Given that x/y = 0.99, express x in terms of y and y in terms of x.

Step-by-step solution

1. 1

   Clear the denominator to get $x$. Multiply both sides by $y$ (allowed since $y\neq0$, or the original ratio would be undefined):

   $x=0.99y$

   Read plainly: $x$ is $99\%$ of $y$, so $x$ is the smaller of the two.

2. 2

   Solve for $y$ by dividing. From $x=0.99y$, divide both sides by $0.99$:

   $y=\frac{x}{0.99}$

3. 3

   Convert the decimal to an exact fraction. Since $0.99=\tfrac{99}{100}$, dividing by it means multiplying by its reciprocal $\tfrac{100}{99}$:

   $y=\frac{100}{99}x$

   Working in fractions avoids the rounding that $1/0.99$ invites, and $\tfrac{100}{99}=1.\overline{01}\approx1.0101$ ✓.

4. 4

   Check the two forms are consistent. Substituting $y=\tfrac{100}{99}x$ into $x=0.99y$ gives $x=\tfrac{99}{100}\cdot\tfrac{100}{99}x=x$ ✓ — the two coefficients are exact reciprocals, as they must be. Confirmed in exact rational arithmetic ✓.

5. 5

   Interpret the asymmetry. A $1\%$ decrease from $y$ to $x$ requires slightly more than a $1\%$ increase to reverse: going back up needs a factor $\tfrac{100}{99}\approx1.0101$, i.e. about $1.01\%$. Percentages are not symmetric because the base changes — a point worth remembering whenever a discount is undone.