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If a and b are positive numbers and \(\frac{100}{a} = \frac{95}{b} \),

prove that:

(i)  \(a >b\)

(ii) \(\frac{105}{a} < \frac{100}{b}\)

Guest Jan 10, 2016

Best Answer 

 #2
avatar+93882 
+15

If a and b are positive numbers and 100/a =95/b,

prove that:

(i)  a > b

 

\(\frac{100}{a}=\frac{95}{b}\\ 100b=95a\\ 95a=100b\\ a=\frac{100b}{95}\\ a=\frac{95b}{95}+\frac{5b}{95}\\ a=b+more\\ a>b \)

   

 

 

i)  prove \(\frac{105}{a}<\frac{100}{b}\)

 

\(\frac{100}{a}=\frac{95}{b}\\ \frac{100}{a}\times 1.05=\frac{95}{b}\times 1.05\\ \frac{105}{a}=\frac{99.75}{b}\\ \frac{105}{a}=\frac{100}{b}-\frac{0.25}{b}\\ \frac{105}{a}=\frac{100}{b}-\;\;a\; bit\\ so\\ \frac{105}{a}<\frac{100}{b} \)

Melody  Jan 10, 2016
 #1
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0

If a and b are positive numbers and 100/a =95/b,

prove that:

(i)  a > b

(ii) 105/a <100/b

Guest Jan 10, 2016
 #2
avatar+93882 
+15
Best Answer

If a and b are positive numbers and 100/a =95/b,

prove that:

(i)  a > b

 

\(\frac{100}{a}=\frac{95}{b}\\ 100b=95a\\ 95a=100b\\ a=\frac{100b}{95}\\ a=\frac{95b}{95}+\frac{5b}{95}\\ a=b+more\\ a>b \)

   

 

 

i)  prove \(\frac{105}{a}<\frac{100}{b}\)

 

\(\frac{100}{a}=\frac{95}{b}\\ \frac{100}{a}\times 1.05=\frac{95}{b}\times 1.05\\ \frac{105}{a}=\frac{99.75}{b}\\ \frac{105}{a}=\frac{100}{b}-\frac{0.25}{b}\\ \frac{105}{a}=\frac{100}{b}-\;\;a\; bit\\ so\\ \frac{105}{a}<\frac{100}{b} \)

Melody  Jan 10, 2016
 #3
avatar+91160 
+15

If a and b are positive numbers and 100/a =95/b,

prove that:

(i)  a > b

(ii) 105/a <100/b

 

(i)  if 100/a = 95/b, this implies that 95a = 100b, which implies that a = (100/95)b......Then a must be greater than b since we have to multiply b by a quantity > 1 to get a

 

(ii)  105/a < 100/b..... Cross-multiplying........

 

105b < 100a........but, by definition, a =(100/95)b....so....

 

105b < 100(100/95)b  .......divide both sibes by 100 →

 

(105/100)b < (100/95)b ........reduce the fractions →

 

(21/20)b < (20/19)b ........divide both sides by b  →

 

(21/20) < (20/19)  →    cross-multiply, again

 

19*21 < 20*20  →

 

399 < 400......and since the left side is less than the right side......then the left side of the original inequality is also less than the right

 

 

cool cool cool

CPhill  Jan 10, 2016

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