Math utility Tool / 02
Ratio and Proportion Calculator
Simplify ratios, test equivalent ratios, and divide a total into two or three proportional shares.
a : b : c
How to Use the Ratio and Proportion Calculator
Use this ratio and proportion calculator by choosing the operation that matches the question before entering any numbers. Simplify a ratio reduces two or three terms to the smallest equivalent whole-number ratio. Check equivalent ratios compares two ordered ratios. Divide a total by a ratio turns ratio units into exact shares that add back to the amount entered.
Enter the first two terms in their original order. The third term is optional;
use it for a ratio such as 2 : 3 : 5. Whole numbers, terminating decimals,
fractions, and mixed numbers are accepted. The result keeps exact fractional
shares where a terminating decimal would lose information.
This page is for ratio structure and allocation. To find one missing value in
a / b = c / d, use the proportion calculator, which leaves any one of
the four positions blank and shows the cross-multiplication work.
Ratio vs. Proportion
A ratio is an ordered comparison of quantities. The ratio 2 : 3 means
that the first quantity has two units for every three units of the second.
Changing the order changes the comparison: 2 : 3 and 3 : 2 are not the
same ratio.
A proportion is an equation stating that two ratios are equal. For example,
2 : 3 = 8 : 12 is true because both ratios simplify to 2 : 3. This
calculator can verify that equivalence, but the homepage solver is the focused
tool for an unknown term in a proportion equation.
Ratios may compare parts to parts, such as red paint to blue paint, or a part to a whole. Keep labels and units aligned when interpreting a result. A numeric ratio alone does not say which quantity each position represents.
Simplify Two-Part and Three-Part Ratios
To simplify a ratio containing whole numbers, divide every term by their
greatest common factor. For 12 : 18, the greatest common factor is 6, so:
12 : 18 = 2 : 3
The same operation applies to a three-part ratio. The terms in
12 : 18 : 30 share a factor of 6, producing 2 : 3 : 5. All terms must be
divided by the same value; simplifying only one pair changes the relationship.
Decimal and fraction ratios need one additional step. First multiply all terms
by a common denominator so they become whole numbers. For
1/2 : 3/4, multiplying both terms by 4 gives 2 : 3. The calculator performs
that conversion with exact rational arithmetic before removing the common
factor.
Zero can be a valid term when at least one other term is positive. An all-zero ratio has no usable scale and is rejected. Negative terms are not accepted because their meaning depends on a specialized context rather than ordinary ratio allocation.
Check Whether Ratios Are Equivalent
Two ratios are equivalent when one can be obtained by multiplying every term of the other by the same nonzero scale factor. Reducing both ratios to a common normal form gives a direct check:
4 : 6 = 10 : 15
Both ratios simplify to 2 : 3.
Order and term count both matter. A two-part ratio cannot be compared directly
with a three-part ratio, and 2 : 3 is not equivalent to 3 : 2. When zero is
present, it must appear in the corresponding position in both ratios.
The comparison result shows each simplified ratio separately, making a failed check auditable instead of returning only “yes” or “no.”
Divide a Total in a Ratio
To split a total in the ratio a : b : c, first add the ratio terms. Call that
sum the number of ratio units:
units = a + b + c
Each share is its ratio term divided by the units, multiplied by the total:
share = total × term / units
For example, divide 120 in the ratio 2 : 3 : 5. The ratio contains
2 + 3 + 5 = 10 units, so one unit is 120 / 10 = 12. The three shares are
2 × 12 = 24, 3 × 12 = 36, and 5 × 12 = 60. Their sum is 120.
This method works even when the total or terms are fractions. Exact results are useful for measurements, recipes, or allocations that will be rounded only at the final practical step. If a currency split produces fractions of a cent, the calculator reports the mathematical shares; deciding how to distribute the rounding remainder is a separate policy choice.
Worked Ratio Examples
Simplify a Ratio with Decimals
Simplify 1.5 : 2.25. Multiplying by 100 gives 150 : 225; dividing both
terms by 75 gives 2 : 3. Clearing fewer decimal places first, such as
6 : 9, leads to the same simplest ratio.
Check a Three-Part Equivalent Ratio
Compare 2 : 3 : 5 with 8 : 12 : 20. The second ratio has every original
term multiplied by 4. Both simplify to 2 : 3 : 5, so they are equivalent.
Allocate a Total Between Two Parts
Divide 350 in the ratio 3 : 2. There are five ratio units. The first share is
350 × 3/5 = 210; the second is 350 × 2/5 = 140. The order of the answers
follows the order of the entered terms.
Supported Ratio Inputs and Limits
The calculator accepts integers such as 12, decimals such as 2.5, fractions
such as 3/4, and mixed numbers such as 1 1/2. Fraction denominators cannot
be zero. Enter two required terms and use the third field only when the problem
actually contains three corresponding quantities.
The tool does not infer units, reorder terms, convert physical measurements, or choose a rounding policy for money. Put corresponding quantities in the same position and convert unlike units before comparing them. For an allocation, the total and ratio terms must be zero or positive and the ratio cannot consist entirely of zeros.
Ratio and Proportion Calculator FAQ
Can I Simplify a Three-Part Ratio?
Yes. Enter all three terms in order. The calculator clears fraction denominators if necessary and divides every term by the same greatest common factor.
How Do I Know if Two Ratios Are Equivalent?
Choose the equivalence mode and enter both ratios. They are equivalent only if their ordered simplest whole-number forms match exactly.
Can I Split a Decimal or Fractional Total?
Yes. Shares are calculated as exact fractions first, then displayed with a decimal when useful. The exact shares always preserve the entered total.
Is a Ratio the Same as a Fraction?
They use related division notation, but context differs. a : b compares the
first quantity with the second; the fraction a/b represents their quotient.
For a part-to-whole fraction, the whole may instead be a + b.
Why Is the Order of a Ratio Important?
Each position refers to a specific quantity. Reversing 2 : 5 to 5 : 2
reverses which quantity receives two units and which receives five.
Ratio Calculation Method and Source
All simplification and allocation operations use exact rational arithmetic. The underlying definitions follow the treatment of ratios, rates, and proportions in OpenStax Prealgebra 2e, section 6.5. Results still depend on entering quantities in the correct order and using compatible units.