Question

Sa se afle numerele intregi a b c d pentru care a/b=2/3,b/c=3/4,c/d=4/5 si abcd =120

Answer 1

$\frac{a}{b} = \frac{2}{3}$

$a = \frac{2b}{3}$

$\frac{b}{c} = \frac{3}{4}$

$c = \frac{4b}{3}$

$\frac{c}{d} = \frac{4}{5}$

$d = \frac{5c}{4}$

$d = \frac{5 \times \frac{4b}{3} }{4}$

$d = \frac{ \frac{5 \times 4b}{3} }{4}$

$d = \frac{ \frac{20b}{3} }{4}$

$d = \frac{20b}{3} \times \frac{1}{4}$

$d = \frac{20b \times 1}{3 \times 4}$

$d = \frac{20b}{12}$

$d = \frac{5b}{3}$

$abcd = 120$

$\frac{2b}{3} \times b \times \frac{4b}{3} \times \frac{5b}{3} = 120$

$\frac{2b \times b \times 4b \times 5b}{3 \times 3 \times 3} = 120$

$\frac{40 {b}^{4} }{27} = 120$

$40 {b}^{4} = 27 \times 120$

$40 {b}^{4} = 3240$

${b}^{4} = \frac{3240}{40}$

${b}^{4} = 81$

$b = 3$

$a = \frac{2b}{3}$

$a = \frac{2 \times 3}{3}$

$a=\frac{6}{3}$

$a=2$

$c = \frac{4b}{3}$

$c = \frac{4 \times 3 }{3}$

$c=\frac{12}{3}$

$c=4$

$d = \frac{5b}{3}$

$d = \frac{5 \times 3}{3}$

$d=\frac{15}{3}$

$d=5$