Find-1-2-1-3-2-16-1-4-2-4-3-4-8-

Question Number 202324 by hardmath last updated on 24/Dec/23

$$\mathrm{Find}:\:\:\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:+\:…\:+\:\mathrm{16}}{\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{2}}{\mathrm{4}}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:+\:…\:+\:\mathrm{8}} \\ $$

Answered by MATHEMATICSAM last updated on 24/Dec/23

$$\frac{\frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:+\:….\:+\:\mathrm{16}}{\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{2}}{\mathrm{4}}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:+\:….\:+\:\mathrm{8}} \\ $$$$=\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\:+\:\frac{\mathrm{2}}{\mathrm{2}}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:+\:\frac{\mathrm{4}}{\mathrm{2}}\:+\:….\:+\:\frac{\mathrm{32}}{\mathrm{2}}}{\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{2}}{\mathrm{4}}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:+\:\frac{\mathrm{4}}{\mathrm{4}}\:+\:….\:+\:\frac{\mathrm{32}}{\mathrm{4}}} \\ $$$$=\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\left[\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:….\:+\:\mathrm{32}\right]}{\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{1}\:+\:\mathrm{2}\:+\:\mathrm{3}\:+\:\mathrm{4}\:+\:….\:+\:\mathrm{32}\right]} \\ $$$$=\:\frac{\mathrm{1}}{\mathrm{2}}\:×\:\mathrm{4} \\ $$$$=\:\mathrm{2}\: \\ $$

Commented by hardmath last updated on 24/Dec/23

$$\mathrm{thankyou}\:\mathrm{dear}\:\mathrm{professor}/ \\ $$

Answered by Rasheed.Sindhi last updated on 25/Dec/23

$$\:\frac{\frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:+\:…\:+\:\mathrm{16}}{\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{2}}{\mathrm{4}}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:+\:…\:+\:\mathrm{8}} \\ $$$${In}\:{order}\:{to}\:{kill}\:{the}\:{denominators} \\ $$$${multiply}\:{numerator}\:\&\:{denominator}\:{by}\:\mathrm{4} \\ $$$$\:=\frac{\mathrm{4}\left(\frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{1}\:+\:\frac{\mathrm{3}}{\mathrm{2}}\:+\:…\:+\:\mathrm{16}\right)}{\mathrm{4}\left(\frac{\mathrm{1}}{\mathrm{4}}\:+\:\frac{\mathrm{2}}{\mathrm{4}}\:+\:\frac{\mathrm{3}}{\mathrm{4}}\:+\:…\:+\:\mathrm{8}\right)} \\ $$$$=\frac{\mathrm{2}\cancel{\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\mathrm{32}\right)}}{\cancel{\left(\mathrm{1}+\mathrm{2}+\mathrm{3}+…+\mathrm{32}\right)}}=\mathrm{2} \\ $$

Commented by MathematicalUser2357 last updated on 26/Dec/23

$$\mathrm{denominators}\:\mathrm{isn}'\mathrm{t}\:\mathrm{humans} \\ $$

Commented by Rasheed.Sindhi last updated on 26/Dec/23

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