Ratio to Decimal & Fraction Calculator

Every form of a two-term ratio at once: the decimal value, the simplified fraction, the mixed number, the 1 : n form and the reverse direction. Recurring decimals are marked, and the exact fraction is always given.

Formula A : B → A ÷ B

A
B
Try an example
Method

Four forms of one number

A ratio, a fraction, a decimal and a unit rate are the same quantity written four ways. Converting between them is mechanical once you see that.

Write the ratio as a fraction

A : B becomes A/B. This is a change of notation, not a calculation — the colon and the fraction bar mean the same thing here.

Reduce it

Divide numerator and denominator by their greatest common divisor. 45/60 has GCD 15, giving 3/4. The reduced fraction is the canonical form, and it is also what tells you whether the decimal will terminate.

Divide for the decimal

3 ÷ 4 = 0.75. If the reduced denominator has any prime factor other than 2 or 5, the decimal recurs and no finite decimal is exact — 2/3 = 0.666…. In that case the fraction is the answer and the decimal is an approximation.

Split off a whole part if the fraction is improper

7/4 contains one whole 4 with 3 left over, so it is 1 3/4. The mixed number is usually the most readable form for a measurement, though the improper fraction is easier to calculate with.

Divide both terms by the first for 1 : n

4 : 10 divided through by 4 gives 1 : 2.5. This is the form to use when you want to read off "two and a half times as much" without doing any arithmetic in your head.

A : B = A/B = A ÷ B

The core identity. One number, three notations. Everything else on this page follows from it.

A : B = 1 : B/A

Unit form. Dividing both terms by A. The second term becomes the multiplier that takes you from the first quantity to the second.

Which decimals terminate

Reduce the fraction and look at the denominator. Only 2s and 5s in its prime factorisation means the decimal stops. So /2, /4, /5, /8, /10, /16, /20, /25 all terminate; /3, /6, /7, /9, /11, /12 all recur. This is because 10 = 2 × 5, so only those two primes divide a power of ten.

Worked examples

Terminating, recurring and improper

Terminating

Convert 3 : 4 to a decimal and a fraction.

As a fraction
3/4
GCD of 3, 4
1 — already reduced
Denominator 4 = 2²
so it terminates
Divide
3 ÷ 4 = 0.75
1 : n form
1 : 1.3333…

0.75 exactly, or 3/4

Recurring

Convert 2 : 3 to a decimal.

As a fraction
2/3
Denominator 3
not 2 or 5, so it recurs
Divide
2 ÷ 3 = 0.666666…
Exact answer
2/3
1 : n form
1 : 1.5

2/3 exactly; 0.6667 rounded

Improper fraction

Convert 7 : 4 to every form.

As a fraction
7/4 — improper
Whole part
7 ÷ 4 = 1 remainder 3
Mixed number
1 3/4
Decimal
1.75
Meaning
A is 1.75 times B

7/4 = 1¾ = 1.75

Back to a ratio

Convert 0.75 back into a ratio.

Over a power of ten
75/100
GCD of 75, 100
25
Reduce
3/4
As a ratio
3 : 4
Check
3 ÷ 4 = 0.75 ✓

3 : 4

Reference

Conversion table

An ellipsis marks a recurring decimal, where the fraction is the only exact form.

Ratio Fraction Decimal Mixed 1 : n B ÷ A
1 : 21/20.5—1 : 22
1 : 31/30.3333…—1 : 33
1 : 41/40.25—1 : 44
2 : 32/30.6667…—1 : 1.51.5
3 : 43/40.75—1 : 1.3333…1.3333…
3 : 53/50.6—1 : 1.6667…1.6667…
5 : 85/80.625—1 : 1.61.6
7 : 47/41.751 3/41 : 0.5714…0.5714…
16 : 916/91.7778…1 7/91 : 0.56250.5625
45 : 603/40.75—1 : 1.3333…1.3333…
5 : 25/22.52 1/21 : 0.40.4
Background

Which form to use when

The four forms are mathematically interchangeable but practically quite different. A ratio is best for instructions you will act on physically — 3 parts to 4 parts tells you what to count out. A fraction is best for further calculation, because it stays exact. A decimal is best for comparison and for entering into a spreadsheet. The 1 : n form is best when you need a multiplier you can apply without thinking.

Choosing badly costs accuracy or clarity. Converting 1/3 to 0.333 early in a calculation introduces an error that compounds; keeping it as a fraction until the end does not. Conversely, reporting a survey result as 17/53 rather than 32% makes it much harder to compare against another number.

Why recurring decimals happen at all

Our decimal system is built on powers of ten, and 10 factorises as 2 × 5. A fraction can be rewritten with a denominator that is a power of ten only if its own denominator divides some power of ten — which means it can contain only 2s and 5s. 1/8 works because 8 = 2³ and 1/8 = 125/1000. 1/3 cannot, because no power of ten is divisible by 3. In base 3, 1/3 would be a clean "0.1" and 1/2 would recur instead — recurrence is a property of the notation, not of the number.

Rounding a recurring decimal safely

If you must use a decimal, round as late as possible and keep more digits than you think you need. A common rule is to carry two more significant figures through intermediate steps than you intend to report. Better still, keep the fraction and convert once at the end — which is how the calculators on this site work internally.

Related tools

For percentages rather than decimals, including the share-of-the-whole reading, use the ratio to percentage calculator. To reduce a ratio to whole numbers first, the simplify ratio calculator handles two to six terms. And for the 1 : n form applied to rates and prices, the unit ratio calculator adds unit labels and the per-unit phrasing.

Questions

Ratio, decimal and fraction — questions

Including which decimals recur and why, since that decides whether your answer is exact.

How do you convert a ratio to a decimal?
Divide the first term by the second. 3 : 4 becomes 3 ÷ 4 = 0.75. That decimal is the ratio's value — the number of units of the first quantity for every one unit of the second. A value above 1 means the first quantity is larger; below 1 means the second is.
How do you write a ratio as a fraction?
The ratio A : B becomes the fraction A/B, then reduce it. 45 : 60 becomes 45/60, which simplifies to 3/4. Note this is A relative to B, not A as a share of the whole — for that you would use A/(A+B), which the ratio to percentage calculator covers.
Why does the decimal sometimes recur?
A fraction terminates as a decimal only when its denominator, in lowest terms, has no prime factors other than 2 and 5. 3/4 terminates because 4 = 2²; 1/3 recurs because 3 is neither. 2 : 3 therefore gives 0.666… forever. The exact answer is the fraction, which is why this calculator shows it first and marks recurring decimals with an ellipsis.
What is a mixed number and when do I get one?
A mixed number combines a whole part with a proper fraction, like 1 3/4. You get one when the ratio's first term exceeds its second, making the fraction improper. 7 : 4 is 7/4, which is 1 3/4 as a mixed number and 1.75 as a decimal. All three describe the same value.
What is the 1 : n form and why is it useful?
It rewrites the ratio so the first term is exactly 1, making the second a direct multiplier. 4 : 10 becomes 1 : 2.5, telling you immediately that the second quantity is two and a half times the first. Map scales, gear ratios and mixing instructions use this form because it removes the mental arithmetic. Divide both terms by the first to get it.
How do I convert a decimal back into a ratio?
Write the decimal as a fraction over a power of ten, then simplify. 0.75 becomes 75/100, which reduces to 3/4, so the ratio is 3 : 4. For a recurring decimal you need the algebraic method, or you can simply recognise it — 0.333… is 1/3, so the ratio is 1 : 3.
Is the decimal of a ratio the same as its percentage?
They differ by a factor of 100 and, more importantly, by which denominator you use. The decimal of 3 : 5 is 0.6, which as a percentage is 60% — that is A as a percentage of B. But A as a share of the whole is 3/8 = 0.375 or 37.5%. Both are legitimate; make sure you know which one the question wants.
Can I convert a ratio with three terms to a decimal?
Not to a single decimal, because a decimal expresses one comparison and a three-term ratio contains several. What you can do is take each term relative to the first, giving the 1 : n : m form — 2 : 3 : 5 becomes 1 : 1.5 : 2.5. This calculator handles two terms; for longer ratios the simplify ratio calculator reports that form.