Averages
The average is the sum divided by the count. The total equals the average times the count. The average of the first n natural numbers is n plus 1 over 2. The average of the first n odd numbers is n and of the first n even numbers is n plus 1. If a new member joins and raises the average by d, the new member equals the old average plus n plus 1 times d, where n is the old count. The weighted average is the sum of each count times its average, divided by the total count. Adding k to every number raises the average by k, and multiplying every number by k multiplies the average by k.
Mixtures and Alligation
If a cheaper item costs c, a dearer item costs d, and the mixture costs m, then the ratio of cheaper to dearer is d minus m to m minus c. If two items are mixed in the ratio x to y, the mixture price is the sum of price times quantity divided by the total quantity. For a milk and water mixture of N litres in the ratio x to y, the milk is N times x over x plus y.
Age Problems
Take present ages as variables or as multiples of a ratio unit. Past or future ages are present ages minus or plus the number of years. Everybody's age changes by the same number of years, so the difference in ages never changes. Form an equation from the given condition.
Shortcuts
Use the constant age difference. If the father is k times the son, the son's age equals the difference divided by k minus one. In averages, find totals first. In alligation, the mixture price must lie between the two prices.
Worked Examples
Example 1: Five numbers have an average of 20. A sixth number 26 is added. Find the new average.
Solution: The total is 100. New total is 126, so the average is 21.
Example 2: A father is 40 and his son is 10. In how many years will the father be twice as old as the son?
Solution: 40 plus t equals 2 times 10 plus t, so t equals 20.
Example 3: In what ratio should rice at 20 and at 30 be mixed to get a mixture at 26?
Solution: The ratio is 4 to 6, which is 2 to 3.
Quick Check
Question 1: What is the average of the first n natural numbers?
Answer: n plus 1 over 2.
Question 2: Does the difference between two people's ages change over time?
Answer: No, it stays constant.
Question 3: What is the alligation rule?
Answer: Cheaper to dearer equals d minus m to m minus c.