A ratio compares parts without saying how big anything is. 2 : 3 fixes the relative sizes and nothing else — it could be 2 and 3, or 200 and 300, or two fifths of a spoonful and three fifths. That is the whole idea, and it is why every part can be multiplied or divided by the same number without cha...
In 2 : 3, the first part is not two thirds of the total. It is two thirds of the second part, and two fifths of the total — 40%, not 67%. Both numbers are correct answers to different questions, and using one where the other was meant is the commonest ratio mistake there is. For every two-part ratio with non-zero parts the two readings are different numbers, and the part-to-part one is always the larger. This page shows both for every part, every time, rather than picking one and leaving you to guess which.
A QUANTITY TO SHARE OUT (optional)
12 : 18 IN ITS SIMPLEST FORM
2 : 3
Every part was divisible by 6. The ratio itself has not changed.
| PART | OF THE WHOLE | % |
|---|---|---|
| 12 | 2/5 | 40% |
| 18 | 3/5 | 60% |
Two fractions, two meanings. The first part is 2/3 of the second part, and 2/5 of the total — 40%. Ask which question you are answering before quoting either.
Scaling keeps a ratio; shifting destroys it. Doubling every part leaves 2 : 3 unchanged, but adding 1 to every part gives 3 : 4, which is a different ratio entirely. Across random two-part ratios, adding the same amount to both parts changed the ratio 96.7% of the time — the survivors were the ones already equal.
SIMPLEST FORM
2 : 3
divided by 6
SHARES IN TOTAL
5
what the simplest form adds up to
LARGEST SHARE
60%
of the whole, not of another part
PART TO PART
2/3
first divided by second
WHAT IS HAPPENING
12 : 18 ÷6 → 2 : 3
Every part is divisible by 6, and dividing them all by the same number is the one operation a ratio does not notice. 2 : 3 says exactly what 12 : 18 says, in the smallest whole numbers that can say it.
part to part: 2/3 · part to whole: 2/5
The first part is 2/3 of the second, and 2/5 of the total. Both are correct answers to different questions, and swapping them is the commonest ratio error there is.
A ratio has no size. 2 : 3 does not say whether there are five things or five thousand — only how they divide. That is why every part can be multiplied or divided by the same number without changing anything, and why a ratio only becomes an amount once a total is supplied. Give it one above and the slices become quantities.
Exact fractions throughout · the slices are drawn from the same numbers printed beside them
Type the ratio with colons between the parts — 12 : 18, or 1 : 2 : 3, or 2.5 : 1. Decimals and fractions are fine; they are multiplied up to whole numbers before anything else happens.
Read both fractions for each part. The table gives each part as a share of the whole, as a percentage, and separately as a fraction of another part. These are different numbers and the page never makes you guess which is which.
Add a quantity to share out if you have one. The amounts are exact fractions, so they always add back to the total you typed rather than drifting by a rounding error.
Switch to the proportion mode to solve a : b = c : d. Fill in three terms, leave the fourth blank, and the cross multiplication is shown along with a check that the two products really match.
Take 12 : 18. Both parts divide by 6, so this is 2 : 3. Nothing has changed about the relationship — dividing every part by the same number is the one move a ratio cannot detect. Drawn as blocks of six, 12 is two blocks and 18 is three, which is the same picture the smaller numbers describe. Now the two fractions, which is where care is needed. Part to part: the first is 12/18 = 2/3 of the second. Part to whole: the total is 12 + 18 = 30, so the first is 12/30 = 2/5 of everything, which is 40%. Both are true. If somebody says "the first is two thirds" they mean of the other part; if they say "40%" they mean of the total. Quoting 2/3 as a percentage of the whole is a 27-point error. Share 120 in 3 : 5. The parts sum to 8, so the shares are 3/8 and 5/8. Multiply: 120 × 3/8 = 45, and 120 × 5/8 = 75. Check: 45 + 75 = 120. Working in fractions rather than decimals matters here — 100 shared in 1 : 1 : 1 is three exact thirds, and a calculator that rounds each to 33.33 loses a penny that has to reappear from somewhere. Solve 3 : 4 = 9 : ? Cross multiply: 3 × ? = 4 × 9, so ? = 36/3 = 12. The check is that the two cross products match: 3 × 12 = 36 and 4 × 9 = 36. Cross multiplying is not a rule to memorise — a/b = c/d is an equation, and multiplying both sides by b and then by d gives a × d = b × c directly. Finally, the trap worth remembering. If a recipe is 2 : 3 and you add one spoonful of each, you have 3 : 4, not 2 : 3. Scaling keeps a ratio; shifting destroys it. The same applies to combining: two classes with boys to girls of 2 : 3 and 3 : 4 do not combine to 5 : 7 unless the classes happen to be the right sizes. Ratios do not average — you need the actual counts.
| Rule | What it says | Why |
|---|---|---|
| What a ratio says | 2 : 3 compares two parts | It fixes the relative sizes and says nothing about how big either is. |
| Part to part | first ÷ second = 2/3 | The first is two thirds of the second. |
| Part to whole | first ÷ total = 2/5 | The first is two fifths of everything. This is the one people mean and rarely say. |
| They are never equal | 2/3 against 2/5 | For any ratio with two non-zero parts the two readings differ, and part-to-part is the larger. |
| Simplifying | divide every part by the gcd | 12 : 18 = 2 : 3. Dividing all parts alike is the one move a ratio ignores. |
| Scaling up | multiply every part alike | 2 : 3 = 4 : 6 = 200 : 300. Same ratio, different amounts. |
| Never add to the parts | 2 : 3 + 1 each = 3 : 4 | A different ratio. Scaling preserves; shifting does not. |
| Order matters | 2 : 3 ≠ 3 : 2 | Ratios are ordered. Boys to girls is not girls to boys. |
| Sharing a quantity | total × part ÷ sum | £120 in 3 : 5 is £45 and £75, because 120 × 3/8 = 45. |
| Three or more parts | 1 : 2 : 3 sums to 6 | Shares are 1/6, 1/3 and 1/2. The method does not change. |
| Decimals and fractions | 2.5 : 1 = 5 : 2 | Multiply every part until they are whole, then reduce. |
| A proportion | a/b = c/d | Two equal ratios. Cross multiply: a × d = b × c. |
| Cross multiplying | is not a new rule | It is multiplying both sides by both denominators, which any equation allows. |
| Ratios do not average | 2:3 and 3:4 is not 5:7 | Combining needs the actual counts, not the ratios. Group sizes matter. |
It says the second part is one and a half times the first, and nothing about how big either is. 2 : 3 describes 2 and 3, 20 and 30, or 0.2 and 0.3 equally well. A ratio fixes proportions and leaves scale open, which is why it only becomes an amount once a total is supplied. It is also ordered: the 2 belongs to whatever was named first.
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Last updated: August 5, 2026 · Exact fractions throughout, never floating point · In 2 : 3 the first part is 2/5 of the whole and 2/3 of the second.