What this tool does
A ratio compares two quantities by division and can be written three ways: 4 to 6, 4:6, or the fraction 4/6. Almost every ratio problem is one of three tasks, simplifying, scaling or solving a proportion, and each uses a different rule. This calculator identifies which task applies and shows the working, whether the numbers come from a recipe, a map scale or homework.
How it works
Simplifying a ratio means dividing every part by the same number, and the largest number that divides them all is their greatest common divisor. The ratio 4:6 has a GCD of 2, so both terms divide by 2 and the result is 2:3, the same relationship written in smaller numbers. A ratio can also be written as a fraction by putting the first value over the second, so 4:6 equals 4/6, which reduces to 2/3, and three-part ratios such as 2:3:5 reduce the same way.
Scaling changes the size of the quantities without changing their relationship, so every part is multiplied or divided by the same factor. Scaling 4:6 by 3 gives 12:18, which still simplifies to 2:3, and that is what happens when a recipe written for four servings is made for six: each ingredient is multiplied by 1.5. Scale factors work the same way for mixtures, dilutions, model kits and map scales, where a ratio such as 1:50,000 means one centimetre on the map represents 50,000 centimetres on the ground.
A proportion sets two ratios equal, written A:B = C:D, and is solved by cross multiplication, where A × D must equal B × C. In 2:3 = x:9 the cross products give 2 × 9 = 3 × x, so 18 = 3x and x = 6. This is the method behind map reading, recipe resizing, unit-price comparisons and every textbook problem that asks for a missing value in a proportion.
Worked example
Simplifying 4:6, converting it to a fraction, then solving 2:3 = x:9.
- gcd(4, 6) = 2, so 4:6 ÷ 2 = 2:3
- 4:6 as a fraction = 4/6 = 2/3
- Scale 4:6 by 3: 4 × 3 = 12 and 6 × 3 = 18, so 12:18
- Cross multiply 2:3 = x:9: 2 × 9 = 3x, so 18 = 3x and x = 6
4:6 simplifies to 2:3 (the fraction 2/3), scales to 12:18, and the proportion 2:3 = x:9 gives x = 6.
Accuracy and limitations
- Both sides of a ratio must use the same unit. Comparing 4 kg to 600 g needs converting to matching units first, or the ratio is meaningless.
- A scale factor applies directly to lengths, but areas scale by the factor squared and volumes by the factor cubed.
- Decimal inputs are turned into whole-number ratios by scaling, so rounding an input early can shift the simplified result.
Frequently asked questions
- How do you simplify a ratio?
- Divide every part by the greatest common divisor of the parts. The ratio 4:6 has a GCD of 2, so both terms divide by 2 and it simplifies to 2:3. The relationship does not change, because both sides are scaled by the same amount.
- How do you convert a ratio to a fraction?
- Write the first number over the second and reduce it, so 4:6 becomes 4/6, which simplifies to 2/3. Read that way, the first quantity is two thirds of the second. As a share of the whole, 4:6 means 4 parts out of 10, or 40%.
- What is a proportion and how do I solve one?
- A proportion is two ratios set equal, such as 2:3 = x:9, and you solve it by cross multiplication. Multiplying across gives 2 × 9 = 3 × x, so 18 = 3x and x = 6. Equal cross products are also how you check whether a proportion is true.
- How does a scale factor calculator work?
- It multiplies every term by the same number so the ratio grows or shrinks without changing. Scaling 4:6 by 3 gives 12:18, which still simplifies to 2:3. For maps and models the factor applies to length, so an area changes by the factor squared.
- What is the difference between a ratio and a fraction?
- A ratio compares two separate quantities, while a fraction usually compares a part to a single whole. The ratio 4:6 can be written as the fraction 4/6 = 2/3, but the reading differs: the ratio describes two amounts side by side, the fraction describes one amount relative to another.