Highest Common Factor: solving equations with fractions.

[SoCentral2]

Problem 19. Evaluate: \(\displaystyle \, \dfrac{3^2\, \times\, 5^5\, +\, 3^3\, \times\, 5^3}{3^4\, \times\, 5^4}\)

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Dividing each term by the HCF (highest common factor) of the three terms, i.e., 3² × 5³, gives:

. . . . .\(\displaystyle \begin{align} \dfrac{3^2\, \times\, 5^5\, +\, 3^3\, \times\, 5^3}{3^4\, \times\, 5^4}\, &=\, \dfrac{\dfrac{3^2\, \times\, 5^5}{3^2\, \times\, 5^3}\, +\, \dfrac{3^3\, \times\, 5^3}{3^2\, \times\, 5^3}}{\dfrac{3^4\, \times\, 5^4}{3^2\, \times\, 5^3}}

\\ \\ &=\, \dfrac{3^{(2-2)}\, \times\, 5^{(5-3)}\, +\, 3^{(3-2)}\, \times\, 5^0}{3^{(4-2)}\, \times\, 5^{(4-3)}}

\\ \\ &=\, \dfrac{3^0\, \times\, 5^2\, +\, 3^1\, \times\, 5^0}{3^2\, \times\, 5^1}

\\ \\ &=\, \dfrac{1\, \times\, 25\, +\, 3\, \times\, 1}{9\, \times\, 5}\, =\, \dfrac{28}{45} \end{align}\)

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This is from Bird - "Understanding Engineering Mathematics" Page 58. Is there an easy way to find that the HCF of each of these terms is 3²x5³​? I can't see how.

 

[Ishuda]

This is from Bird - "Understanding Engineering Mathematics" Page 58. Is there an easy way to find that the HCF of each of these terms is 3²x5³​? I can't see how.

I'm not sure I know what you are asking but maybe this will help: In looking at the expression
\(\displaystyle \frac{3^2 5^5 + 3^3 5^3}{3^4 5^4}\)
we note that, in the three terms, 2 is the minimum exponent of 3 and 3 is the minimum exponent of 5. Thus the HCF of all three terms is 3²5³. After enough practice, we can do the intermediate steps mentally without writing them down and just write down the final answer.

 

[SoCentral2]

Hi,

Sorry, yes. I didn't explain it very well because I wasn't sure I understood what was going on. After an afternoon of doing more exercises and scratching my head, I can see there are three expressions here all with a 3^x and a 5^x - is that what makes them all candidates for dividing by the HCFs?

Thanks,
Paul.

 

[SoCentral2]

This one's also from Bird "Understanding Engineering Mathematics" :-

I got through the solution to the part circled in red, where my understanding fails me. I would have thought that the fraction I've highlighted in yellow would also need to be divided by 2⁸. Why isn't it? Could you point me towards more resources about this?

Thanks,
Paul.

 

[pka]

This is from Bird - "Understanding Engineering Mathematics" Page 58. Is there an easy way to find that the HCF of each of these terms is 3²x5³​? I can't see how.

\(\displaystyle \begin{cases}\dfrac{3^2\cdot 5^5+3^3\cdot 5^3}{3^4\cdot 5^4}\\\dfrac{ 5^5+3\cdot 5^3}{3^2\cdot 5^4} &:\text{ divide by least power of }3 \\ \dfrac{ 5^2+3}{3^2\cdot 5} &:\text{ divide by least power of }5\\\dfrac{28}{45}\end{cases}\)

 

[Ishuda]

This one's also from Bird "Understanding Engineering Mathematics" :-

I got through the solution to the part circled in red, where my understanding fails me. I would have thought that the fraction I've highlighted in yellow would also need to be divided by 2⁸. Why isn't it? Could you point me towards more resources about this?

Thanks,
Paul.

This is like the 3²5⁵ term in the previous question but like they decided to just do the 3² part of the HCF first.

About resources, you might look around archive.org a bit such as a general search
https://archive.org/search.php?query=description:(Mathematics practice problems)
and a specific book
https://archive.org/stream/ERIC_ED327707#page/n1/mode/2up