Learn first: Common and decimal fractions
Ratio and proportion are the “language and grammar” of word problems. Map scales, recipes, workers and days, sharing money between partners, the make-up of an alloy — all rely on this topic. In the exam it appears both as a separate question and inside problems on percentages, geometry (similarity) and physics. In this lesson we cover simplifying ratios, the main property of a proportion, direct and inverse proportion, dividing a number in a given ratio, scale and mixture problems.
The ratio of two numbers is their quotient: a : b = a/b. A ratio shows how many times one number is larger than another, or what part of another it is. For example, if a class has 12 boys and 18 girls, the ratio of boys to girls is 12 : 18 = 2 : 3.
The ratio 2 : 3 also means “2/5 and 3/5 of the whole”: there are 2 + 3 = 5 parts in total. This idea underlies all the problems in §4.
A proportion is an equality of two ratios: a : b = c : d. a and d are the outer terms (extremes), b and c the inner terms (means).
Main property: a·d = b·c (product of the extremes = product of the means)
The property comes from multiplying both sides of a/b = c/d by bd. It gives the rule for an unknown term: an extreme = product of the means / the known extreme (x = (b·c)/d); a mean = product of the extremes / the known mean.
In a proportion the means, or the extremes, may be swapped: from a : b = c : d it follows that a : c = b : d and d : b = c : a. A “derived proportion” is also useful: if a : b = c : d, then (a + b) : b = (c + d) : d.
Find x from x : 12 = 5 : 3.
Direct: x₁/x₂ = y₁/y₂; inverse: x₁/x₂ = y₂/y₁
Before solving, always ask: “if one quantity increases, does the other increase or decrease?”. If the quantities are not proportional at all (for example, a person’s age and height), no proportion can be set up.
6 workers do a job in 10 days. How many days will 15 workers take?
To divide S in the ratio a : b : c, first find one part:
one part q = S/(a + b + c), parts: a·q, b·q, c·q
For chained ratios (a : b and b : c), make the common term b the same number. For example, if a : b = 3 : 4 and b : c = 6 : 5, make b equal to 12: 9 : 12 and 12 : 10, so a : b : c = 9 : 12 : 10.
With “times more/less” conditions, it helps to let the unknown be the smallest or most convenient quantity.
A shop sold 175 kg of potatoes in 3 days. On the second day it sold 1.5 times as much as on the third, and on the first day 2.4 times less than on the second. How many kg were sold on the first day?
A scale 1 : n means 1 unit on the map corresponds to n units on the ground. For example, at 1 : 50 000, 1 cm = 50 000 cm = 500 m.
The ratio of areas is the square of the ratio of lengths: at scale 1 : n areas are in the ratio 1 : n², and volumes 1 : n³. For example, on a 1 : 10 000 map 1 cm = 100 m, so 1 cm² = 10 000 m² = 1 hectare; a plot of 3 cm² on the map covers 3 hectares on the ground.
If the components are given as a ratio, first find “one part”. When a substance is added to or removed from a mixture, the problem becomes easier if you follow the component that does not change.
A 40 kg alloy of copper and zinc contains copper and zinc in the ratio 3 : 2. How much zinc must be added so that there is equal copper and zinc?