Ratio Calculator
Enter three values of a proportion a:b = c:d to find the fourth. This calculator solves for d given a, b, and c. It also shows simplified ratio values. The content below covers ratio simplification, unit rates, and real-world proportion problems like scaling recipes and maps.
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Formula
a/b = c/d → d = b×c / a
A proportion states that two ratios are equal: a/b = c/d. Cross-multiplying gives a×d = b×c, so d = b×c / a. This is the fundamental 'cross-multiply and divide' rule. To check your answer, substitute back: verify that a×d equals b×c.
How to use the Ratio Calculator
- 1
Enter your a (first ratio, part 1)
- 2
Enter your b (first ratio, part 2)
- 3
Enter your c (second ratio, part 1)
- 4
Read your results instantly
Results update in real time as you type.
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Ratios vs. fractions vs. rates
A ratio compares two quantities of the same type: 3 red balls to 4 blue balls is a 3:4 ratio. A fraction expresses a part out of a whole: 3/7 of the balls are red. A rate compares quantities of different types: 60 miles per hour, $5 per kilogram.
Ratios can be written three ways: 3:4, 3/4, or '3 to 4.' They're equivalent — choose whichever form fits the context. A ratio of 3:4 does not mean 3 out of 4; it means for every 3 of one thing there are 4 of another.
To express a ratio as a fraction of the total: in a 3:4 mixture, there are 3+4=7 parts total, so the first component is 3/7 of the mixture.
Simplifying ratios
A ratio is in simplest form when the two numbers share no common factor other than 1. To simplify, divide both parts by their GCF.
12:18 → GCF(12, 18) = 6 → 2:3.
If one or both values are decimals, first multiply both by a power of 10 to make them integers, then simplify. 1.5:2.5 = 15:25 = 3:5.
When working with three-part ratios (a:b:c), find the GCF of all three and divide each. 6:9:12 → GCF = 3 → 2:3:4.
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Proportions in everyday problems
Proportions appear everywhere. Scaling a recipe: if 3 cups of flour makes 12 cookies and you want 20, set up 3/12 = x/20 → x = 5 cups.
Map scales: a 1:50,000 map means 1 cm on the map represents 50,000 cm (500 m) in reality. A 3 cm line on the map covers 1.5 km.
Similar triangles: if two triangles are similar with side ratio 3:5, all corresponding sides are in that ratio. If one short side is 9 cm, the corresponding side in the larger triangle is 9 × (5/3) = 15 cm.
Currency exchange: if $1 USD = 1.08 EUR, you have a ratio 1:1.08. To convert $350: 1/1.08 = 350/x → x = 350 × 1.08 = 378 EUR.
Tips & Insights
Cross-multiplication is your main tool
Any proportion a:b = c:d can be solved by cross-multiplying: a×d = b×c. Rearrange to isolate whichever variable is unknown. This works equally well whether d, c, a, or b is the unknown.
Unit rate simplifies comparisons
Reduce a ratio to a unit rate (denominator = 1) for easy comparison. $4.50 for 3 cans vs. $3.20 for 2 cans: $1.50/can vs. $1.60/can. The three-pack is cheaper per unit. Unit rates are the core of best-value comparisons.
Part-to-part vs. part-to-whole
A 3:2 ratio of boys to girls is part-to-part. The fraction of boys in the group (3/5) is part-to-whole. Mistaking one for the other is a common error: 3:2 does NOT mean boys are 3/2 of the total — it means they're 3/5.
Worked Examples
Scaling a recipe
d = 8 × 12 / 2 = 48. Wait — re-read the proportion: a:b = c:d means 2:8 = 12:d, so d = (8×12)/2 = 48? Let's reframe: if 2 cups makes 8 servings, how much for 12? 2/8 = x/12 → x = (2×12)/8 = 3 cups of flour.
Map distance
d = (150 × 7) / 3 = 350 km. A 7 cm distance on the map represents 350 km in real life. The map scale is 1 cm : 50 km.
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Frequently Asked Questions
What is a proportion?
A proportion is a statement that two ratios are equal: a/b = c/d. It means the relationship between the first pair of quantities is the same as the relationship between the second pair.
How do I simplify a ratio?
Divide both parts by their greatest common factor. GCF(12, 18) = 6, so 12:18 simplifies to 2:3. You can also divide both parts by any common factor repeatedly until none remain.
What is a unit ratio?
A unit ratio has 1 as one of its parts, obtained by dividing through by one of the values. 3:4 as a unit ratio is 1:1.333... or 0.75:1. Unit ratios make comparison easy.
What is the difference between a ratio and a rate?
A ratio compares quantities of the same unit (apples to oranges, or dollars to dollars). A rate compares quantities of different units (miles per hour, dollars per kilogram). Rates are expressed with 'per' and have units attached.
How do I check if two ratios are equivalent?
Cross-multiply: a/b = c/d if and only if a×d = b×c. For 3/4 and 6/8: 3×8 = 24 and 4×6 = 24. Equal, so the ratios are equivalent. You can also simplify each ratio to lowest terms and check if they match.
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