$$The sum of the reciprocals of three consecutive integers is 47/60. What is the sum of the three integers?$$

The sum of the reciprocals of three consecutive integers is 47/60. What is the sum of the three integers?

Mellie
Jun 22, 2015

#1**+15 **

This must be of the form a/60 + b/60 + c/60 = 47/60

But since it must also be of the form 1/p + 1/q + 1/r = 47/60 then a, b and c must be factors of 60 (so that each term on the LHS cancels to leave a 1 in the numerator).

$${factor}{\left({\mathtt{60}}\right)} = {{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}$$

Clearly the 3 consecutive numbers are 3, 4 and 5, so their sum is 12.

Check

$${\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{5}}}} = {\frac{{\mathtt{47}}}{{\mathtt{60}}}} = {\mathtt{0.783\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

.

Alan
Jun 22, 2015

#1**+15 **

Best Answer

This must be of the form a/60 + b/60 + c/60 = 47/60

But since it must also be of the form 1/p + 1/q + 1/r = 47/60 then a, b and c must be factors of 60 (so that each term on the LHS cancels to leave a 1 in the numerator).

$${factor}{\left({\mathtt{60}}\right)} = {{\mathtt{2}}}^{{\mathtt{2}}}{\mathtt{\,\times\,}}{\mathtt{3}}{\mathtt{\,\times\,}}{\mathtt{5}}$$

Clearly the 3 consecutive numbers are 3, 4 and 5, so their sum is 12.

Check

$${\frac{{\mathtt{1}}}{{\mathtt{3}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{4}}}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{1}}}{{\mathtt{5}}}} = {\frac{{\mathtt{47}}}{{\mathtt{60}}}} = {\mathtt{0.783\: \!333\: \!333\: \!333\: \!333\: \!3}}$$

.

Alan
Jun 22, 2015