nick1816 wrote:
\(X= \frac{7^3 -1}{49^3-7^3 +1}\)
\(Y = \frac{7^3 +1}{49^3+7^3 +1}\)
\(Z = \frac{7^3 +1}{49^3-7^3 +1}\)
Which of the following is the correct order?
A. X>Z>Y
B. X>Y>Z
C. Z>Y>X
D. Z>X>Y
E. Y>X>Z
We can see that any two in the given three fractions are similar to one another either in numerator or in denominator.
Rules
1. If denominator is same, then the fraction with higher numerator will be higher. \(X= \frac{7^3 -1}{49^3-7^3 +1}\)
\(Z = \frac{7^3 +1}{49^3-7^3 +1}\)
Denominators are the same but \(7^3+1>7^3-1\), so Z>X
1. If numerator is same, then the fraction with lower denominator will be higher. \(Y= \frac{7^3 +1}{49^3+7^3 +1}\)
\(Z = \frac{7^3 +1}{49^3-7^3 +1}\)
Denominators are the same but \(49^3+7^3+1>49^3-7^3+1\), so Z>Y
So Z is the greatest and we have to check for relation in X and Y
Multiply both the terms with product of both denominators to remove fraction.
1) \(X *(49^3-7^3 +1)* (49^3+7^3 +1 )= \frac{7^3 -1}{49^3-7^3 +1}*(49^3-7^3 +1)* (49^3+7^3 +1)\)
\(X *(49^3-7^3 +1)* (49^3+7^3 +1 )= (7^3 -1)* (49^3+7^3 +1)=7^3*49^3+49^3+7^3-49^3-7^3-1=7^3*49^3-1\)
2) \(Y *(49^3-7^3 +1)* (49^3+7^3 +1 )= \frac{7^3 +1}{49^3+7^3 +1} *(49^3-7^3 +1)* (49^3+7^3 +1)\)
\(Y *(49^3-7^3 +1)* (49^3+7^3 +1 )= (7^3+1)* (49^3-7^3 +1)=7^3*49^3-49^3+7^3+49^3-7^3+1=7^3*49^3+1\)
Y>X
So Z>Y>X
C