Find the fraction.Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 1/3; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 1/10. Find the fraction.My work:Let x / y be the fraction. Thus y + 1 = x 2 . Also, ( x + 2 ) / ( y + 2 ) &gt; 1 / 3 and 0 &lt; ( x − 3 ) / ( y − 3 ) &lt; 1 / 10Substituting the first equality in the second inequality, 0 &lt; ( x − 3 ) / ( x − 2 ) ( x + 2 ) &lt; 1 / 10How do I proceed after this?

Question

Find the fraction.Let us consider a fraction whose denominator is smaller than the square of the numerator by unity. If we add 2 to the numerator and the denominator, the fraction will exceed 1/3; now if we subtract 3 from the numerator and the denominator, the fraction remains positive but smaller than 1/10. Find the fraction.My work:Let   x      /    y be the fraction. Thus   y  +  1  =      x    2  . Also,  (  x  +  2  )      /    (  y  +  2  )  &amp;gt;  1      /    3 and   0  &amp;lt;  (  x  −  3  )      /    (  y  −  3  )  &amp;lt;  1      /    10Substituting the first equality in the second inequality,   0  &amp;lt;  (  x  −  3  )      /    (  x  −  2  )  (  x  +  2  )  &amp;lt;  1      /    10How do I proceed after this?

escampetaq5 · Accepted Answer

HINT From             x      +      2              y      +      2        &amp;gt;      1    3   you get             x      +      2                      x        2            +      1        &amp;gt;      1    3   and from here       x    2    −  3  x  −  5  &amp;lt;  0  ,  x  ∈      N

antennense · Accepted Answer

You have an answer already. To add some detail to it, for   x  ∈      Z              x      +      2                      x        2            +      1        &amp;gt;      1    3      ⟹        x    2    −  3  x  −  5  &amp;lt;  0    ⟹              (      x      −              3        2            )        2    &amp;lt;      29    4      ⟹    x  ⩽  4Similarly,       0    &amp;lt;                  x        −        3                              x          2                −        4              &amp;lt;          1      10            ⟹      Case       x    2    &amp;gt;  4: gives from   0  &amp;lt;                    x        −        3                              x          2                −        4                ⟹    x  &amp;gt;  3Case       x    2    &amp;lt;  4: gives from                              (                      ×                    )                                                                x              −              3                                                      x                2                            −              4                                      &amp;lt;                              1            10                                  ⟹                (        x        −        5                  )          2                &amp;lt;        −        9            So we are left with   x  =  4 as the only possibility.

Find the fraction. Let us consider a fraction whose denominator is smaller than the square of the n

Answered question

Answer & Explanation

New Questions in Pre-Algebra