A ratio compares two or more quantities. They appear in cooking recipes, map scales, financial analysis, mixing concrete, and countless everyday situations. Once you understand how ratios work — how to simplify them, scale them, and use them to divide quantities — you have a powerful problem-solving tool that applies across almost every field.
What Is a Ratio?
A ratio expresses the relative size of two or more quantities. The ratio 3:5 means "for every 3 of one thing, there are 5 of another." Ratios can be written in three equivalent ways:
Colon notation: 3 : 5
Fraction notation: 3/5
Word form: "3 to 5"
All three mean the same thing — for every 3 units of the first quantity, there are 5 units of the second.
Types of Ratios
Compares one part of a group to another part. Example: boys to girls in a class.
Compares one part to the total. Same as a fraction. Example: passes out of total attempts.
Ratio of two different units. Example: speed, price per unit, pay per hour.
Used in maps and models. 1 cm on the map = 50,000 cm in real life.
How to Simplify a Ratio
Simplifying a ratio means reducing it to its lowest terms — the smallest whole numbers that maintain the same relationship. The method is identical to simplifying a fraction: divide both parts by their Greatest Common Divisor (GCD).
Find the largest number that divides evenly into both a and b, then divide both parts by it.
Find the GCD of the two numbers
List the factors of both numbers and find the largest factor they share. For larger numbers, use the Euclidean algorithm: divide the larger by the smaller, take the remainder, and repeat until the remainder is zero — the last non-zero divisor is the GCD.
Divide both parts by the GCD
Divide every term of the ratio by the GCD. The result is the simplified ratio. All terms should now be whole numbers with no common factor other than 1.
Verify — check that the simplified ratio is equivalent
Cross-multiply to verify: a/b = c/d is true if a × d = b × c. If the cross products are equal, the ratios are equivalent.
Simplifying — Worked Examples
🧮 Ratio Calculator
Enter any two values to simplify, convert, and visualize the ratio.
How to Scale a Ratio
Scaling a ratio means multiplying or dividing all parts by the same number. The ratio stays equivalent — only the actual quantities change. This is used in recipes, map reading, mixing, and proportion problems.
Multiply every part of the ratio by k to scale up
Divide every part by k to scale down
How to Divide a Quantity in a Given Ratio
This is one of the most common ratio problems: you have a total amount and need to split it according to a given ratio.
Find the total number of parts
Add all the numbers in the ratio. For ratio a : b, total parts = a + b. For a : b : c, total parts = a + b + c.
Find the value of one part
Divide the total quantity by the total number of parts: one part = total ÷ (a + b).
Multiply each ratio number by the value of one part
Each person's or portion's share = their ratio number × value of one part.
Converting Ratios
Ratio to Fraction
A ratio a : b equals the fraction a/b. The first term is the numerator, the second is the denominator.
Ratio to Percentage
To express part A as a percentage of the total (A + B): divide A by (A + B), then multiply by 100.
Proportion — Solving for an Unknown
A proportion states that two ratios are equal: a/b = c/d. If three of the four values are known, you can solve for the fourth using cross-multiplication.
Then solve for the unknown variable.
⚖️ Proportion Solver
Enter three values. Leave the unknown as blank — type x or leave empty.
Ratio vs Proportion — What's the Difference?
A ratio compares two quantities: 3 : 5
A proportion states that two ratios are equal: 3/5 = 6/10
Every proportion contains two ratios. Not every ratio is part of a proportion. When you say "this recipe scales proportionally," you mean the ratio of ingredients stays the same as the quantity changes.
Practice Problems
| Problem | Answer |
|---|---|
| Simplify 24 : 36 | 2 : 3 |
| Simplify 45 : 60 : 75 | 3 : 4 : 5 |
| Divide ₱5,000 in ratio 1:4 | ₱1,000 and ₱4,000 |
| Convert ratio 7:3 — what % is Part A? | 70% |
| If 3 : x = 9 : 15, find x | x = 5 |
| Map scale 1:25,000. Map distance = 4 cm. Real distance? | 1 km |
| Concrete mix: cement:sand:gravel = 1:2:3. For 12 bags cement, how much sand and gravel? | 24 bags sand, 36 bags gravel |
1. A ratio compares quantities — written as a:b, a/b, or "a to b"
2. Simplify by dividing both parts by their GCD
3. Scale by multiplying all parts by the same factor k
4. Divide a quantity: total parts = a+b → one part = total ÷ (a+b) → each share = ratio × one part
5. Ratio to fraction: a:b = a/b
6. Ratio to %: a/(a+b) × 100
7. Proportion: a/b = c/d → solve with cross-multiplication a×d = b×c
8. Always verify by checking that shares add back to the original total
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Simplify, scale, convert and split quantities using our free ratio calculator.
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