The rate of change is an indicator that allows you to measure the variation between an initial value and a final value. Its analysis allows for conclusions to be drawn about the possible causes of its variation in order to provide potential solutions. What is the formula for calculating this rate? Mathematics offers us one. Discover in this article the rate of change formula in mathematics.
Positive rate of change.
To calculate the rate of change of a quantity or an index, you need to have its initial value and its final value. The general formula for calculating the rate of change in mathematics is presented in the following form: let X be the initial value, X2 the final value, and T the rate of change.
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T= [(X2-X1) /X1] ×100
Thus, the result of T that you find is the percentage change of the quantity or index from one year to another or from its initial value to its final value. For example, let’s assume: X1 is the number of students in 2021 and X2 is the number of students in 2022: X1 1500= students and X2 =2000 students.
T= [(2000-1500) /1500] ×100 which gives T= 33.33%
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We see that the number of students increased from 2021 to 2022 by 33.33%. This implies that the rate of change is positive. Furthermore, we can also determine the initial value and the final value if we know the rate of change.
The initial value.
The rate of change and the final value can be used to determine the final value if they are known. The mathematical formula below can be applied. X2 being the final value, X1 the initial value, and T the rate of change, we have:
X2= X1 +T×X1/100 X2= X1(1+)
Let’s take a case as an illustration. Suppose the number of students in 2021 is 1000 and it increased by 40% during the year. To find the final value, the last formula will be applied.
X2= 1000( 1+) X2= 1400. This means that the final value is 1400.
The initial value
Similarly, the initial value can be determined if you know the final value and the rate of change. The formula below will be applied in this case.
X2= X1(1+) which leads to X1=
Let’s assume that the number of students in 2022 is 2000 students and that during the year this number decreased by 30%. According to the last formula, we will have:
X1= X2= 1538.4615 which means that the initial value is 1538.4615.
Negative rate of change
The general formula for calculating the rate of change applies even when it comes to a negative rate. Indeed, it is the variation of the quantity or index that determines the sign of the rate of change. If the variation is negative, the rate of change is also negative. Let’s take a case to illustrate. Suppose the number of students in 2021 is 1500 students and the number of students in 2022 is 1000 students. The general formula will be applied to find the rate of change.
T=[(X2-X1) /X1] × 100
By applying the formula, we obtain:
T= [(1000-1500) /1500] T= -33.33% We note a decrease of 33.33% in the number of students from 2021 to 2022. The rate of change in this case is negative. The formulas for initial value and final value can also be used to determine the initial and final data. Furthermore, there are other formulas aside from the general rate of change formula.
Reciprocal rate of change
The formula for the reciprocal rate of change is obtained by interchanging the initial and final values of the rate of change formula. It is presented as follows.
T= [(X1-X2) /X2] ×100
Average rate of change
Consider a quantity or index that goes from an initial value to a final value over n periods (n can be in days, months, or years).
TM=
By applying this formula, you obtain the average rate which, if it were the same over each of the n periods, would allow the transition from the initial value X1 to the final value X2.