Ratio Calculator – Simplify and Scale Ratios
Solve a proportion A : B = C : D for any missing term, simplify a ratio with 2–4 parts to its lowest whole-number form, or scale a ratio to a given factor, part value, or total. Enter your values and the result appears instantly.
Solve a Proportion or Simplify and Scale a Ratio
Ratio Formula
Where:
- \(A, B, C, D\) — the four terms of the proportion \(A : B = C : D\). Two ratios form a true proportion exactly when their cross products are equal.
- \(a_1, a_2, \dots, a_n\) — the parts of a ratio (2 to 4 numbers)
- \(g\) — the greatest common divisor of all parts, used to reduce a ratio to its simplest form
- \(k\) — the scale factor; multiplying every part by the same \(k > 0\) keeps the ratio equivalent
The first line is the cross-product rule for solving a proportion. The last line shows simplification: dividing every part by the gcd gives the smallest whole-number ratio with the same proportions, while \(k \cdot a_1 : k \cdot a_2 : \dots\) for any \(k > 0\) is an equivalent ratio.
Ratio – Calculation Example
Example: a recipe uses flour and sugar in the ratio 3 : 4. You have 3 cups of flour and 6 cups of sugar left — how much sugar matches the 3 : 4 ratio? You want to solve \(3 : 4 = 6 : D\) for the missing term \(D\).
Leave \(D\) empty, enter \(A = 3\), \(B = 4\), \(C = 6\), and apply the cross-product rule:
\[ 3 \times D = 4 \times 6 \quad\Rightarrow\quad D = \frac{B \times C}{A} = \frac{4 \times 6}{3} = 8 \]
Check: \(A \times D = 3 \times 8 = 24\) and \(B \times C = 4 \times 6 = 24\) ✓
Result: \(D = 8\), so the complete proportion is \(3 : 4 = 6 : 8\). With 6 cups of sugar you would need 8 cups of flour to keep the 3 : 4 ratio.
How to use the calculator
- Open the “What do you want to do?” dropdown and pick a mode: Solve A : B = C : D (find the missing value), Simplify a ratio, or Scale a ratio. Only the fields of the active mode are shown.
- To solve a proportion: fill in First term (A), Second term (B), Third term (C) and Fourth term (D). Leave exactly one field empty — it will be found by cross-multiplication. If you fill all four, the calculator checks whether the proportion is true.
- To simplify: type 2–4 numbers into Ratio parts (2–4 numbers), separated by colons, spaces, commas or semicolons (e.g.
18 : 24 : 30). Decimals such as0.5 : 0.75are allowed. - To scale: enter the parts, choose a Scaling method — multiply/divide by a Factor, force one part to a Target value (select the part in Part that equals the target), or make the total equal a target.
- Read the Answer, the table of each part as a fraction and percentage of the total, and the step-by-step Steps section.
What a ratio is and how to read the result
A ratio compares two or more quantities of the same kind. The ratio \(3 : 4\) says that for every 3 units of the first quantity there are 4 units of the second. Ratios do not fix the actual amounts — 3 : 4, 6 : 8 and 30 : 40 all describe the same relationship, because each is the original multiplied by some positive factor.
The proportion \(A : B = C : D\) states that two ratios are equivalent. The cross-product rule \(A \times D = B \times C\) gives a quick test for equivalence and lets you find any missing term: multiply the two values you have that are opposite the unknown, and divide by the remaining one. This is the workhorse behind unit conversions, recipe resizing, map scales, and similar-triangle problems.
A ratio is in simplest form when all parts are whole numbers with no common divisor larger than 1 — so \(18 : 24 : 30\) reduces to \(3 : 4 : 5\). The calculator also shows each part as a fraction and a percentage of the total, which is often the most useful interpretation: in \(3 : 5\), the parts hold \(3/8\) (37.5%) and \(5/8\) (62.5%) of the whole.
When you scale a ratio, every part is multiplied by the same factor \(k\). “Make the total equal 40” for \(3 : 5\) means \(k = 40 / 8 = 5\), giving \(15 : 25\), which still totals 40 and preserves the exact same shares. This is also how you split money, ingredients, or work time among people proportionally.
Assumptions, limitations and tips
- Only 2 to 4 parts are supported per ratio. Parts must not be negative, and at least one part must be greater than zero.
- For solving a proportion, leave exactly one of A, B, C, D empty; the value across from the unknown in the cross product must not be zero, otherwise no unique solution exists.
- Decimal parts are converted to whole numbers by multiplying by a power of ten. Keep decimals to at most 8 decimal places, or the calculator cannot finish the conversion.
- Simplified results are rounded only for display (4 significant figures); the underlying values are exact whenever the arithmetic allows.
- To reduce a ratio by hand, the GCF calculator finds the greatest common divisor you need to divide out. When scaling up to equal totals, the LCM calculator helps find the smallest common total.
- Proportions describe direct variation — both quantities grow together at a constant rate. For the full equation behind it, see the direct variation calculator.
Frequently Asked Questions
How do I simplify a ratio by hand?
Divide every part of the ratio by the greatest common divisor (GCD) of all the parts. For example, with 18 : 24 : 30 the GCD of 18, 24 and 30 is 6, so dividing each part by 6 gives 3 : 4 : 5. If the parts contain decimals, first multiply everything by a power of ten to make whole numbers, then divide by the GCD.
Is the order of the numbers in a ratio important?
Yes, order matters. The ratio 3 : 5 says the first quantity is smaller, while 5 : 3 reverses the relationship — they are not the same ratio. Always keep the parts in the order in which the corresponding quantities are named. For example, "flour to sugar is 3 : 4" means 3 units of flour for every 4 units of sugar, not the other way around.
Can I simplify a ratio with decimals or fractions?
Yes. First convert everything to whole numbers by multiplying each part by the same power of ten or common denominator. For example, 0.5 : 0.75 becomes 50 : 75 after multiplying by 100, which simplifies to 2 : 3. Similarly, 1/2 : 1/3 becomes 3 : 2 after multiplying both parts by 6. The calculator handles this conversion automatically for decimals.
What is the difference between a ratio and a proportion?
A ratio compares two or more quantities, like 3 : 4. A proportion is a statement that two ratios are equal, like \(3 : 4 = 6 : 8\). You use a ratio to describe a relationship and a proportion to solve for an unknown value when the relationship must stay the same, for instance when scaling a recipe or solving for a distance on a map.
How do I split an amount in a given ratio?
Add up the parts of the ratio to find the total number of shares, then divide the amount by that total to get the value of one share, and multiply by each part. For example, splitting $120 in the ratio 2 : 3 : 5 gives 10 shares, so one share is $12 and the amounts are $24, $36 and $60. The "Make the total equal a value" scaling mode does exactly this.
What do ratio results like "1 : x" or percentages of the total mean?
A ratio expressed as 1 : x shows how many units of the second quantity correspond to one unit of the first; 3 : 4 becomes 1 : 1.333. The percentages show each part's share of the whole: in 3 : 5 the total is 8 parts, so the shares are 37.5% and 62.5%. This interpretation is useful when comparing shares of a total rather than relative size.
Do ratios have units? What mistakes should I avoid?
Ratios themselves are unitless, but the quantities you compare must be expressed in the same unit — compare 20 cm to 1 m by converting one of them first. A common mistake is adding to one part instead of multiplying every part by the same factor, which breaks the ratio. Also avoid mixing up the order of terms and mixing up scaling a ratio with scaling each quantity independently.
What happens if one part of a ratio is zero?
A zero part is mathematically valid in a comparison, such as 0 : 5, but it behaves oddly: it can never be scaled to a positive target, and it contributes 0% to the total. If the number opposite an unknown in a proportion is zero, the missing value has no unique solution, which is why the calculator requires at least one positive part and warns when division by zero would occur.