Numbers, Quantification and Numerical Applications

About this unit

This Applied Mathematics unit uses arithmetic to model everyday situations. Modulo arithmetic deals with remainders and cycles, such as days of the week or clock times. Alligation finds the mixing ratio that produces a desired mean price. Boats and streams, pipes and cisterns, and races and games are classic rate problems built on the relations distance = speed × time and work = rate × time. Numerical inequalities express real-life limits such as budgets, minimum marks or capacity using the signs <, ≤, > and ≥.

Key ideas

• a mod n is the remainder (from 0 to n − 1) when a is divided by n; a ≡ b (mod n) means n divides a − b.
• Rule of alligation: cheaper : dearer = (dearer price − mean) : (mean − cheaper price).
• Downstream speed = boat speed + stream speed; upstream speed = boat speed − stream speed.
• A pipe that fills a tank in n hours does 1/n of the job per hour; an outlet contributes a negative rate.
• In races, equal times mean distances are in the ratio of speeds.
• Multiplying or dividing an inequality by a negative number reverses its sign.

How to prepare

Work through many short numericals and learn the "per unit time" approach, which unifies pipes, work and speed problems. Memorise the alligation cross diagram and the boats and streams formulas. Always state the units in the final answer.