the foundation of proportionality, which you'll see everywhere: prices, recipes, blueprints, speeds.
Ratio between two quantities
In a class with 12 girls and 18 boys, the girls-to-boys ratio is $\frac{12}{18} = \frac{2}{3}$: for every 2 girls there are 3 boys.
Scales
A scale is the ratio between a measurement on the drawing and the real measurement. A scale of $1 : 300,000$ means that 1 cm on the map is $300,000$ cm = 3 km in real life. On a road map of Jalisco at that scale, the 15 cm between Guadalajara and Chapala would be $15 \times 3 = 45$ km.
Calculate the value of the ratio of each pair of quantities and interpret the result: a) $3\ \text{kg}$ of cherries cost $\$210$. b) A gravel cyclist covers $95\…
Calculate the value of the ratio of each pair of quantities and interpret the result:
- a) $3\ \text{kg}$ of cherries cost $\$210$.
- b) A gravel cyclist covers $95\ \text{km}$ in $4\ \text{h}$.
- c) A bottle contains $2\ \text{L}$ and a drum, $25\ \text{L}$.
Two metal rods measure $8\ \text{dm}$ and $12\ \text{dm}$. a) Calculate the ratio of the length of the first to the second, and simplify it. b) Calculate the ratio of…
Two metal rods measure $8\ \text{dm}$ and $12\ \text{dm}$.
- a) Calculate the ratio of the length of the first to the second, and simplify it.
- b) Calculate the ratio of the second to the first.
- c) How are the two ratios obtained related to each other? Does this always happen?
A road map is drawn to a scale of $1 : 300,000$. a) Explain the meaning of this ratio. b) What real-life distance is represented by $5\ \text{cm}$ on the map? c) Two…
A road map is drawn to a scale of $1 : 300,000$.
- a) Explain the meaning of this ratio.
- b) What real-life distance is represented by $5\ \text{cm}$ on the map?
- c) Two towns are $21\ \text{km}$ apart in reality. How far apart are they on the map?
Proportions
Finding the missing term
Thanks to this property, if you know three terms, you can find the fourth (the fourth proportional):
$$\frac{x}{4} = \frac{9}{12} \ \Rightarrow\ 12x = 36 \ \Rightarrow\ x = 3$$And if the means are equal, $\frac{a}{x} = \frac{x}{b}$, then $x^2 = ab$: $x = \sqrt{ab}$ is the mean proportional (or geometric mean). For example, between 4 and 9: $x = \sqrt{36} = 6$.
Check, by calculating the product of the extremes and the product of the means, whether the following pairs of ratios form a proportion: a) $\frac{6}{4}$ and…
Check, by calculating the product of the extremes and the product of the means, whether the following pairs of ratios form a proportion:
- a) $\frac{6}{4}$ and $\frac{15}{10}$
- b) $\frac{5}{8}$ and $\frac{7}{12}$
- c) $\frac{2.5}{4}$ and $\frac{5}{8}$
Find the fourth proportional in each case: a) $\frac{x}{4} = \frac{9}{12}$ b) $\frac{5}{x} = \frac{15}{21}$ c) $\frac{8}{6} = \frac{20}{x}$
Find the fourth proportional in each case:
- a) $\frac{x}{4} = \frac{9}{12}$
- b) $\frac{5}{x} = \frac{15}{21}$
- c) $\frac{8}{6} = \frac{20}{x}$
Find the mean proportional in each case, that is, the positive value of $x$ that satisfies the continuous proportion: a) $\frac{x}{4} = \frac{25}{x}$ b)…
Find the mean proportional in each case, that is, the positive value of $x$ that satisfies the continuous proportion:
- a) $\frac{x}{4} = \frac{25}{x}$
- b) $\frac{3}{x} = \frac{x}{27}$
- c) $\frac{x}{2} = \frac{50}{x}$