Ratios, Averages and Mixtures

Ratios and proportion

A ratio a : b compares two quantities; multiplying both terms by the same number gives the same ratio. Four quantities are in proportion if a : b = c : d, that is ad = bc.

• Sharing: to divide N in the ratio a : b, the parts are N·a/(a + b) and N·b/(a + b). Dividing 1200 in the ratio 3 : 5 gives 450 and 750.
• Combining ratios: if A : B = 2 : 3 and B : C = 4 : 5, make B equal in both: A : B = 8 : 12 and B : C = 12 : 15, so A : B : C = 8 : 12 : 15.
• Partnership: profit is shared in the ratio of (investment × time). Investments of 2000, 3000 and 5000 for the same period share a profit of 1500 as 300, 450 and 750.

Averages

• Average = sum / count, so sum = average × count: most average problems are solved by working with sums.
• The average of 5 numbers is 20 (sum 100). After one is removed, the average of the rest is 18 (sum 72), so the removed number is 100 − 72 = 28.
• A weighted average combines groups of different sizes: (n₁a₁ + n₂a₂)/(n₁ + n₂).

Average speed

Average speed = total distance / total time, never the plain average of the speeds (unless equal times are spent at each speed).

For equal distances at speeds u and v, the average is the harmonic mean 2uv/(u + v): going at 40 km/h and returning at 60 km/h averages 2 × 40 × 60 / 100 = 48 km/h, not 50.

Mixtures and alligation

When two ingredients with values c₁ (cheaper) and c₂ (dearer) are mixed to get a mean value m, the quantities must be in the ratio

(c₂ − m) : (m − c₁)

Example: rice at ₹40/kg and ₹60/kg mixed to cost ₹46/kg needs (60 − 46) : (46 − 40) = 14 : 6 = 7 : 3.

Another common type: 60 litres of milk and water are in the ratio 3 : 1 (45 L milk, 15 L water). To make the ratio 1 : 1, add water until it equals the milk: 30 litres.

Age problems

Translate each sentence into an equation in the unknown multiplier. "Five years ago, A and B's ages were in the ratio 3 : 4; five years from now they will be in the ratio 5 : 6." Let their ages five years ago be 3x and 4x. Then (3x + 10)/(4x + 10) = 5/6, so 18x + 60 = 20x + 50 and x = 5: they are now 20 and 25.

from fractions import Fraction

print("A:B:C =", (2 * 4, 3 * 4, 3 * 5))
print("removed number =", 5 * 20 - 4 * 18)
print("average speed =", 2 * 40 * 60 / (40 + 60), "km/h")
print("alligation ratio =", Fraction(60 - 46, 46 - 40))
x = next(x for x in range(1, 50) if Fraction(3 * x + 10, 4 * x + 10) == Fraction(5, 6))
print("present ages:", 3 * x + 5, "and", 4 * x + 5)

A:B:C = (8, 12, 15)
removed number = 28
average speed = 48.0 km/h
alligation ratio = 7/3
present ages: 20 and 25

Practice questions