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Two numbers are in the ratio of 4 : 9. If their sum is 26, then find the mean proportional between them.
11
12
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12
Since the ratio is 4:9, the mean proportional is 4x├Ч9xтАЛ=36x2тАЛ=6x. Given 4x+9x=13x=26, so x=2. Thus, 6├Ч2=12.
Ratio of two numbers is 4 : 9, and the sum of the numbers is 26.
Mean┬аProportional=a├ЧbтАЛ
Since the ratio is 4:9, the mean proportional is 4x├Ч9xтАЛ=36x2тАЛ=6x. Given 4x+9x=13x=26, so x=2. Thus, 6├Ч2=12.
Students often calculate the individual numbers (8 and 18) and then try to find the mean proportional, which is correct but time-consuming; the trap is in miscalculating the sum constant (13x = 26).
Define the numbers
Let the two numbers be 4x and 9x based on the given ratio of 4:9.
4x,9x
Solve for the common ratio constant x
Given that their sum is 26, set up the equation 4x+9x=26 to find x.
13x=26тЯ╣x=2
Find the individual numbers
Substitute x=2 back into the expressions to find the numbers: 4(2)=8 and 9(2)=18.
8,18
Calculate the mean proportional
The mean proportional of a and b is defined as a├ЧbтАЛ. Substituting our values: 8├Ч18тАЛ=144тАЛ.
Mean=8├Ч18тАЛ=12
B is correct because the calculation of the mean proportional 8├Ч18тАЛ results in 12.
The mean proportional abтАЛ is the geometric mean of two numbers; in geometry, this is frequently used to find the altitude to the hypotenuse in a right-angled triangle.
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