Ratio Calculator

Let nonnegative quantities be $q_1,\ldots,q_n$, with at least one positive part, and let $S=\sum_i q_i$. For integer quantities, divide every part by their common positive greatest divisor. For fractional sources, first multiply every part by a common denominator. This changes the scale of the tuple while preserving its relative relationship; it does not add that common denominator to the final quantities.

Here $s_i$ is a part's fraction of the total, $p_i$ is its percentage, and $y_i$ is the exact amount when a fixed total $T$ is divided proportionally. To scale from a known target $t$ for a positive source part $q_j$, use $k=t/q_j$ and multiply every part by $k$. A target for the total uses $k=T/S$ instead. These formulas apply to quantities in compatible units.

1. Reduce the default quantities and identify the denominator

   Start with $12:18$. The common divisor is $\gcd(12,18)=6$, giving $2:3$ and $5$ simplified units. The entered quantities themselves sum to $30$. The first share is $12/30=2/5=40\%$ and the second is $18/30=3/5=60\%$. The direct first-per-second quotient is $12/18=2/3$, which answers a different question from the first share of the total.

2. Anchor a scale or divide a total

   For those entered quantities, a target first amount of $30$ gives $k=30/12=5/2$, so the output is $30:45$ and totals $75$. Alternatively, dividing a fixed total of $100$ in the reduced ratio $2:3$ gives $100(2/5)=40$ and $100(3/5)=60$. The scale factor relative to the original entered amounts is $100/30=10/3$; using the reduced units as if they were the original batch would describe a different factor.

3. Allocate a total that cannot match the ratio exactly

   For $101$ whole items in the ratio $2:3$, the ideals are $40.4$ and $60.6$. Flooring gives $40$ and $60$, leaving one item. The second part has the larger remainder and receives it, producing $40$ and $61$ with total $101$. The achieved counts conserve the total but are not exactly in the original ratio. For three equal parts and two items, equal remainders are resolved in input order, producing $1,1,0$.

4. Solve a missing proportion and verify its domain

   For $2/3=10/x$, the cross-product equation is $2x=30$, so $x=15$. Substitution gives $2/3=10/15$, with both denominators positive. In contrast, $2/3=0/x$ would suggest $x=0$, which is excluded and therefore gives no solution. For $0/3=0/x$, every positive $x$ works. The tool distinguishes this nonunique case from an invalid known denominator and from a uniquely solved value.