Divide 56 into four parts which …

Let the four numbers be {tex}a - 3d, a - d, a + d, a + 3d{/tex}
{tex}\therefore{/tex} {tex}a - 3d + a - d + a + d + a + 3d {/tex}= 56
 or, {tex}4a = 56{/tex}
or, {tex}a = 14{/tex}
Hence numbers are {tex}14 - 3d, 14 - d, 14 + d, 14 + 3d{/tex}
Now, according to question,
{tex}\frac { ( 14 - 3 d ) ( 14 + 3 d ) } { ( 14 - d ) ( 14 + d ) } = \frac { 5 } { 6 }{/tex}
or, {tex}\frac { 196 - 9 d ^ { 2 } } { 196 - d ^ { 2 } } = \frac { 5 } { 6 }{/tex}
or, {tex}6(196 - 9d^2) = 5(196 - d^2){/tex}
or, {tex}6 \times 196 - 54 d ^ { 2 } = 5 \times 196 - 5 d ^ { 2 }{/tex}
or, {tex}6 \times 196 - 5 \times 196 = 54 d ^ { 2 } - 5 d ^ { 2 }{/tex}
or, {tex}( 6 - 5 ) \times 196 = 49 d ^ { 2 }{/tex}
or, {tex} 196 = 49 d ^ { 2 }{/tex}
or, {tex}d ^ { 2 } = \frac { 196 } { 49 } = 4{/tex}
or, {tex}d = \pm 2{/tex}
Therefore, Numbers are
{tex}14 - 3 \times 2 , \quad 14 - 2 , \quad 14 + 2 , \quad 14 + 3 \times 2{/tex}
{tex}14 - 6, 12, 16, 14 + 6{/tex}
{tex}8, 12, 16, 20{/tex}