Question Number 1335 by 123456 last updated on 23/Jul/15
$$\mathrm{if}\:{a}\leqslant{x}\leqslant{b}\:\mathrm{and}\:{c}\leqslant{y}\leqslant{d}\:\mathrm{then}\:\mathrm{did} \\ $$ $$\frac{{a}}{{d}}\leqslant\frac{{x}}{{y}}\leqslant\frac{{b}}{{c}}? \\ $$ $${a},{b},{c},{d},{x},{y}\in\mathbb{R} \\ $$ $$\mathrm{0}<{c} \\ $$ $$\mathrm{0}<{a} \\ $$
Answered by prakash jain last updated on 23/Jul/15
$${a}\leqslant{x} \\ $$ $${ay}\leqslant{xy}\:\:\:\:\:\because{y}>\mathrm{0} \\ $$ $${ay}\leqslant{xd}\:\:\:\:\:\because{d}\geqslant{y} \\ $$ $$\frac{{a}}{{d}}\leqslant\frac{{x}}{{y}}\:\:\:\:\:\:\because{dy}>\mathrm{0} \\ $$ $${x}\leqslant{b} \\ $$ $${xc}\leqslant{bc}\:\:\:\:\:\:\:\because{c}>\mathrm{0} \\ $$ $${xc}\leqslant{by}\:\:\:\:\:\:\:\:\because{y}\geqslant{c} \\ $$ $$\frac{{x}}{{y}}\leqslant\frac{{b}}{{c}}\:\:\:\:\:\:\:\:\because{yc}>\mathrm{0} \\ $$