Geometric Mean
Problems to do with means arise when we wish to replace a quantity that varies with a constant that would produce an equivalent result. They also occur in geometrical contexts.
For example, when a car journey is undertaken there are many variations in speed along the way but we find it convenient to think of the average speed - the constant speed that would result in the same time being taken for the journey.
Another example, involving just a finite number of values, is the set of examination scores for a student studying five different subjects. The actual scores could vary considerably but it is convenient to give the mean score as a measure of the student's achievement. This is the score that the student would have obtained in every subject if his or her performance had been equally good in all of them.
Several kinds of means were identified by ancient Greek mathematicians. We frequently use the arithmetic mean, obtained by summing a number of values and dividing the result by the number of values summed. The example about examination results is of this kind. We use the tools of calculus to extend this idea of a mean to quantities that vary continuously, such as in the car journey example.
A different approach is needed when we deal with varying quantities that are to be combined by multiplication rather than by addition. For example, to calculate the volume of a rectangular prism we multiply together three mutually perpendicular lengths. If we wished to replace three different lengths by three identical lengths - making a cube - such that the volume of the cube was the same as that of the prism, we would find the cube root of the prism volume. This mean is called the geometric mean.
Example 1
A rectangular prism has sides $5.4\ m$5.4 m , $1.8\ m$1.8 m and $0.9\ m$0.9 m . What cube has the same volume?
The volume of the prism is $5.4\times1.8\times0.9=8.748\ m^3$5.4×1.8×0.9=8.748 m3. The cube with the same volume must have sides $\sqrt[3]{5.4\times1.8\times0.9}\approx2.06\ m$3√5.4×1.8×0.9≈2.06 m.
Note that if we took the arithmetic mean of the three lengths and used it to construct a cube, the volume of the resulting cube would be much greater than that of the prism. The arithmetic mean is $\frac{5.4+1.8+0.9}{3}=2.7$5.4+1.8+0.93=2.7 and $2.7^3=19.683\ m^3$2.73=19.683 m3.
In this case, the geometric mean was approximately $2.06$2.06 and the arithmetic mean was $2.7$2.7. It is always the case that the geometric mean of a collection of positive numbers is smaller than the arithmetic mean. This can be shown algebraically for pairs of numbers by observing that
| 0 | $\le$≤ | $\left(\sqrt{x}-\sqrt{y}\right)^2$(√x−√y)2 |
| $=$= | $x-2\sqrt{x}\sqrt{y}+y$x−2√x√y+y |
Therefore,
| $2\sqrt{x}\sqrt{y}$2√x√y | $\le$≤ | $x+y$x+y |
| $\sqrt{x}\sqrt{y}$√x√y | $\le$≤ | $\frac{x+y}{2}$x+y2 |
| $\sqrt{x\times y}$√x×y | $\le$≤ | $\frac{x+y}{2}$x+y2 |
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The following diagram illustrates three kinds of mean.
- Triangle $PRQ$PRQ is inscribed in a semicircle with diameter $PQ$PQ. (We will make use of the fact that the angle, $\angle PRQ$∠PRQ, in a semicircle is always a right angle.)
- The line segment $RS$RS is perpendicular to the diameter.
- Similarly, $ST$ST is perpendicular to the radius $OR$OR.
A point $S$S on the diameter divides $PQ$PQ into segments $PS$PS and $SQ$SQ with lengths $a$a and $b$b respectively. Since, $\overline{PQ}=a+b=2r$PQ=a+b=2r, we have
$r=\frac{a+b}{2}$r=a+b2.
Thus, the radius $OR$OR is the arithmetic mean of segments $PS$PS and $SQ$SQ.
By Pythagoras, $g^2+a^2=x^2$g2+a2=x2 and $g^2+b^2=y^2$g2+b2=y2. Also, $x^2+y^2=\left(a+b\right)^2$x2+y2=(a+b)2. Therefore, by substitution,
$\left(g^2+a^2\right)+\left(g^2+b^2\right)=\left(a+b\right)^2$(g2+a2)+(g2+b2)=(a+b)2.
On simplifying this, we have $2g^2=2ab$2g2=2ab and hence,
$g=\sqrt{ab}$g=√ab.
Thus, segment $RS$RS is the geometric mean of segments $PS$PS and $SQ$SQ.
In the diagram, the segment $RT$RT represents a third type of mean, called the harmonic mean. The triangles $RTS$RTS and $RSO$RSO are similar (equal angles). Therefore, we can form the ratios $\frac{h}{g}=\frac{g}{r}$hg=gr. That is, $h=\frac{g^2}{r}$h=g2r.
On substituting the expressions for $r$r and $g$g, and rearranging, we have
$h=\frac{2}{\frac{1}{b}+\frac{1}{a}}.$h=21b+1a.
This is the formula that defines the harmonic mean of two numbers. Thus, segment $RT$RT is the harmonic mean of segments $PS$PS and $SQ$SQ.
It is apparent from the diagram that the arithmetic mean is greater than the geometric mean, which is greater than the harmonic mean.
More examples
Example 2
The geometric mean of $20$20 and $29$29 is $\sqrt{20\times29}$√20×29. In symbols, we could write this as $g\left(20,29\right)$g(20,29). Explore whether $g\left(20,29,41\right)$g(20,29,41) is the same as $g\left(g\left(20,29\right),41\right)$g(g(20,29),41). That is, is the operation of finding this geometric mean associative?
$g\left(20,29\right)=\sqrt{580}\approx24.0832$g(20,29)=√580≈24.0832
$g\left(20,29,41\right)=\sqrt[3]{20\times29\times41}\approx28.7566$g(20,29,41)=3√20×29×41≈28.7566
$g\left(24.0832,41\right)=\sqrt{24.0832\times41}\approx31.4231$g(24.0832,41)=√24.0832×41≈31.4231
Since the last two results are not equal, we see that associativity fails!
Question 1
A $\$100$$100 investment in the company RocketZ grows by $40$40% in the first year, $10$10% in the second year, and $30$30% in the third year. The same investment in the company Lesta grows by $p$p% every year. At the end of $3$3 years the value of investments in RocketZ and Lesta are exactly the same.
Find $p$p. Give your answer to one decimal place.
Let $a%$a%, $b%$b% and $c%$c% be the interest rates for three consecutive years on an initial investment of $\$I$$I.
What information does the geometric mean $\sqrt[3]{\left(100+a\right)\left(100+b\right)\left(100+c\right)}$3√(100+a)(100+b)(100+c) provide?
The constant interest rate that has the same return after three years on an initial investment of $\$I$$I.
AThe return after three years on an initial investment of $\$I$$I.
B$\$I$$I plus the constant interest rate that has the same return after three years on an initial investment of $\$I$$I.
CThe initial investment with a return of $\$I$$I after three years.
D
Question 2
Question 3
Find the exact value of $h$h in the following diagram, given that the two smaller triangles are similar.
A right-angled triangle with the right angle is indicated by a square box at the lower left corner. This right angle is split into two by an internal segment labeled as $h$h which divides the original triangle into two triangles, thus making this internal segment as a shared side of the two triangles. The two triangles inside the original triangle differ in size: The one on the upper part is larger compared to the one on the bottom part.
The larger triangle has a side measuring 8 m which is opposite to the divided angle. The smaller triangle has a side measuring 3 m which is also opposite to the divided angle.