Short answer
- 4 divided by 3 is 1.333..., which is the fraction 4/3 or the mixed number 1 and 1/3, with a remainder of 1. 10 divided by 3 is 3.333..., which is the fraction 10/3 or the mixed number 3 and 1/3, also with a remainder of 1. 12 divided by 3 is 4 exactly, with a remainder of 0, because 12 is a multiple of 3. 16 divided by 3 is 5.333..., which is the fraction 16/3 or the mixed number 5 and 1/3, with a remainder of 1. 3 divided by 3 is 1 exactly, with a remainder of 0, because any nonzero number divided by itself equals 1. The bonus question in this set, what is 1 divided by 1/2, has the answer 2, because dividing by a fraction means multiplying by its reciprocal, and two halves fit into one whole. The pattern behind the first five answers is the remainder, which can only be 0, 1 or 2 when dividing by 3. A remainder of 0 gives a whole number. A remainder of 1 gives a fractional part of 1/3, ending in .333... A remainder of 2 gives a fractional part of 2/3, ending in .666... Once you can see that rule, you can predict the form of any division by 3 before you finish the working.
By Emma Hudson
Dividing by 3 looks simple, and that is exactly why this small set of divisions is so useful to learn together. Within five short questions you meet every kind of division outcome: an answer that comes out as a whole number with nothing left over, an answer that repeats forever as a decimal, and a remainder that is best kept as a fraction. This guide gives the full long division working for each result in the set, shows the fraction, decimal and percentage forms side by side, checks every answer by multiplying back, and explains how rounding works when a decimal will not stop. A related question about dividing by a fraction is covered at the end, because it follows a different rule and surprises many learners.
The Parts of a Division
Every division has three named parts and sometimes a fourth. The dividend is the number being divided. The divisor is the number you divide by. The quotient is the result. The remainder is what is left over when the divisor does not fit into the dividend an exact whole number of times. In 10 divided by 3, the dividend is 10, the divisor is 3, the quotient is 3 and 1/3, and the remainder, before you turn it into a fraction, is 1.
There are two useful ways to read the same division, and both give the same quotient. The sharing reading splits the dividend equally: share 12 cookies among 3 children and each child gets 4 cookies. The grouping reading counts how many groups of the divisor size fit into the dividend: how many groups of 3 fit into 12 is also 4. Switching between the readings is a quick sense check.
A fraction is a division written in another form. The fraction 4/3 means 4 divided by 3, with the numerator as the dividend and the denominator as the divisor. A mixed number such as 1 and 1/3 is the same value as a whole part plus a proper fraction part. A percentage is the same value scaled so the whole is 100: multiply a quotient by 100 to get its percentage. So 4 divided by 3 can be written as 4/3, as 1 and 1/3, as 1.333…, or as 133.333… percent, and all four forms name the same amount.
Results at a Glance
| Division | Whole part | Remainder | Fraction | Mixed number | Decimal | Percentage |
|---|---|---|---|---|---|---|
| 4 divided by 3 | 1 | 1 | 4/3 | 1 and 1/3 | 1.333… | 133.333… percent |
| 10 divided by 3 | 3 | 1 | 10/3 | 3 and 1/3 | 3.333… | 333.333… percent |
| 12 divided by 3 | 4 | 0 | 12/3 | 4 | 4 | 400 percent |
| 16 divided by 3 | 5 | 1 | 16/3 | 5 and 1/3 | 5.333… | 533.333… percent |
| 3 divided by 3 | 1 | 0 | 3/3 | 1 | 1 | 100 percent |
| 1 divided by 1/2 | 2 | 0 | 2/1 | 2 | 2 | 200 percent |
Read the remainder column first, because it predicts the rest of the row. A remainder of 0 means the decimal stops at a whole number. A remainder of 1 with a divisor of 3 means the leftover is one third of the divisor, so the decimal and the percentage repeat the digit 3. The three dots after a decimal mean the pattern continues without end. They are part of the exact answer, not a sign that the working was left unfinished.
4 Divided by 3: Full Working
Start with the whole number part. Ask how many times 3 fits into 4. It fits once, because 1 times 3 is 3, and 2 times 3 is 6, which is already larger than 4. Subtract: 4 minus 3 is 1. That 1 is the remainder. So far the answer is 1 remainder 1.
Now turn the remainder into a fraction. The remainder is still measured against the divisor, so a remainder of 1 with a divisor of 3 becomes the fraction 1/3. The full answer is therefore 1 and 1/3. As a single fraction, the dividend over the divisor, it is simply 4/3, and you can check the conversion: 1 and 1/3 means 1 plus 1/3, and 1 written in thirds is 3/3, so 3/3 plus 1/3 is 4/3. The two forms agree.
To get the decimal, continue the long division past the whole number part. The remainder of 1 becomes 10 tenths. 3 fits into 10 three times, because 3 times 3 is 9, leaving a remainder of 1. Bring down another zero and the same step happens again. The remainder is 1 at every stage, so the digit 3 is produced at every stage, forever. That is why the decimal is 1.333…, with the 3 repeating without end. Written with the dots, 1.333… is exact, exactly as the fraction 4/3 is exact.
Rounded forms are different from the exact answer and should be labeled as rounded. To two decimal places, 4 divided by 3 is 1.33, because the third decimal digit is 3, which is less than 5, so the digit before it stays unchanged. To three decimal places it is 1.333. To one decimal place it is 1.3, because the second decimal digit is 3, which is less than 5.
Check by multiplying back. Using the fraction: 4/3 times 3 is 4, because the divisor cancels. Using the mixed number: 1 and 1/3 times 3 is 3 plus 1, which is 4. Using the rounded decimal shows why rounding must come last: 1.33 times 3 is 3.99, not 4. The missing 0.01 is rounding error introduced by cutting the decimal short, not a mistake in the division itself.
A concrete picture helps. Share 4 pizzas equally among 3 people: each person gets 1 whole pizza plus 1/3 of the last pizza, a share of 1 and 1/3 pizzas. The fraction describes that fair share exactly, even though it cannot be cut as a finite decimal slice.
As a percentage, 4 divided by 3 is 133.333… percent. A result above 100 percent makes sense here, because the dividend is larger than the divisor: 4 is more than one whole lot of 3, by exactly one third of a lot.
10 Divided by 3: Full Working
Ask how many times 3 fits into 10. It fits three times, because 3 times 3 is 9, and 4 times 3 is 12, which is larger than 10. Subtract: 10 minus 9 is 1. The remainder is 1 again, so the answer is 3 remainder 1, which becomes 3 and 1/3 as a mixed number, or 10/3 as a single fraction. Check the conversion: 3 written in thirds is 9/3, and 9/3 plus 1/3 is 10/3. The forms agree.
Continue into decimals in the same way as before. The remainder of 1 becomes 10 tenths, 3 fits three times with a remainder of 1, and the step repeats at every decimal place. The decimal is therefore 3.333…, exact when written with the dots. To two decimal places it is 3.33, because the third decimal digit is 3, which is less than 5. To one decimal place it is 3.3.
A shortcut confirms this result. 10 is 4 plus 6, and 6 divided by 3 is exactly 2, so 10 divided by 3 equals 4 divided by 3 plus 2, giving 3 and 1/3. Adding a multiple of the divisor to the dividend raises the whole part and leaves the fractional part untouched, because it does not change the remainder.
Check by multiplying back. 10/3 times 3 is 10. In mixed number form, 3 and 1/3 times 3 is 9 plus 1, which is 10. With the rounded decimal, 3.33 times 3 is 9.99. Once again the shortfall of 0.01 is pure rounding error, and it disappears when the check uses the fraction or the full repeating decimal.
In everyday terms, this division appears whenever 10 is split three ways: a bill shared by 3 people, or 10 meters of ribbon cut into 3 equal pieces of 3 and 1/3 meters each. A third of a meter is 33.333… centimeters, so a maker working to the nearest millimeter cuts about 3 meters and 33.3 centimeters and accepts a tiny difference in practice. The fraction stays exact on paper even when the physical cut cannot be.
As a percentage, 10 divided by 3 is 333.333… percent, because 10 is three whole lots of 3 plus one third of a lot more.
12 Divided by 3: The Exact Case
Ask how many times 3 fits into 12. It fits exactly four times, because 4 times 3 is 12. Subtract: 12 minus 12 is 0. There is no remainder, no fractional part, and no decimal places to continue into. The answer is simply 4.
This is the tidy case in the set. 12 is a multiple of 3. The multiples of 3 are 3, 6, 9, 12, 15 and 18, and dividing any of them by 3 gives a whole number: 1, 2, 3, 4, 5 and 6 respectively. Whenever the dividend is a multiple of the divisor, the quotient is a whole number, because there is nothing left over to share out.
A quick test for multiples of 3 is the digit sum rule: if the digits add to a multiple of 3, the number is a multiple of 3. For 12, 1 plus 2 is 3, so 12 is divisible by 3. For 16, the digits add to 7, so 16 cannot divide exactly by 3.
Check by multiplying back: 4 times 3 is 12. No fraction, no rounding, and no error term. As a percentage, 12 divided by 3 is 400 percent, because 12 is exactly four lots of 3.
This result also anchors its neighbors. 11 divided by 3 is one third less than 4, which is 3 and 2/3. 13 divided by 3 is 4 and 1/3, 14 divided by 3 is 4 and 2/3, and 15 divided by 3 is exactly 5. Working outward from a known multiple gives a built in check: neighboring answers must differ by exactly 1/3 each step.
16 Divided by 3: Full Working
Ask how many times 3 fits into 16. It fits five times, because 5 times 3 is 15, and 6 times 3 is 18, which is larger than 16. Subtract: 16 minus 15 is 1. The remainder is 1, so the answer is 5 remainder 1, which is 5 and 1/3 as a mixed number, or 16/3 as a single fraction. Check the conversion: 5 written in thirds is 15/3, and 15/3 plus 1/3 is 16/3.
Continue the long division and the familiar pattern returns. The remainder of 1 becomes 10 tenths, produces the digit 3, leaves a remainder of 1, and repeats at every place. The decimal is 5.333…, exact with the dots. To two decimal places it is 5.33, because the third decimal digit is 3, which is less than 5. To one decimal place it is 5.3.
Check by multiplying back. 16/3 times 3 is 16. In mixed number form, 5 and 1/3 times 3 is 15 plus 1, which is 16. With the rounded decimal, 5.33 times 3 is 15.99, short by 0.01 for the same rounding reason as before. Notice that the shortfall is the same 0.01 seen in the checks for 4 divided by 3 and 10 divided by 3. That is not a coincidence: all three answers carry the same fractional part of 1/3, so cutting them at two decimal places discards the same amount, and multiplying by 3 gives the same total shortfall.
A practical form of this answer is often more useful than the decimal. Sixteen hours shared across 3 days is 5 and 1/3 hours per day, and one third of an hour is exactly 20 minutes, because 60 divided by 3 is 20, so the share is 5 hours and 20 minutes per day. Converting the fractional part into a smaller unit works whenever that unit divides evenly by the divisor: thirds of a minute are 20 seconds, and thirds of a dozen are 4. Thirds of a dollar, by contrast, are 33.333… cents, which is why money split three ways so often leaves a cent unaccounted for.
As a percentage, 16 divided by 3 is 533.333… percent, because 16 is five whole lots of 3 plus one third of a lot more.
3 Divided by 3: Dividing a Number by Itself
Ask how many times 3 fits into 3. It fits exactly once. Subtract: 3 minus 3 is 0. The answer is 1 exactly, with a remainder of 0. As a fraction it is 3/3, which simplifies to 1, and as a percentage it is 100 percent, which fits the picture: 3 is exactly one whole lot of 3.
This is the simplest result in the set, and it states a general rule: any nonzero number divided by itself equals 1. 7 divided by 7 is 1, 25 divided by 25 is 1, and one half divided by one half is 1. The rule needs the word nonzero, because 0 divided by 0 has no defined answer: there is no number that multiplied by 0 gives a single agreed result, since every number multiplied by 0 gives 0. Division by zero in general is undefined, which is why the divisor in any of these problems can never be 0.
Check by multiplying back: 1 times 3 is 3. In longer work, a step that should reduce to a number divided by itself is a built in test: if it does not give 1, an earlier slip has happened. 3 is also the first multiple of 3, the base of the landmark sequence in this guide: 3 divided by 3 is 1, 6 divided by 3 is 2, 9 divided by 3 is 3, and each step up adds exactly 1 to the quotient.
What Is 1 Divided by 1/2?
This question belongs to a different family, because the divisor is a fraction rather than a whole number, and its answer surprises many people: 1 divided by 1/2 is 2, which is larger than the number you started with.
The rule is that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is the fraction turned over: the reciprocal of 1/2 is 2/1, which is 2. So 1 divided by 1/2 becomes 1 times 2, which is 2.
The grouping reading explains why the answer grows. Dividing by 1/2 asks how many halves fit into 1 whole, and two halves make a whole. In general, dividing by a number smaller than 1 gives a result larger than the dividend, because you are counting how many small pieces fit inside it. Dividing by exactly 1 leaves the dividend unchanged.
Check by multiplying back: 2 times 1/2 is 1, which returns the original dividend, as a correct check must. The common wrong answer here is 1/2, and it comes from multiplying by 1/2 instead of dividing by it. One half of 1 is indeed 1/2, but that answers a different question. Keeping the grouping picture in mind, how many halves fit into this whole, prevents the mix up.
The same method extends to other unit fractions. 1 divided by 1/3 is 3, because three thirds fit into a whole. 1 divided by 1/4 is 4. 2 divided by 1/2 is 4, because four halves fit into 2 wholes. In each case the multiply back check returns the dividend exactly.
As a percentage, 1 divided by 1/2 is 200 percent: the whole contains two lots of one half.
The Remainder Pattern and Neighboring Divisions
Dividing whole numbers by 3 can only end in one of three ways, because the remainder must be smaller than the divisor. The remainder is 0, 1 or 2, and each remainder fixes the fractional part of the answer. Remainder 0 gives a whole number. Remainder 1 gives a fractional part of 1/3, ending in .333… Remainder 2 gives a fractional part of 2/3, ending in .666…, because 2 divided by 3 produces the digit 6 at every decimal place: the remainder of 2 becomes 20 tenths, 3 fits six times to make 18, the remainder is 2 again, and the step repeats.
That pattern fills in the divisions around the main set without any new long division:
| Division | Answer | Decimal |
|---|---|---|
| 1 divided by 3 | 1/3 | 0.333… |
| 2 divided by 3 | 2/3 | 0.666… |
| 5 divided by 3 | 1 and 2/3 | 1.666… |
| 6 divided by 3 | 2 | 2 |
| 7 divided by 3 | 2 and 1/3 | 2.333… |
| 8 divided by 3 | 2 and 2/3 | 2.666… |
| 9 divided by 3 | 3 | 3 |
| 11 divided by 3 | 3 and 2/3 | 3.666… |
| 13 divided by 3 | 4 and 1/3 | 4.333… |
| 14 divided by 3 | 4 and 2/3 | 4.666… |
| 15 divided by 3 | 5 | 5 |
| 17 divided by 3 | 5 and 2/3 | 5.666… |
| 18 divided by 3 | 6 | 6 |
| 19 divided by 3 | 6 and 1/3 | 6.333… |
| 20 divided by 3 | 6 and 2/3 | 6.666… |
Every row can be checked the same way. 14 divided by 3 gives 4 and 2/3, and 4 and 2/3 multiplied by 3 is 12 plus 2, which is 14. 19 divided by 3 gives 6 and 1/3, and 6 and 1/3 multiplied by 3 is 18 plus 1, which is 19.
Why does the decimal repeat at all? A decimal terminates only when the fraction, in its lowest terms, has a denominator whose only prime factors are 2 and 5, the prime factors of 10. The denominator 3 is neither, so a remainder always survives, and the same remainder returning at every step forces the same digit to repeat. By contrast, 1 divided by 4 is 0.25 exactly, because 4 is 2 times 2, and 1 divided by 5 is 0.2 exactly. That is why this guide keeps fractions as the main form: a fraction is always exact, while a decimal for thirds is exact only when written with the dots.
Rounding Repeating Decimals
Rounding replaces a number with a nearby number that has fewer digits, and a rounded value must never be presented as exact. The standard rules are simple. Look at the first digit you are discarding. If it is less than 5, the last digit you keep stays the same. If it is 5 or more, in ordinary school rounding the last digit you keep goes up by 1. Formal rounding guidance, including the rules set out by the National Institute of Standards and Technology in its Guide for the Use of the International System of Units, treats the exact halfway case with a tie rule so that the final kept digit ends up even, which avoids a steady upward bias over many roundings. For the thirds in this guide the tie case never arises, because the discarded digits are 3s or 6s, never an exact 5 followed by zeros.
Worked examples make the rules concrete. 1.333… to two decimal places is 1.33, because the first discarded digit is 3, which is less than 5. 5.333… to two decimal places is 5.33 for the same reason. 0.666… to two decimal places is 0.67, because the first discarded digit is 6, which is more than 5, so the kept digit rises. Thirds ending in .333… round down at any stated precision, while thirds ending in .666… round up.
Two habits keep rounding honest. First, state the precision: 1.33 to two decimal places is a complete, honest answer, while a bare 1.33 offered as the answer to 4 divided by 3 claims an exactness it does not have. Second, round only at the final step. Keep the fraction through the working and convert at the end, so the rounding error appears once, at a known size, instead of adding up through a chain of calculations.
Frequently Asked Questions
What is 4 divided by 3?
4 divided by 3 is 1.333…, which is the fraction 4/3 or the mixed number 1 and 1/3. The remainder is 1. Multiplied back, 4/3 times 3 is 4. Rounded to two decimal places, the answer is 1.33, but that rounded form is not exact.
What is 10 divided by 3?
10 divided by 3 is 3.333…, which is the fraction 10/3 or the mixed number 3 and 1/3. The remainder is 1. Multiplied back, 10/3 times 3 is 10. Rounded to two decimal places, the answer is 3.33.
What is 12 divided by 3?
12 divided by 3 is 4 exactly, with a remainder of 0. 12 is a multiple of 3, since 4 times 3 is 12, so there is no fractional part and no decimal places. As a percentage, the answer is 400 percent.
What is 16 divided by 3?
16 divided by 3 is 5.333…, which is the fraction 16/3 or the mixed number 5 and 1/3. The remainder is 1. Multiplied back, 16/3 times 3 is 16. In time units, the answer is 5 hours and 20 minutes, because one third of an hour is 20 minutes.
What is 3 divided by 3?
3 divided by 3 is 1 exactly, with a remainder of 0. Any nonzero number divided by itself equals 1. Multiplied back, 1 times 3 is 3. As a percentage, the answer is 100 percent.
What is 1 divided by 1/2?
1 divided by 1/2 is 2. Dividing by a fraction means multiplying by its reciprocal, and the reciprocal of 1/2 is 2, so 1 times 2 is 2. In grouping terms, two halves fit into one whole. Multiplied back, 2 times 1/2 is 1.
Why do divisions by 3 so often repeat forever?
Because the denominator 3 has a prime factor other than 2 and 5. A fraction in lowest terms gives a terminating decimal only when its denominator is built solely from 2s and 5s, the factors of 10. With a denominator of 3, the long division remainder never reaches 0 and the same remainder returns at every step, so the same digit, 3 or 6, repeats without end.
What is the difference between a remainder and a fraction?
They describe the same leftover in two forms. In 10 divided by 3, the remainder form says 3 remainder 1: three whole groups with 1 unit left over. The fraction form shares that leftover out too, writing it as 1/3 of the divisor, giving 3 and 1/3, the complete exact quotient.
Is 1.33 the same as 4 divided by 3?
No. 1.33 is 4 divided by 3 rounded to two decimal places. The exact answer is the fraction 4/3 or the repeating decimal 1.333… You can see the difference by multiplying back: 1.33 times 3 is 3.99, not 4. The missing 0.01 is rounding error.
How do you check a division answer?
Multiply the quotient by the divisor. If the division was correct, the product is the dividend. For 16 divided by 3, 16/3 times 3 is 16. Do the check with the fraction or the full repeating decimal, not with a rounded decimal, or the check will appear to fail by a small rounding amount.
What is 2 divided by 3?
2 divided by 3 is 0.666…, which is the fraction 2/3. The remainder is 2, and a remainder of 2 with a divisor of 3 gives the fractional part 2/3. Rounded to two decimal places, the answer is 0.67, because the first discarded digit is 6, which is more than 5.
Keep reading: How Many Weeks in a Year? 52 Weeks Plus the Leftover Days
Keep reading: How Many Seconds in an Hour? The Simple Answer, Worked Out
Sources & references
- National Institute of Standards and Technology, NIST Guide to the SI, Appendix B: Conversion Factors, sections B.7, B.7.1 and B.7.2, Rounding numbers and rounding converted numerical values of quantities. Used for the principal rules for rounding numbers: when the digits to be discarded begin with a digit less than 5, the preceding digit is unchanged; when they begin with a 5 followed by a nonzero digit, the preceding digit is increased by 1; and when they begin with an exact 5 followed only by zeros, the preceding digit is left unchanged if even and increased by 1 if odd, so the final digit is always even. Also used for the principle that a converted or rounded value should be stated to a precision consistent with its accuracy. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors
