Ratios in simple words
A ratio compares two amounts. If a bag has 2 red sweets for every 3 green sweets, the ratio of red to green is 2 : 3, said “2 to 3”.
You can simplify a ratio by dividing both numbers by the same number, just like a fraction. 12 : 18 becomes 2 : 3 when you divide both by 6. The ratio still means the same thing.
You can also scale a ratio up. For a drink mixed 1 : 4, one cup of juice goes with four cups of water. For three cups of juice, multiply both by 3 to get 3 : 12, so you need twelve cups of water.
To find a missing number, such as 2 : 3 = 8 : ?, ask what 2 was multiplied by to make 8. It was 4, so multiply 3 by 4 too. The answer is 12.
Formula and method
A ratio a:b compares two quantities. Divide positive integer terms by their greatest common factor to simplify. Multiply both terms by the same factor to scale. A proportion a:b = c:d is solved by cross multiplication, ad = bc.
Worked example
12:18 simplifies to 2:3 after dividing by 6. Scaling 2:3 by 4 gives 8:12. In 12:18 = 8:d, d = 18 × 8 / 12 = 12.
Simplifying ratios with fractions and decimals
Clear fractions first: multiply every term by the least common denominator (LCD). For 1/2 : 3/4 : 5/6 the LCD is 12, giving 6 : 9 : 10, which has no common factor, so that is the simplest whole-number form. Mixed numbers become improper fractions first: 1 1/2 : 2 = 3/2 : 2/1, and × 2 gives 3 : 4.
Decimals are scaled by a power of ten: 0.5 : 1.25 × 100 = 50 : 125. Then divide every term by the greatest common factor, written term by term as 50 = 25 × 2 and 125 = 25 × 5, so the ratio is 2 : 5. Simplify and 3+ terms both accept whole numbers, decimals, fractions and mixed numbers.
Inputs and limitations
The percentage shown is a/b × 100, not a/(a+b) × 100. A 2:3 mixture contains 40% of the first ingredient, even though the ratio a/b is 66.6667%. Simplification accepts positive integers up to 10¹², decimals with up to 8 places, fractions and mixed numbers; the other modes accept positive decimals.
Part-to-whole, unit price and the golden ratio
The part-to-whole mode divides each term by the sum of the terms: 12:18 has a whole of 30, so the parts are 12/30 = 2/5 (40%) and 18/30 = 3/5 (60%). A ratio with three or more terms simplifies the same way as two: 12:18:24 shares the factor 6 and becomes 2:3:4.
Unit price is price ÷ quantity. At 4.99 for 12 oz the price is about 0.4158 per oz; at 7.49 for 20 oz it is 0.3745 per oz, so the second offer is about 9.94% cheaper per ounce.
The golden ratio φ = (1 + √5) ÷ 2 ≈ 1.6180339887 splits a length so that the whole is to the longer part as the longer part is to the shorter: (a + b)/a = a/b = φ. A longer part of 10 gives a shorter part of 10 ÷ φ ≈ 6.18034 and a whole of about 16.18034.