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

Answer:

$(\frac{3}{4})*(\frac{8}{9})*(\frac{15}{16})*(\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49})*(\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}) = \frac{11}{20}$

Step-by-step explanation:

Given

$(\frac{3}{4})*(\frac{8}{9})*(\frac{15}{16})*(\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49})*(\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100})$

Required

Simplify

For clarity, group the expression in threes

$((\frac{3}{4})*(\frac{8}{9})*(\frac{15}{16}))*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

Evaluate the first group [Divide 8 by 4]

$((\frac{3}{1})*(\frac{2}{9})*(\frac{15}{16}))*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

[Divide 9 by 3]

$((\frac{1}{1})*(\frac{2}{3})*(\frac{15}{16}))*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

$((\frac{2}{3})*(\frac{15}{16}))*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

[Divide 15 by 3]

$((\frac{2}{1})*(\frac{5}{16}))*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

[Divide 16 by 2]

$((\frac{1}{1})*(\frac{5}{8}))*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

$(\frac{5}{8})*((\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

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

$(\frac{5}{8})*((\frac{24}{5})*(\frac{7}{36})*(\frac{48}{49}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

[Divide 49 by 7]

$(\frac{5}{8})*((\frac{24}{5})*(\frac{1}{3})*(\frac{4}{7}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

[Divide 24 by 3]

$(\frac{5}{8})*((\frac{8}{5})*(\frac{1}{1})*(\frac{4}{7}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

$(\frac{5}{8})*((\frac{8}{5})*(\frac{4}{7}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

Merge the first and second group

$((\frac{5}{8})*(\frac{8}{5})*(\frac{4}{7}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

$(1*(\frac{4}{7}))*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

$(\frac{4}{7})*((\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}))$

Evaluate the last group [Divide 99 by 9]

$(\frac{4}{7})*((\frac{63}{64})*(\frac{80}{9})*(\frac{11}{100}))$

[Divide 63 by 9]

$(\frac{4}{7})*((\frac{7}{64})*(\frac{80}{1})*(\frac{11}{100}))$

[Divide 64 and 80 by 8]

$(\frac{4}{7})*((\frac{7}{8})*(\frac{10}{1})*(\frac{11}{100}))$

[Divide 10 and 4 by 2]

$(\frac{4}{7})*((\frac{7}{4})*(\frac{5}{1})*(\frac{11}{100}))$

[Divide 100 by 5]

$(\frac{4}{7})*((\frac{7}{4})*(\frac{1}{1})*(\frac{11}{20}))$

$(\frac{4}{7})*((\frac{7}{4})*(\frac{11}{20}))$

$(\frac{4}{7})*(\frac{7}{4})*(\frac{11}{20})$

$1*(\frac{11}{20})$

$\frac{11}{20}$

Hence;

$(\frac{3}{4})*(\frac{8}{9})*(\frac{15}{16})*(\frac{24}{25})*(\frac{35}{36})*(\frac{48}{49})*(\frac{63}{64})*(\frac{80}{81})*(\frac{99}{100}) = \frac{11}{20}$