Fractions Explained: Adding and Subtracting Different Denominators Online

The homework question sounds short: add 2/3 and 3/4. A confident person reaches for a calculator and gets the answer in a second, but a student learning the topic has to show the working, because that is what gets marked. And then it gets harder: 2/3 + 3/4 turns into a problem with different denominators, where nothing can be added as it stands, and the fractions have to be brought to a common form first.

Here is the whole route: how to reduce fractions, how to find a common denominator, how to add and subtract step by step, what multiplication and division look like, how a mixed number differs from an ordinary fraction, and how fractions, decimals and percentages fit together. With an online calculator the whole thing takes less than a minute, and the understanding stays.

A fraction: numerator, denominator and the one rule

A fraction is written as two numbers separated by a bar, with the numerator on top and the denominator below. The denominator shows how many equal parts the whole was divided into, and the numerator shows how many of those parts were taken. Three quarters is 3/4, twelve twentieths is 12/20, and the notation 12/20 describes exactly the same quantity as 3/4, just less conveniently.

• proper fraction: the numerator is smaller than the denominator, for example 3/4;
• improper fraction: the numerator is larger than the denominator, for example 7/3, which gets converted into a mixed number;
• reducible fraction: numerator and denominator share a divisor greater than one, for example 6/8;
• irreducible fraction: there is no common divisor, for example 3/4, and nothing more can be cancelled;
• the denominator can never be zero, because a whole cannot be divided into zero parts, and in calculations such a value causes a division error;
• zero in the numerator is fine: 0/5 is zero, since zero parts out of five is zero.

A common fraction, a decimal, a mixed number and a percentage are four notations for the same quantity, and nearly every mistake in this topic comes not from not knowing the rules but from failing to switch between the notations in your head. There is a separate piece on ordinary fractions that goes deeper into that, while here the focus is on different denominators.

Step 0: reduce what can be reduced

This step is often skipped and it saves the most time. Reducing a fraction means dividing the numerator and the denominator by the same divisor: the quantity does not change, but the notation becomes shorter and easier to work with.

What you need is the greatest common divisor, the GCD. For small numbers the quick route is to factor both numbers into primes and see what they share.

18 = 2 × 3²
24 = 2³ × 3

Shared factors: 2 and 3
GCD = 2 × 3 = 6

18 ÷ 6 = 3
24 ÷ 6 = 4

18/24 = 3/4

Check: 3/4 = 0.75   and   18/24 = 0.75   matches

You can check the reduction in your head: if multiplying the new numbers by the original GCD gives the original numbers back, the work was correct. A reduced fraction saves time later on, because the common denominator ends up smaller.

Step 1: find a common denominator

To add two fractions the denominators have to match. When they differ, the fractions are brought to a common denominator: a number that divides both denominators without a remainder. The one to use is the least common multiple, the LCM, because it is smaller than any other number that works and smaller numbers are simply easier to handle.

• LCM(3, 4) = 12, because 12 divides by both 3 and 4;
• LCM(6, 4) = 12;
• LCM(2, 6) = 6, and this is the most common case in exercises: one denominator already divides the other;
• LCM(5, 8) = 40;
• if the denominators are coprime you can simply multiply them: LCM(3, 4) = 3 · 4 = 12;
• if the denominators share a factor, multiplying overshoots: 6 · 4 = 24 instead of the correct 12, and the extra cancelling happens later.

The relation between LCM and GCD is short and extremely useful: the product of two numbers equals their GCD multiplied by their LCM. For 6 and 8 that is 6 · 8 = 48, and 2 · 24 = 48. So once the GCD is known, the LCM is a division, and the other way round. An even faster route is to factor both denominators into primes, take the highest power of each prime, and multiply those.

LCM(3, 4):
  factorisation: 3 = 3,  4 = 2²
  take the highest powers: 3 and 2²
  LCM = 3 · 4 = 12

LCM(6, 8):
  factorisation: 6 = 2 · 3,  8 = 2³
  highest powers: 3 and 2³
  LCM = 3 · 8 = 24

Through the GCD:  GCD(6, 8) = 2,  so LCM = 6 · 8 / 2 = 24

LCM(2, 6) = 6, since 6 already divides by 2.

Step 2: adding and subtracting step by step

Once the denominators match, everything comes down to three actions, and the algorithm is the same for addition and subtraction.

1. find the LCM of the denominators and bring each fraction to it by multiplying numerator and denominator by the same number;
2. add or subtract the numerators, leaving the common denominator as it is;
3. reduce the result if it is reducible, and convert it to a proper fraction or a mixed number;
4. optionally convert the answer to a decimal and check it by adding the numerators in the opposite order.

LCM(3, 4) = 12

Bring to a common denominator:
  2/3 = 8/12    (numerator multiplied by 4, denominator as well)
  3/4 = 9/12    (numerator multiplied by 3)

Add the numerators:
  8/12 + 9/12 = 17/12

17 divided by 12 gives 1 with a remainder of 5:
  17/12 = 1 5/12

Answer: 1 5/12, which is about 1.4167

Check in decimals:
  2/3 ≈ 0.6667,  3/4 = 0.75
  0.6667 + 0.75 ≈ 1.4167   matches

Notice the crucial detail: when bringing a fraction to a common denominator, both the numerator and the denominator are multiplied, by the same number. Multiplying only the numerator changes the value of the fraction, and that is the single most common mistake in the topic.

Subtraction follows exactly the same path, with one change at the end: the numerators are subtracted.

5/6 − 1/4:
  LCM(6, 4) = 12
  5/6 = 10/12,   1/4 = 3/12
  10/12 − 3/12 = 7/12
  7 and 12 are coprime, nothing to cancel
  answer: 7/12 ≈ 0.5833

1/2 − 1/6:
  LCM(2, 6) = 6, the denominator is already common
  1/2 = 3/6
  3/6 − 1/6 = 2/6
  2/6 cancels by 2
  answer: 1/3 ≈ 0.3333

When the denominators cooperate, the example simplifies by itself. For 1/2 and 1/6 the common denominator 6 falls out without any LCM at all, because 2 divides 6 exactly. In cases like that the algorithm is short: 1/2 is 3/6, and from there everything is routine.

Multiplication and division

No common denominator is needed here, which is good news, because the rules are different and noticeably simpler.

• when multiplying, multiply the numerators and multiply the denominators, then reduce: 2/3 × 3/4 = 6/12 = 1/2;
• when dividing by a fraction, swap the denominator and numerator of the divisor: 2/3 ÷ 3/4 = 2/3 × 4/3 = 8/9;
• dividing by a fraction is always better than multiplying by the reciprocal, because the cancelling happens earlier;
• dividing by zero is impossible in any system, and in fractions that restriction applies literally.

2/3 × 3/4:
  numerators: 2 · 3 = 6
  denominators: 3 · 4 = 12
  cancel 6/12 → 1/2

2/3 ÷ 3/4:
  invert the divisor: 3/4 → 4/3
  numerators: 2 · 4 = 8
  denominators: 3 · 3 = 9
  8 and 9 are coprime
  answer: 8/9 ≈ 0.8889

Mixed numbers and going back

A mixed number is the same fraction written with the whole part standing on its own and the fraction part staying proper. It is easier to read: 17/12 turns into 1 5/12, and it becomes obvious at once that the whole part is one and the remainder is no bigger than one.

To go back, divide the numerator by the denominator, write down the whole part, and keep the remainder as the numerator over the old denominator. It is worth remembering that a mixed number is not a separate thing but a convenient notation, and that mixed numbers are brought to a common denominator just like ordinary fractions, after being split into whole and fractional parts.

17/12 → mixed number:
  17 ÷ 12 = 1, remainder 5
  1 5/12

1 5/12 → ordinary fraction:
  whole part multiplied by the denominator: 1 · 12 = 12
  add the remainder to the numerator: 12 + 5 = 17
  answer: 17/12

Check by adding mixed numbers:
  1 5/12 + 2 7/12 = 3 12/12 = 4

Decimals, percentages and converting back

A decimal is the same quantity written with a comma or a point. Converting a fraction to a decimal is easiest by dividing the numerator by the denominator, while going back requires care: you cannot simply shift the decimal point, you have to multiply the numerator and the denominator by a power of ten.

To a decimal: 3 ÷ 8 = 0.375

To a percentage: 0.375 · 100 % = 37.5 %

Back from decimal to fraction:
  0.375 = 375 / 1000
  cancel by 125: 375 ÷ 125 = 3,  1000 ÷ 125 = 8
  answer: 3/8

Useful correspondences:
  1/2 = 0.5   = 50 %
  1/4 = 0.25  = 25 %
  3/4 = 0.75  = 75 %
  1/10 = 0.1  = 10 %
  1/8 = 0.125 = 12.5 %

Going back is easiest in two steps: first get rid of the decimal point, then reduce. For 0.375 that is 375/1000, and cancelling by 125 gives 3/8. The cancelling continues until the numerator and the denominator are coprime.

There is one technical detail worth knowing: not every common fraction turns into a terminating decimal. The ones that do have only twos and fives in the factorisation of the denominator, such as 1/2, 1/4, 1/5, 1/8, 1/10, 1/20, 1/25 and 1/100. Fractions such as 1/3, 1/6 and 1/7 never terminate, and the answer is an endless decimal: 1/3 = 0.3333..., 1/6 = 0.1666..., 1/7 = 0.142857142857...

Percentages are calculated separately, and there is a convenient calculator for that: the percentage calculator finds a percentage of a number, works out what percentage one number is of another, shows the change between two values, and handles markups and discounts. When values have to be averaged over a list, the statistics calculator helps: it computes the mean, median, mode and range, and exports the result as CSV.

Typical mistakes

• when bringing fractions to a common denominator only the numerator is multiplied, and the fraction grows several times over;
• the product of the denominators is used instead of the LCM, giving a common denominator twice the size needed, and the cancelling afterwards gets forgotten;
• the denominators get added along with the numerators instead of adding the numerators only;
• the result is not reduced: 17/12 becomes 1 5/12 and 8/12 becomes 2/3, and an unreduced answer looks careless;
• when subtracting, the order of the terms gets confused and the result comes out negative although the answer should be positive;
• nobody checks whether an ordinary fraction can be larger than one, and 7/3 gets treated as a proper fraction;
• when dividing, the divisor is not inverted, and the answer comes out several times too small;
• percentages get converted to fractions by dividing instead of multiplying, so 25 % turns into 0.25 rather than 1/4.

Nearly all of these show up in the final check. Add the resulting fractions in decimal form: 17/12 should come to about 1.4167 and 7/12 about 0.5833. If the decimal value disagrees, the mistake is in the LCM, in bringing the fractions to a common denominator, or in the action itself.

Common questions

What is the fastest way to add fractions with different denominators?

Three actions: find the LCM of the denominators, multiply the numerator and denominator of each fraction by the factor it needs, add the numerators and reduce. The fraction calculator does all of it, showing the LCM, the GCD, the multiplier for each fraction and the step by step solution, and it returns the answer in three forms at once: as an irreducible fraction, as a decimal and as a mixed number.

Why can fractions not simply be added as they are?

Because fractions with different denominators measure different parts of a whole. A third and a quarter cannot be added directly: one fraction splits the whole into three parts, the other into four. Bringing them to a common denominator is the way of switching to one and the same unit of measurement, and only after that does adding make sense.

How do you convert a fraction to a decimal without a calculator?

Divide the numerator by the denominator in long division. There are handy shortcuts for the convenient denominators: 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/8 = 0.125, 1/10 = 0.1. For the others long division produces an endless decimal, and in school work it is normal to stop at a few digits or to mark the repeating part.

What is the difference between 3/4 and 0.75?

There is none apart from the notation. They are the same number, and moving between the two is exact, with no rounding. A difference only appears in two situations: when a fraction does not convert to a terminating decimal, as with 1/3 = 0.333..., and the digits get rounded; and when an already terminating fraction gets rounded, as when 0.666... is rounded to 0.67.

Why reduce a fraction if it can be computed as it is?

Reduction makes the work shorter and lowers the chance of a mistake. Reducing 18/24 to 3/4 means the common denominator with other fractions comes out smaller, and smaller numbers in a calculation mean fewer mistakes. On top of that, the irreducible fraction is the canonical form that answers get compared against.