Solving Continued Proportion Problem with Componendo Dividendo

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In summary, the conversation is about solving a mean proportion problem where a, b, and c are in continued proportion. The goal is to prove that (a+b+c)^2/(a^2 +b^2 + c^2)=(a+b+c)/(a-b+c). The person suggests breaking the numerator into (a^2 +b^2+c^2+2ab+2bc+2ac) and using componendo dividendo. However, it is not working out and they are seeking help. Another person suggests using the fact that ac=b^2 to simplify the equation.
  • #1
kuheli
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Re: mean proportion problem

if a,b,c are in continued proportion

prove that
(a+b+c)^2/(a^2 +b^2 + c^2)=(a+b+c)/(a-b+c) i break the part -> (a+b+c)^2 into (a^2 +b^2+c^2+2ab+2bc+2ac) and then use componendo dividendo to the numerator and denominator to the problem,,, but its still not working out .wondering how to solve it . please help ! :confused:
 
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  • #2
Re: mean proportion problem

kuheli said:
if a,b,c are in continued proportion

prove that
(a+b+c)^2/(a^2 +b^2 + c^2)=(a+b+c)/(a-b+c) i break the part -> (a+b+c)^2 into (a^2 +b^2+c^2+2ab+2bc+2ac) and then use componendo dividendo to the numerator and denominator to the problem,,, but its still not working out .wondering how to solve it . please help ! :confused:

Hi kuheli, :)

Note that,

\[(a-b+c)(a+b+c)=a^2-b^2+c^2+2ac\]

Since \(\frac{a}{b}=\frac{b}{c}\Rightarrow ac=b^2\) we have,

\[(a-b+c)(a+b+c)=a^2+b^2+c^2\]

Hope you can continue. :)
 

FAQ: Solving Continued Proportion Problem with Componendo Dividendo

What is a continued proportion problem?

A continued proportion problem is a type of mathematical problem that involves finding the missing term in a proportion, where the ratio between the first and second terms is equal to the ratio between the second and third terms. It is often used in real-life situations such as scaling and resizing objects.

How do you solve a continued proportion problem?

To solve a continued proportion problem, you first need to set up a proportion with four terms. Then, you can cross-multiply and solve for the missing term. It is important to remember to keep the ratios in the same order when setting up the proportion.

What are the key terms used in continued proportion problems?

The key terms used in continued proportion problems are "antecedent" and "consequent." The antecedent refers to the first term in the proportion, while the consequent refers to the second term. These terms are used to differentiate between the two ratios in the proportion.

Can you give an example of a continued proportion problem?

Yes, an example of a continued proportion problem is: If it takes 3 hours for 4 workers to complete a project, how many hours will it take for 6 workers to complete the same project? The proportion would be set up as 3/4 = x/6, where x represents the missing term. Solving for x would give the answer of 4.5 hours.

What are some real-life applications of continued proportion problems?

Continued proportion problems can be applied in various real-life situations such as resizing images, finding the missing side of a similar shape, or calculating ingredient measurements for a recipe. They can also be used in business and finance to calculate profit margins or determine the most cost-effective option.

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