Solve and verify ratios A:B = C:D with step-by-step solutions and comprehensive ratio analysis
A ratio is a comparison of two or more quantities showing the relative sizes of these quantities. Ratios are fundamental in mathematics and appear everywhere in real life—from cooking recipes and map scales to financial analysis and scientific research.
A ratio compares two quantities by division. If you have 2 apples and 3 oranges, the ratio of apples to oranges is written as:
2 : 3
Read as "2 to 3"
This can also be written as the fraction 2/3 or using the word "to" (2 to 3). All three notations represent the same relationship.
All three forms express the same relationship between quantities.
In 5 : 8, the antecedent is 5 and the consequent is 8.
A proportion states that two ratios are equal. When we write A : B = C : D, we're saying that the ratio of A to B is the same as the ratio of C to D.
A : B = C : D
This means:
Two ratios A : B and C : D are equal if and only if A × D = B × C.
Example: Check if 2 : 3 = 4 : 6
Cross multiply: 2 × 6 = 12 and 3 × 4 = 12
Since 12 = 12, the ratios are EQUAL
Reduce both ratios to their simplest form. If they simplify to the same ratio, they are equal.
Example: Check if 4 : 6 = 6 : 9
Simplify 4 : 6 → Divide by GCD(4,6) = 2 → 2 : 3
Simplify 6 : 9 → Divide by GCD(6,9) = 3 → 2 : 3
Both simplify to 2 : 3, so they are EQUAL
Question: Is 3 : 4 = 9 : 12?
Solution using cross multiplication:
3 × 12 = 36
4 × 9 = 36
Answer: Yes, since 36 = 36, the ratios are equal (TRUE)
Question: Is 2 : 5 = 3 : 7?
Solution using cross multiplication:
2 × 7 = 14
5 × 3 = 15
Answer: No, since 14 ≠ 15, the ratios are not equal (FALSE)
Question: Simplify the ratio 18 : 24
Solution:
Find GCD(18, 24) = 6
18 ÷ 6 = 3
24 ÷ 6 = 4
Answer: 18 : 24 = 3 : 4 (simplest form)
Question: Find x if 5 : 8 = x : 24
Solution:
Using cross multiplication: 5 × 24 = 8 × x
120 = 8x
x = 120 ÷ 8
Answer: x = 15
Compares one part to another part of the whole.
Example: In a class of 12 boys and 15 girls, the ratio of boys to girls is 12 : 15 or 4 : 5.
Compares one part to the total amount.
Example: 12 boys out of 27 total students gives a ratio of 12 : 27 or 4 : 9.
Recipes use ratios to maintain the correct proportions of ingredients. If a recipe calls for a 2:1 ratio of flour to sugar, doubling the recipe requires maintaining that same ratio.
Map scales use ratios to represent distances. A scale of 1:50,000 means 1 cm on the map represents 50,000 cm (500 m) in real life.
Financial ratios like debt-to-equity ratio, profit margins, and price-earnings ratios are crucial for analyzing business performance and making investment decisions.
Speed is a ratio of distance to time (miles per hour). Density is a ratio of mass to volume. These ratios help us understand and compare different physical properties.
Ratio to Fraction:
The ratio a : b can be written as the fraction a/b
Example: 3 : 4 = 3/4
Fraction to Ratio:
The fraction a/b can be written as the ratio a : b
Example: 5/8 = 5 : 8
Ratio to Percentage:
Convert to fraction, then multiply by 100%
Example: 3 : 4 = 3/4 = 0.75 = 75%
1. Cross Product Property:
If A : B = C : D, then A × D = B × C
2. Alternation Property:
If A : B = C : D, then A : C = B : D
3. Inversion Property:
If A : B = C : D, then B : A = D : C
4. Addition Property:
If A : B = C : D, then (A + B) : B = (C + D) : D
Ratios can compare more than two quantities. For example, A : B : C = 2 : 3 : 5 means that for every 2 parts of A, there are 3 parts of B and 5 parts of C.
A recipe calls for ingredients in the ratio 2 : 3 : 5 (flour : sugar : milk).
If you use 6 cups of flour:
While ratios and fractions are closely related, they have different meanings. A fraction represents a part of a whole, while a ratio compares two separate quantities. For example, 3/4 as a fraction means 3 parts out of 4 total parts. As a ratio (3 : 4), it means for every 3 of one thing, there are 4 of another. However, ratios can be written as fractions mathematically.
Use cross multiplication: for A : B = C : D to be true, A × D must equal B × C. Alternatively, simplify both ratios to their lowest terms—if they simplify to the same ratio, they're equivalent. Our calculator above does both checks automatically.
Yes! Ratios can be any positive number. A ratio of 5 : 2 means the first quantity is 2.5 times the second. Ratios greater than 1 (when written as fractions) simply mean the first quantity is larger than the second.
Simplifying a ratio means reducing it to its smallest whole number terms by dividing both parts by their greatest common divisor (GCD). For example, 10 : 15 simplifies to 2 : 3 by dividing both by 5. The simplified ratio represents the same relationship but with smaller, easier-to-work-with numbers.
Use cross multiplication. If you have A : B = C : x and need to find x, cross multiply: A × x = B × C, then solve for x by dividing: x = (B × C) / A. For example, in 3 : 4 = 6 : x, we get 3x = 24, so x = 8.
The order in a ratio matters because it specifies which quantity comes first. A ratio of boys to girls of 2 : 3 is different from a ratio of girls to boys of 3 : 2. Always make sure you understand what the first and second quantities represent in the context of the problem.
While ratios can technically involve decimals (like 2.5 : 3.7), it's standard practice to convert them to whole numbers for simplicity. Multiply both terms by the same power of 10 to eliminate decimals. For example, 2.5 : 3.7 becomes 25 : 37 when multiplied by 10.
Ratios appear everywhere: cooking recipes (ingredient proportions), map scales (1:50,000), finance (debt-to-equity ratios), photography (aspect ratios like 16:9), medicine (dosage calculations), construction (concrete mix ratios), and sports statistics (win-loss ratios). Understanding ratios is essential for many practical applications.