Averages, Mixtures & Alligation

An average shares a total equally. A mixture combines quantities; larger quantities have more influence on the final average.

Track the conserved ingredient. Repeated removal from a well-mixed container multiplies the remaining concentration.

Three children collect 6, 9 and 12 stones. What is the average? 1. Add the stones: 6 + 9 + 12 = 27. 2. There are three children, so divide by 3. 3. 27 ÷ 3 = 9 stones per child.

The bars show 6, 9 and 12 stones; the dashed line marks the mean of 9. Moving three stones from the third child to the first would bring every bar to that line without changing the total. For groups of unequal size, use weighted totals rather than an unweighted average of group means.

A 40 litre container is full of milk. Remove 10 litres and replace with water; repeat once after mixing. Each replacement retains 30/40=3/4 of the existing milk. Milk remaining=40 x (3/4)^2=22.5 litres; water=17.5 litres. After n identical replacements the milk is V(1-r/V)^n, because the same retained fraction acts each time.

Do not average two group averages unless their group sizes are equal. This shortcut needs complete mixing and restoration to the same volume. Simply subtracting 20 litres of milk ignores water in the second removal.

Two students score 10 each and three score 20 each. Total marks = 2 x 10 + 3 x 20 = 80. There are 5 students, so the average is 80 / 5 = 16. Taking (10 + 20)/2 would ignore the different group sizes.

Alligation is a shortcut for mixing two values. The target average must lie between them; otherwise no positive mixture of those two values can produce it.

Track the conserved ingredient. Repeated removal from a well-mixed container multiplies the remaining concentration.

A 40 litre container is full of milk. Remove 10 litres and replace with water; repeat once after mixing. Each replacement retains 30/40=3/4 of the existing milk. Milk remaining=40 x (3/4)^2=22.5 litres; water=17.5 litres. After n identical replacements the milk is V(1-r/V)^n, because the same retained fraction acts each time.

This shortcut needs complete mixing and restoration to the same volume. Simply subtracting 20 litres of milk ignores water in the second removal.

Arithmetic mean = total/count for a nonempty collection. For unequal groups, weight each mean by its group size: combined mean = (n₁a₁+n₂a₂)/(n₁+n₂).

Twenty students average 60 marks and thirty average 70. Their totals are 1200 and 2100, so the combined mean is 3300/50=66, rather than 65. If a group of five people has average age 18, its total age is 90. Adding a person aged 30 changes the mean to 120/6=20. Age problems After t years, every remaining person's age rises by t, so the mean age of an unchanged group rises by t too. If people enter or leave, adjust the total and the count separately. Age differences between two people stay constant, but age ratios usually change.

An average need not equal an observed value. A group's count is the weight, not an extra quantity to add to its marks or ages.

Track the amount of the conserved ingredient. When mixing two concentrations a<c<b, the amounts required at a and b have ratio (b−c):(c−a), provided concentration and quantity use compatible bases.

To mix 20% and 50% solutions into 30%, use amounts in ratio (50−30):(30−20)=2:1. Two litres at 20% and one litre at 50% contain 0.4+0.5=0.9 litre of solute in 3 litres: 30%. Replacement Start with V litres of pure milk. Remove r litres, replace with water to restore V, and mix completely before repeating. Milk remaining after n replacements is V(1−r/V)^n for 0≤r≤V. For V=40, r=10 and n=2, milk remaining is 40×(3/4)²=22.5 litres, with 17.5 litres water.

A positive mixture's concentration lies between its source concentrations. Use the same type of concentration on both sides: mass percentages cannot be combined using volume weights without density information. Repeated replacement needs complete mixing and restoration to the original volume.