pls. explain how to get this:
A number x is the harmonic mean of two
numbers a and b if 1/x
is the mean of 1/a
and 1/b.
a) Write an equation to represent the harmonic
mean of a and b.
b) Determine the harmonic mean of 12 and 15.
c) The harmonic mean of 6 and another
number is 1.2. Determine the other number.
Connecting
Problem Solving
Reasoning and Proving
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184 MHR
a) To write an equation representing the harmonic mean of two numbers, we can use the given information: "1/x is the mean of 1/a and 1/b."
Using the definition of the mean, we know that the mean of two numbers is their sum divided by 2. Therefore, we can express the mean of 1/a and 1/b as:
(1/a + 1/b) / 2
According to the given information, this mean is equal to 1/x. Therefore, we can set up the following equation:
(1/a + 1/b) / 2 = 1/x
This equation represents the harmonic mean of a and b.
b) To determine the harmonic mean of 12 and 15, we need to substitute the values of a and b into the equation we derived in part a) and solve for x.
Substituting a = 12 and b = 15, we have:
(1/12 + 1/15) / 2 = 1/x
To find x, we can start by simplifying the left side of the equation:
(5/60 + 4/60) / 2 = 1/x
(9/60) / 2 = 1/x
Multiplying both sides by 2 to eliminate the fraction on the left:
9/60 = 2/x
Cross-multiplying, we get:
9x = 120
Dividing both sides by 9:
x = 120/9
x = 13.33
Therefore, the harmonic mean of 12 and 15 is approximately 13.33.
c) To determine the other number when the harmonic mean of 6 and that number is 1.2, we can follow a similar process as in part b).
Substituting a = 6 and the other number as b, we have:
(1/6 + 1/b) / 2 = 1/1.2
To simplify the equation, we first multiply both sides by 2:
1/6 + 1/b = 2/1.2
Next, we can find a common denominator on the left side:
(b + 6) / 6b = 2/1.2
To eliminate the fractions, we can cross-multiply:
1.2(b + 6) = 12b
1.2b + 7.2 = 12b
Subtracting 1.2b from both sides:
7.2 = 10.8b
Dividing both sides by 10.8:
b = 7.2/10.8
b = 0.6667
Therefore, the other number when the harmonic mean of 6 and that number is 1.2 is approximately 0.6667.
To find the harmonic mean, we follow these steps:
a) Write an equation to represent the harmonic mean of a and b.
Let x be the harmonic mean of two numbers a and b.
The reciprocal of x, which is 1/x, is the mean of the reciprocals of a and b.
So, we can write the equation as:
1/x = (1/a + 1/b)/2
b) Determine the harmonic mean of 12 and 15.
We need to find x, which is the harmonic mean of 12 and 15, following the equation from part (a):
1/x = (1/12 + 1/15)/2
To simplify this equation, we need to find the common denominator, which is 60:
1/x = (5/60 + 4/60)/2
1/x = (9/60) /2
1/x = 9/120
1/x = 3/40
To find x, we take the reciprocal of both sides:
x = 40/3
So, the harmonic mean of 12 and 15 is 40/3.
c) The harmonic mean of 6 and another number is 1.2. Determine the other number.
We have the harmonic mean (x) of 6 and another number.
We can write the equation as:
1/x = (1/6 + 1/y)/2
Substituting the given value, 1.2, for x:
1/1.2 = (1/6 + 1/y)/2
To simplify this equation, we need to find the common denominator:
1/1.2 = (2/12 + 1/y)/2
1/1.2 = (2 + 12/y)/24
To eliminate the denominator, we multiply both sides of the equation by 24:
24/1.2 = 2 + 12/y
Simplifying,
20 = 2 + 12/y
To isolate y, we subtract 2 from both sides:
20 - 2 = 12/y
18 = 12/y
To find y, we take the reciprocal of both sides:
y = 12/18
y = 2/3
So, the other number is 2/3.