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Math Lesson 4.2.3 - Rate of a Quantity Change

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Welcome to our Math lesson on Rate of a Quantity Change, this is the third lesson of our suite of math lessons covering the topic of Rates. Applications of Ratios and Rates in Practice, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Rate of a Quantity Change

Earlier we discussed rate examples where time was involved. In such situations, we want to calculate how the other quantity change over time. Therefore, the rates expressing these changes are 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 Y for the rate of a quantity change:

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

Example 3

The price of an item in 2019 was $80 and in 2024 the same item was priced at $120. What is the yearly (annual) rate of the price increase?

Solution 3

Here, the quantity Y represents the price. Thus, using the general formula of rate of change

Rate of Y change = Yfinal - Yinitial/tfinal - tinitial

we obtain

Annual rate of price change = Final price - Initial price/Final year - Initial year
= $120 - $80/2024 - 2019
= $40/5 years
= $8/year

The rate of a quantity change can be also negative. This means the final value of the quantity involved is smaller than the initial value because the quantity decreases. Look at the example below.

Example 4

The price of petrol during lockdown caused by the COVID-19 pandemic dropped uniformly from $2.15/gallon to $1.73/gallon in two weeks due to the reduced use of cars. What was the rate of price change during this period?

Solution 4

Price reduction here represents the rate of price change. We have

Rate of Y change = Yfinal - Yinitial/tfinal - tinitial

Here, Y represents the price and tfinal - tinitial = 2 weeks = 14 days. Hence,

Rate of price change = $1.73 - $2.15/14 days
= -$0.03/day

This result means the price dropped by a rate of $0.03 per day.

More Rates. Applications of Ratios and Rates in Practice Lessons and Learning Resources

Ratio and Proportion Learning Material
Tutorial IDMath Tutorial TitleTutorialVideo
Tutorial
Revision
Notes
Revision
Questions
4.2Rates. Applications of Ratios and Rates in Practice
Lesson IDMath Lesson TitleLessonVideo
Lesson
4.2.1What Are Rates? Clarifying Misconceptions
4.2.2Two Associated Unit Rates
4.2.3Rate of a Quantity Change
4.2.4Applications of Rates in Practice
4.2.5Applications of Ratios in Practice - The Golden Ratio and Fibonacci Numbers

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