Real-Life Ratio Applications and Best-Buys

A ratio compares quantities in a stated order. In a real problem, first identify what each number represents and what is being compared: ingredient with ingredient, an ingredient with the whole mixture, or one quantity with another measured in a different unit. Equivalent ratios keep the same relationship because both sides have been multiplied or divided by the same number.

For example, a conversion rate of £1 to €1.20 can be written as £1 : €1.20. At that rate, £5 corresponds to €6 because both amounts are multiplied by 5. The numbers are linked; they are not amounts to add together. A conversion rate may change in practice, so use the rate given in the question.

Scaling recipes and other quantities

A recipe works only if its ingredients stay in the same proportions. Suppose 6 scones need 250 g of flour, 50 g of butter and 150 ml of milk. To make 15 scones, the scale factor is $15\div6=2.5$. Multiply every ingredient by 2.5: 625 g of flour, 125 g of butter and 375 ml of milk. Scaling just one ingredient would change the recipe.

The same idea works when comparing quantities. You can scale two ratios to a common quantity, or calculate how much there is for one unit. For instance, if one drink contains 3 parts juice to 5 parts lemonade and another contains 2 parts juice to 3 parts lemonade, compare juice as a fraction of each whole drink: $3/8$ and $2/5$. Since $2/5$ is larger, the second drink has the greater proportion of juice. Comparing the first numbers, 3 and 2, alone would not tell you this because the totals differ.

Mixing and concentration

Concentration describes how much of a particular ingredient there is in a mixture. If squash and water are mixed in the ratio $1:6$, there are 7 parts of drink altogether, so squash makes up $1/7$ of it. For 2.8 litres of drink, one part is $2.8\div7=0.4$ litres. Use 0.4 litres of squash and 2.4 litres of water. This is why a ratio of squash to water must not be read as squash to the whole drink.

When two mixtures are combined, work out their actual ingredient amounts before finding the new ratio. For example, a 60 kg alloy with copper to zinc in the ratio $3:2$ contains 36 kg of copper and 24 kg of zinc. A 40 kg alloy with copper to zinc in the ratio $1:3$ contains 10 kg of copper and 30 kg of zinc. Together they contain 46 kg of copper and 54 kg of zinc, giving a new ratio of $46:54=23:27$. The original ratios cannot simply be added because they describe different total masses.

Finding the best buy

Pack sizes and prices often differ, so compare them on the same basis. A 750 g pack costing £2.40 has a unit price of $£2.40\div0.75=£3.20$ per kg. A 1.2 kg pack costing £3.60 has a unit price of $£3.60\div1.2=£3.00$ per kg. The 1.2 kg pack is the better value per kilogram, although it costs more at the till. A question may also impose a budget or require a particular amount; in that case, check that the better-value option meets those conditions.