Ratio and proportion
In a Transition Year class there are 5 boys and 7 girls. In a recipe, for every 2 cups of rice, 3 cups of water are used. In both cases we compare two quantities by dividing: that is a ratio, and if two ratios are equal, they form a proportion.
Ratio between two quantities
In a video game tournament there are 16 strategy players and 24 sports players. The strategy-to-sports ratio is $\frac{16}{24} = \frac{2}{3}$: for every 2 strategy players, there are 3 sports players.
On a social network, an influencer's video has received $450$ "likes" after being viewed by $3000$ people. Calculate the ratio of the number of "likes" to the number of…
On a social network, an influencer's video has received $450$ "likes" after being viewed by $3000$ people. Calculate the ratio of the number of "likes" to the number of views, simplify the result, and interpret what the value obtained means.
A video game shop offers a pack of $3$ controller covers for $12.60$ €. Express the ratio of the price to the number of covers. What would we usually call this result in…
A video game shop offers a pack of $3$ controller covers for $12.60$ €. Express the ratio of the price to the number of covers. What would we usually call this result in everyday language?
Simplifying a ratio
A ratio is simplified in the same way as a fraction: by dividing both terms by their highest common factor, until it cannot be simplified any further.
In a Maths exam of $20$ questions, Nerea answered $15$ correctly and got $5$ wrong. Write down the ratio of: a) The number of correct answers to the number of incorrect…
In a Maths exam of $20$ questions, Nerea answered $15$ correctly and got $5$ wrong. Write down the ratio of:
- a) The number of correct answers to the number of incorrect answers.
- b) The number of correct answers to the total number of questions.
State in each case which is the antecedent and which is the consequent.
A graphic designer is working with two image formats. The first format has dimensions of $12$ cm wide by $18$ cm high. The second format is $20$ cm wide by $30$ cm high.…
A graphic designer is working with two image formats. The first format has dimensions of $12$ cm wide by $18$ cm high. The second format is $20$ cm wide by $30$ cm high.
- a) Calculate the ratio of width to height for each of the formats.
- b) Are these dimensions proportional? Justify your answer by comparing the ratios.
Proportions: the equality of ratios
If you know three terms of the proportion, you can find the fourth by solving for it:
$$\frac{3}{8} = \frac{15}{x} \ \Rightarrow\ 3x = 120 \ \Rightarrow\ x = 40$$On a streaming platform, a moderator counted $1350$ messages in the stream's chat over the last $45$ minutes. a) Calculate the ratio of the number of messages to the…
On a streaming platform, a moderator counted $1350$ messages in the stream's chat over the last $45$ minutes.
- a) Calculate the ratio of the number of messages to the time elapsed in minutes.
- b) Explain what the result obtained means in the context of the problem.
- c) If the ratio remains constant, how many messages will be received in a full hour?
Population density is the ratio of the number of inhabitants of a territory to its area expressed in $km^2$. Observe the data for these two fictional provinces: Province…
Population density is the ratio of the number of inhabitants of a territory to its area expressed in $km^2$. Observe the data for these two fictional provinces:
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Province A: $450\,000$ inhabitants and $15\,000$ $km^2$.
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Province B: $280\,000$ inhabitants and $7\,000$ $km^2$.
- a) Calculate the population density of each province and interpret the result.
- b) Which of the two provinces is more "populated" relative to its size?
Comparing ratios and rates
In a ratio $a : b$, the first term $a$ is called the antecedent and the second term $b$ is the consequent. In "3 tablets to 5 laptops", 3 is the antecedent and 5 is the consequent.
When you compare two quantities of the same kind, both must be in the same unit. A ratio of two quantities in different units, such as euro per kilogram or km per hour, is called a rate. A rate with a consequent of 1 is a unit rate: it tells you the amount for one unit.
To compare two ratios, write each as a decimal or put them over a common denominator. For example, $\frac{7}{10} = 0.7$ and $\frac{9}{12} = 0.75$, so $\frac{9}{12}$ is the bigger ratio.
If you know two linked ratios, multiply them to chain them. If $A : B = \frac{2}{3}$ and $B : C = \frac{3}{5}$, then $A : C = \frac{2}{3} \cdot \frac{3}{5} = \frac{2}{5}$. Check: with 6 litres of A, B is 9 litres and C is 15 litres, and $\frac{6}{15} = \frac{2}{5}$.
When lengths are in the ratio $1 : k$, areas are in the ratio $1 : k^2$, because area is length times length.
For an end-of-term party, students are comparing two offers on bags of crisps to see which is more economical. The "Crunch" brand offers a $250$ g bag for $1.75$ €,…
For an end-of-term party, students are comparing two offers on bags of crisps to see which is more economical. The "Crunch" brand offers a $250$ g bag for $1.75$ €, while the "Snack" brand sells a $400$ g bag for $2.60$ €.
- a) Calculate the ratio of price to weight (in euros per gram) for each brand.
- b) Using the ratios obtained, state which of the two brands is cheaper.
A scale model of a new municipal sports centre has been built. The real building has a total length of $75$ metres, while on the model that same distance measures $30$…
A scale model of a new municipal sports centre has been built. The real building has a total length of $75$ metres, while on the model that same distance measures $30$ centimetres.
- a) Calculate the ratio of the model measurement to the actual measurement (remember to express both in the same unit).
- b) What is the scale of the plan used for the model?
In an archery competition, an archer's score is analysed using the ratio of "arrows in the centre" to "total arrows shot". Archer A achieved a ratio of $7:12$ in $48$…
In an archery competition, an archer's score is analysed using the ratio of "arrows in the centre" to "total arrows shot".
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Archer A achieved a ratio of $7:12$ in $48$ shots.
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Archer B achieved a ratio of $5:8$ in $40$ shots.
- a) Which of the two archers was more effective? Justify your answer by comparing the ratios using a common denominator.
- b) If Archer A wants to match Archer B's success ratio in their next round of $24$ arrows, how many of those new arrows must hit the centre?
Back to the beginning: In the Transition Year class the ratio of boys to girls is $\frac{5}{7}$ (whereas boys make up $\frac{5}{12}$ of the total); in the recipe, the rice-to-water ratio is $\frac{2}{3}$, so for 6 cups of rice you would need 9 cups of water because $\frac{2}{3} = \frac{6}{9}$.