Algebra-easy mode

Number Theory Level 1

If 1 p + 1 q + 1 r = 1 \frac{1}{p}+\frac{1}{q}+\frac{1}{r}=1 and pq = s then p+q = ?

s r ( r − 1 ) \frac{s}{r}(r-1) s r ( s − r ) \frac{s}{r}(s-r) s r + 1 \frac{s}{r}+1 s r ( q − s ) \frac{s}{r}(q-s)

Sudoku Subbu by sudoku subbu , 19, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Alex Delhumeau
May 30, 2015

1 p + 1 q = p + q p q & 1 − 1 r = r − 1 r \large{\frac{1}{p}+\frac{1}{q}}=\large{\frac{p+q}{pq}} \text{ \& } \large{1-\frac{1}{r}}=\large{\frac{r-1}{r}} .

⟹ p + q s = r − 1 r , so p + q = s r ( r − 1 ) \implies \large{\frac{p+q}{s}} = \large{\frac{r-1}{r}}, \text{ so } \large{p+q}=\large{\frac{s}{r}(r-1)} .

0 Helpful 0 Interesting 0 Brilliant 0 Confused

1/p + 1/q + 1/r = 1 => (p+q)/pq + 1/r = 1 => (p+q)/pq = 1 - 1/r = (r-1)/r => (p+q)/s = (r-1)/r => p+q = s(r-1)/r

0 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...