Simplify 12 : 18
- I want to
- Simplify a ratio
- Ratio
- 12 : 18
- Result
- 2 : 3
- Scale factor
- 6
- First term ÷ second term
- 0.666667
Checked against: gcd(12, 18) = 6
Ratio and proportion calculator
Simplify a ratio, solve a proportion a : b = c : x for the missing term, scale a ratio to a target, or split an amount in a given ratio, exactly.
By AdminDrafted with AI writing toolsUpdated Checked examples: 8
Result
2 : 3
Result: 2 : 3
- Scale factor
- 6
- First term ÷ second term
- 0.666667
12 : 18 simplifies to 2 : 3; each original term is 6 times the matching term of the simple form.
Parts of the whole
Term 12Term 23
How it's calculated S
Greatest common divisor
Divide every term by it
About the ratio and proportion calculator
A ratio compares quantities by division, so 12 : 18 and 2 : 3 describe the same proportion. Simplifying divides every term by their greatest common divisor, after multiplying through by the least common multiple of any denominators so that decimal and fraction terms become whole numbers. A proportion a : b = c : d holds exactly when a × d = b × c, which is how cross-multiplication finds a missing term; splitting an amount divides it by the sum of the terms to get the size of one share.
Recipes, paint and concrete mixes, map scales and dividing money between partners are the usual uses. Split 1,000 in the ratio 2 : 3 : 5 and the 10 shares are worth 100 each, giving 200, 300 and 500; solve 3 : 4 = 9 : x and x = 4 × 9 ÷ 3 = 12.
Answers stay exact: shares that do not divide evenly appear as fractions such as 100/3 rather than rounded decimals. Terms must be zero or positive.
How to simplify a ratio
Divide every term by the greatest common divisor of the terms. In 24 : 36 : 60 the GCD is 12, so the ratio simplifies to 2 : 3 : 5. Clear decimals or fractions first by multiplying every term by the same number: 1.5 : 2.5, doubled, is 3 : 5.
| Ratio | Simplest whole-number ratio |
|---|---|
| 12 : 18 | 2 : 3 |
| 1.5 : 2.5 | 3 : 5 |
| 1/2 : 3/4 : 5/6 | 6 : 9 : 10 |
| 0 : 6 | 0 : 1 |
| 24 : 36 : 60 | 2 : 3 : 5 |
For 1/2 : 3/4 : 5/6, multiplying by the least common denominator, 12, gives 6 : 9 : 10. No whole number above 1 divides all three terms, so that is already the simplest form. A zero term stays zero, which is why 0 : 6 becomes 0 : 1.
How to solve a proportion
A proportion says two ratios are equal, a : b = c : d. It holds exactly when the cross products match, a × d = b × c, so a missing term is found by cross-multiplying and dividing.
In 5 : 8 = x : 20, the cross products give 8x = 5 × 20 = 100, so x = 12.5. Check it by dividing: 5/8 = 0.625 and 12.5/20 = 0.625.
Map scales work the same way. At a scale of 1 : 25,000, a 4 cm distance on the map solves 1 : 25,000 = 4 : x, so x = 100,000 cm, which is 1 km on the ground.
If the term diagonally opposite x is 0, the cross products cannot pin x down, and the proportion has no single answer.
How to divide an amount in a ratio
Add the terms to count the shares, divide the amount by that total, then multiply each term by one share. Splitting 750 in the ratio 3 : 2 gives 5 shares of 150, so the parts are 450 and 300.
When the amount does not divide evenly, the exact parts are fractions: 100 split 1 : 1 : 1 is 100/3 each. Rounded to cents that is 33.33 three times, one cent short of 100, so one share has to be 33.34 for the total to balance.
How to scale a ratio
Multiply every term by the same factor. A dough with flour and water in the ratio 5 : 3 needs, for 400 g of flour, a factor of 400 ÷ 5 = 80, so the water is 3 × 80 = 240 g. The scaled ratio 400 : 240 simplifies back to 5 : 3.
Ratios, fractions and percentages
A ratio compares parts with each other; a fraction compares a part with the whole. In a 2 : 3 mix the first part is 2 of 5 shares, so it is 2/5 of the total, or 40%.
The fraction calculator does exact arithmetic with those fractions, and the percentage calculator turns a part and a whole into a percentage.
Common ratio mistakes
- Changing only one term. Doubling the first term of 2 : 3 gives 4 : 3, a different ratio. Every term needs the same factor.
- Adding the same amount to each term. Adding 1 to both terms of 2 : 3 gives 3 : 4, and 2/3 ≈ 0.667 is not 3/4 = 0.75.
- Mixing units. 1 m to 50 cm is 100 cm : 50 cm = 2 : 1, not 1 : 50. Convert to one unit before simplifying.
- Reversing the order. 2 : 3 and 3 : 2 are different ratios. The order follows the wording, so "sand to cement" puts sand first.
- Reading a part-to-part ratio as a share. In 2 : 3 the first quantity is 2/5 of the whole, not 2/3.
Entering ratios
- Separate terms with colons, as in 12 : 18 or 2 : 3 : 5. Writing 2 to 3 also works.
- Terms can be whole numbers, decimals or fractions. A ratio takes 2 to 10 terms, each zero or positive, with at least one above zero.
- In a proportion, type x for the one unknown term.
Worked examples
Decimal terms 1.5 : 2.5
- I want to
- Simplify a ratio
- Ratio
- 1.5 : 2.5
- Result
- 3 : 5
- Scale factor
- 0.5
Checked against: ×10 → 15 : 25, gcd 5
Fraction terms 1/2 : 3/4 : 5/6
- I want to
- Simplify a ratio
- Ratio
- 1/2 : 3/4 : 5/6
- Result
- 6 : 9 : 10
Checked against: LCD 12 → 6 : 9 : 10, gcd 1
Scale 2 : 3 so the first term is 30
- I want to
- Scale a ratio to a target
- Ratio
- 2 : 3
- Target value for that term
- 30
- Result
- 30 : 45
- Scale factor
- 15
Checked against: 30 / 2 = 15; 3 × 15 = 45
Questions
How do you simplify a ratio?
Divide every term by the greatest common divisor of the terms. For 12 : 18 the GCD is 6, giving 2 : 3. Decimals and fractions are cleared first by multiplying every term by the same number: 1.5 : 2.5 × 2 = 3 : 5, and 1/2 : 3/4 : 5/6 × 12 = 6 : 9 : 10, which is already in simplest form because its GCD is 1.
How do you solve a proportion for x?
Cross-multiply, then divide. In 3 : 4 = 9 : x the products of the diagonal terms are equal, so 3x = 4 × 9 = 36 and x = 12. The unknown can sit in any position: in x : 5 = 6 : 15, 15x = 5 × 6 = 30, so x = 2. If the term diagonally opposite x is 0, there is no single answer.
How do you divide an amount in a ratio?
Add the ratio terms, divide the amount by that total to get one share, then multiply each term by the share. For 1,000 split 2 : 3 : 5 the terms add to 10, one share is 100, and the parts are 200, 300 and 500. The parts always add back to the amount; 100 split 1 : 2 gives 100/3 and 200/3, about 33.33 and 66.67.
What is the difference between a ratio and a fraction?
A ratio compares parts with each other; a fraction compares a part with the whole. In a class of 12 girls and 18 boys the ratio of girls to boys is 12 : 18 = 2 : 3, but girls make up 12/30 = 2/5 of the class. A two-term ratio a : b becomes the fraction a/(a + b) of the whole.
How do you scale a ratio up or down?
Multiply or divide every term by the same factor. To make the first term of 2 : 3 equal 30, the factor is 30 ÷ 2 = 15, so the scaled ratio is 30 : 45. Scaling a recipe works the same way: a 2 : 3 mix made 15 times larger needs 30 and 45 units of the two ingredients, and the proportions stay 40% and 60%.
About this calculator
Sources
Checked against references
This calculator includes 8 worked examples with independently sourced answers. They run in the test suite, and you can run them here too.