| Simplify \[\left( 1+\frac{1}{4} \right)\left( 1+\frac{1}{4} \right)\left( 1+\frac{1}{5} \right)\left( 1-\frac{1}{5} \right)\]\[\left( 1+\frac{1}{6} \right)\left( 1-\frac{1}{6} \right).\] |
A) \[\frac{3}{8}\]
B) \[\frac{5}{8}\]
C) \[\frac{7}{8}\]
D) \[\frac{1}{8}\]
Correct Answer: C
Solution :
| \[\left( 1+\frac{1}{4} \right)\left( 1-\frac{1}{4} \right)\left( 1+\frac{1}{5} \right)\left( 1-\frac{1}{5} \right)\left( 1+\frac{1}{6} \right)\left( 1-\frac{1}{6} \right)\] |
| \[=\left( \frac{4+1}{5} \right)\left( \frac{4-1}{5} \right)\left( \frac{5+1}{5} \right)\left( \frac{5-1}{5} \right)\left( \frac{6+1}{6} \right)\left( \frac{6-1}{6} \right)\] |
| \[=\left( \frac{5}{4} \right)\left( \frac{3}{4} \right)\left( \frac{6}{5} \right)\left( \frac{4}{5} \right)\left( \frac{7}{6} \right)\left( \frac{5}{6} \right)\] |
| \[=\frac{5}{4}\times \frac{3}{4}\times \frac{6}{5}\times \frac{4}{5}\times \frac{7}{6}\times \frac{5}{6}=\frac{3\times 7}{4\times 6}=\frac{7}{8}\] |
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