Rates. Applications of Ratios and Rates in Practice - Revision Notes

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4.2Rates. Applications of Ratios and Rates in Practice


In these revision notes for Rates. Applications of Ratios and Rates in Practice, we cover the following key points:

  • What are rates? Where do they differ from ratios?
  • What is a unit rate?
  • What are the associated unit rates? How many are there in a rate?
  • What is the rate of a quantity change? What is the common quantity in all rates of change?
  • What are some applications of rates in practice?
  • What is the golden ratio? How to find it?
  • What is the Fibonacci sequence? Why it is so important?
  • What are some uses of the golden ratio in practice?

Rates. Applications of Ratios and Rates in Practice Revision Notes

By definition, rate is a ratio-like structure involving two different types of quantities. A rate is usually expressed as a fraction.

Rates are very common in physics. Thus, the rate of position change (Δx) represents the velocity v (v = Δx / t where t is the time elapsed); the rate of velocity change (Δv) gives the acceleration a (a = Δv / t); the volumetric rate of water flow R is given by R = ΔV / t, where ΔV is the volume of water flowing through the pipe, and so on.

It is possible to write two unit rates for each situation involving rates. If both of them belong to the same event, they are called associated unit rates.

The rate of a quantity change is obtained by dividing two differences: the difference in the given quantity and the difference in time, which in many cases is not evident because we often take the initial time as zero and therefore, the difference in time simply gives the final time.

In general, we have for the rate of a quantity Y change:

Rate of Y change = Yfinal - Yinitial/tfinal - tinitial = Y2 - Y1/t2 - t1

The rate of a quantity change can also be negative. This occurs when the quantity Y decreases over time.

There are numerous applications of rate in practice. Some of them include:

  1. In banking: interest rates are a very fundamental term, when banks offers extra money to people who deposit their capitals in them.
  2. In Physics: kinematics, electromagnetism, dynamics, etc.
  3. In medicine: heart rate monitors, annual incidence rate of an acute disease, etc.
  4. In civil engineering: when expressing material quantities that are to be mixed with another material to form strong structures, etc.

As for ratios, the golden ratio is a very important application of ratios in practice. When we have two numbers a and b (where a is the biggest) which meet the condition, the golden ratio φ is given by

φ = a + b/a = a/b

Fibonacci discovered a special sequence of numbers, where the next number is obtained by the sum of the previous two. It helps calculate the golden ratio with a good approximation, as the terms in this sequence are used for this purpose.

The golden ratio is a constant number, whose value is

φ = 1 + √5/2 = 1.6180339887

Since √5 is irrational, the digits after the decimal point continue on forever without repeating.

The numbers forming a golden ratio must meet the condition "sum × difference = product".

Items that form a golden ratio between them produce a very fascinating view that has been used since antiquity in art and construction. This is because the building structures that form a golden ratio are stronger and more stable. We can find the golden ratio in a number of applications, both natural and human made.

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