Question

Compute (3/4)*(8/9)*(15/16)*(24/25)*(35/36)*(48/49)*(63/64)*(80/81)*(99/100) Express your answer in the simplest way possible. (Suggestion: First, try computing 3/4*8/9 then 3/4*8/9*15/16 and so on. Look for patterns.

Answer 1

`((3)/(4))*((8)/(9))*((15)/(16))*((24)/(25))*((35)/(36))*((48)/(49))*((63)/(64))*((80)/(81))*((99)/(100)) = (11)/(20)`

Explanation:

Given

`((3)/(4))*((8)/(9))*((15)/(16))*((24)/(25))*((35)/(36))*((48)/(49))*((63)/(64))*((80)/(81))*((99)/(100))`

Required

Simplify

For clarity, group the expression in threes

`(((3)/(4))*((8)/(9))*((15)/(16)))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

Evaluate the first group [Divide 8 by 4]

`(((3)/(1))*((2)/(9))*((15)/(16)))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

[Divide 9 by 3]

`(((1)/(1))*((2)/(3))*((15)/(16)))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

`(((2)/(3))*((15)/(16)))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

[Divide 15 by 3]

`(((2)/(1))*((5)/(16)))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

[Divide 16 by 2]

`(((1)/(1))*((5)/(8)))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

`((5)/(8))*(((24)/(25))*((35)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

Evaluate the second group [Divide 35 and 25 by 5]

`((5)/(8))*(((24)/(5))*((7)/(36))*((48)/(49)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

[Divide 49 by 7]

`((5)/(8))*(((24)/(5))*((1)/(3))*((4)/(7)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

[Divide 24 by 3]

`((5)/(8))*(((8)/(5))*((1)/(1))*((4)/(7)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

`((5)/(8))*(((8)/(5))*((4)/(7)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

Merge the first and second group

`(((5)/(8))*((8)/(5))*((4)/(7)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

`(1*((4)/(7)))*(((63)/(64))*((80)/(81))*((99)/(100)))`

`((4)/(7))*(((63)/(64))*((80)/(81))*((99)/(100)))`

Evaluate the last group [Divide 99 by 9]

`((4)/(7))*(((63)/(64))*((80)/(9))*((11)/(100)))`

[Divide 63 by 9]

`((4)/(7))*(((7)/(64))*((80)/(1))*((11)/(100)))`

[Divide 64 and 80 by 8]

`((4)/(7))*(((7)/(8))*((10)/(1))*((11)/(100)))`

[Divide 10 and 4 by 2]

`((4)/(7))*(((7)/(4))*((5)/(1))*((11)/(100)))`

[Divide 100 by 5]

`((4)/(7))*(((7)/(4))*((1)/(1))*((11)/(20)))`

`((4)/(7))*(((7)/(4))*((11)/(20)))`

`((4)/(7))*((7)/(4))*((11)/(20))`

`1*((11)/(20))`

`(11)/(20)`

Hence;

`((3)/(4))*((8)/(9))*((15)/(16))*((24)/(25))*((35)/(36))*((48)/(49))*((63)/(64))*((80)/(81))*((99)/(100)) = (11)/(20)`