Common Fractions: Adding, Reducing and Practising Them

Homework on fractions looks harmless until the exercise suddenly has three different denominators in it. Now you are asked to find a common divisor, bring the fractions to a common form and then check yourself. Adult tasks look exactly the same: split the restaurant bill four ways, work out how much of the cake is left, turn a 15 % discount into a fraction and add it to another discount. Below all of that is worked out step by step, with numbers you can verify by hand.

A quick answer to most of these tasks comes from the fraction calculator. It handles addition, subtraction, multiplication and division, shows the solution step by step with the LCM and the GCD, and returns the answer in three shapes at once: the fraction in lowest terms, the decimal value and the mixed number. A zero denominator and a division by zero are caught with a separate message.

What a fraction is and what kinds exist

A fraction is written as two numbers separated by a bar: a/b. The numerator a sits on top, the denominator b below. The denominator tells you how many equal parts the whole was divided into, and the numerator tells you how many of those parts were taken. The fraction 3/4 means three quarters. If the numerator is smaller than the denominator the fraction is called proper, and if it is larger, improper.

The denominator cannot be zero. A fraction with a zero denominator has no meaning, because you cannot divide a whole into parts by zero, and in calculation such a value also triggers a division error. Zero is fine in the numerator: 0/5 is zero, because zero parts out of five is zero.

• a common fraction: 3/4, the numerator above the bar and the denominator below it;
• a decimal fraction: 0.75, the same value written with a comma or a point;
• a mixed number: 1 3/4, a whole part and a proper fraction side by side;
• a percentage: 75 %, which is 75 out of 100, and as a fraction 3/4.

All four notations describe the same number. It helps to know that in advance, because almost every mistake in this topic comes not from not knowing the rules but from failing to switch between two notations in your head.

Reducing a fraction: finding the common divisor

To reduce a fraction means to divide the numerator and the denominator by the same number. The value does not change, the notation just gets shorter. What you are looking for is the greatest common divisor (GCD), because that is what turns 12/18 into 2/3.

12 = 2² × 3
18 = 2 × 3²

Common factors: 2 and 3, so the GCD = 2 × 3 = 6

12 ÷ 6 = 2
18 ÷ 6 = 3

12/18 = 2/3

A couple more examples to train on: 24/36 divides by 12 and gives 2/3, while 45/60 divides by 15 and gives 3/4. A fraction is called irreducible when the GCD of numerator and denominator equals 1, and then there is nothing left to reduce.

Finding the GCD without a calculator

• if both numbers are even, divide both by 2 and keep going until odd numbers remain;
• factor both numbers into primes and cross the two lists: the common factors are the basis for the GCD;
• for numbers up to a hundred, checking divisibility by 2, 3, 5, 7, 11 and 13 is enough;
• if the denominator is prime, check only that: the GCD with 17 is 17 when the numerator divides by 17, and 1 in every other case.

Bringing fractions to a common denominator

To add two fractions the denominators have to match. When they differ, both fractions are brought to a common denominator. That denominator is the least common multiple (LCM): it divides by both without a remainder and it is smaller than any other number that would work.

• LCM(3, 4) = 12, because 12 divides by both 3 and 4;
• LCM(4, 6) = 12;
• LCM(6, 8) = 24;
• LCM(6, 10) = 30;
• sometimes it is easier to take the product of the denominators, but the result comes out larger and then has to be reduced.

The link between the LCM and the GCD is short: 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, everything agrees. The formula lets you find the LCM once you know the GCD, and the other way round.

Adding and subtracting: step by step

1. Find the LCM of the denominators and bring each fraction to that denominator.
2. Add or subtract the numerators, keeping the common denominator.
3. Reduce the result if it is reducible, and turn it into a proper fraction or a mixed number.

LCM(3, 4) = 12

2/3 = 8/12   (numerator multiplied by 4)
3/4 = 9/12   (numerator multiplied by 3)

8/12 + 9/12 = 17/12

17 divides by 12 once with a remainder of 5:
17/12 = 1 5/12 ≈ 1.4167

Subtraction follows the same algorithm, except that the numerators are subtracted instead of added. Take 1/3 + 1/6: the common denominator is already there, it is 6, so 1/3 = 2/6, and 2/6 + 1/6 = 3/6 = 1/2. Do not forget to reduce at the end: without that step the answer 3/6 looks sloppy even though it is right.

The frequent case of equal denominators is handled separately: there the numerators are simply added and the denominator is left alone. 1/5 + 2/5 = 3/5, and there is no LCM to work out.

Multiplication and division

Multiplication is easier: multiply the numerators, multiply the denominators, then reduce the fraction you get. Dividing by a fraction is taught as multiplication by the reciprocal, because dividing by a/b is the same as multiplying by b/a. That is exactly why 2/3 ÷ 3/4 gives 8/9 and not 6/12.

2/3 × 3/4 = (2 × 3) / (3 × 4) = 6/12 = 1/2

2/3 ÷ 3/4 = 2/3 × 4/3 = (2 × 4) / (3 × 3) = 8/9

Check by multiplying: 8/9 × 3/4 = 24/36 = 2/3

Reducing intermediate products is worth doing. In the multiplication example the threes cancel straight away: 2/3 × 3/4 = 2/1 × 1/4 = 2/4 = 1/2, so the answer arrives faster and without an extra reduction at the end. The trick pays off most with large numbers, where both the numerator and the denominator grow fast.

Mixed numbers and conversion to a decimal

A mixed number is convenient because the whole part is visible at once, but it is awkward in calculation. Converting it to a common fraction is simple: multiply the whole part by the denominator, add the numerator, and write the result over the denominator.

1 5/12 = (1 × 12 + 5) / 12 = 17/12

17/12 = 1 5/12   (17 = 1 × 12 + 5)

3/8 = 0.375   (3 divided by 8)
5/16 = 0.3125

Converting to a decimal means dividing the numerator by the denominator. If the division ends, the fraction is finite: 3/8 = 0.375. A fraction is still finite when its denominator factors only into twos and fives. But 1/3 = 0.333... is infinite, and the answer is written with an ellipsis or with a recurring digit in brackets, 0,(3).

Going back from a decimal to a common fraction follows a single rule: move the decimal point to the left to reach a whole number, that becomes the numerator, and the denominator is 10 raised to the power of the number of decimal places. So 0.75 turns into 75/100, and after reducing by 25 you get 3/4.

Percentages, remainders and the fractional part

• 15 % = 15/100 = 3/20;
• 1/4 = 25 %;
• 3/4 = 75 %;
• 1/8 = 12.5 %;
• one third = 33.3 %.

Converting to percentages is useful for mental arithmetic: 12.5 % is exactly an eighth, so of 800 roubles it is 100 roubles. The way back is quick too: to find the percentage of a sum, multiply it by the percentage divided by 100. Calculations of that kind are covered in the article about percentages without mistakes, and the percentage calculator handles four variants: a percentage of a number, what percentage one number is of another, the change in percent, and increasing or decreasing by a given percentage.

When one whole number is divided by another, you get a quotient, a remainder and a fractional part. The remainder is always smaller than the divisor, otherwise the division could have gone on. And the fractional part is that same remainder written as a fraction with the divisor in the denominator.

17 = 5 × 3 + 2

Quotient: 3
Remainder: 2
Fractional part: 2/5

17/5 = 3 2/5 = 3.4

The same remainder is what shows up in division with a remainder, in problems about divisibility, and in the tests for divisibility by 2, 3 and 5. A statistics calculator is useful when fractions have to be estimated: for a sample it computes the mean, the median, the mode, the minimum, the maximum and the range, and reduced fractions in those results read more precisely than rounded decimals do.

Common mistakes

• Numerators and denominators were added together: 1/2 + 1/3 turned into 2/5. The right answer is 5/6, because the fractions are brought to a common denominator first.
• The fraction was reduced by a partial divisor: 12/18 became 6/9, and those divide by 3 again. You need the GCD, not just any common factor.
• In a division the numerators were multiplied together and the denominators too, and the second fraction was never turned upside down.
• When converting a mixed number it was not clear what to multiply: 1 5/12 came out as 17/2 instead of 17/12, because 12 was multiplied by 5 instead of by 1 with 5 added afterwards.
• Fractions with different denominators were compared by their numerators: 3/7 is larger than 2/7, but you cannot compare 3/7 with 2/5 that way, because 3/7 is about 0.43 while 2/5 is 0.4. Bring them to a common denominator first.
• It was assumed that 0/0 equals zero. That division is not defined at all, and no calculator will produce an answer for it.

Common questions

What is the fastest way to reduce a fraction?

Factor the numerator and the denominator into primes and divide both parts by their product. For 12/18 that is 2² × 3 and 2 × 3², with common factors 2 and 3, giving 6. The second route is shorter when the denominator is prime: then it is enough to check whether the numerator divides by it, and there is nothing else to remove.

Why can you not divide by zero?

Because dividing by b/c is the same as multiplying by c/b, which puts a zero in the denominator. The fraction 0/0 has no value: it is not a number but an indeterminacy, and you can approach it from different directions and get different results. Calculators catch this case in advance and show an error message instead of an answer.

How is a fraction different from a percentage?

They are two notations for the same value. A percentage is a share out of a hundred, while a fraction states the same parts with an arbitrary denominator. 20 % is 20/100, and after reducing it becomes 1/5. Percentages are convenient when you talk about discounts, taxes and growth, while fractions are convenient in exact calculation and in problems about parts. The difference is precision: percentages get rounded, a fraction keeps the value exactly.

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

Divide the numerator by the denominator in long division. 3/8 gives 0.375, because 3 divides by 8 into zero point three, and the remainder is multiplied by 10 again. You get a finite fraction whenever the denominator factors only into twos and fives. If the remainder never ends, the fraction is infinite and the answer is written with an ellipsis or with a recurring digit in brackets.

How do I cut a round cake into 8 equal pieces?

Three ways. First: cut it in half, then cut each half in half, three times over, and you get 2³ = 8 pieces. Second: draw four diameters through the centre at angles of 45 degrees, and you get 8 equal sectors of 45 degrees each. Third: mark eight arcs along the circumference, each of 1/8 of the radius, and join the points to the centre, but that is easy to get wrong, so the first two are the safer methods.

What does the fraction calculator do, and what does it not do?

It computes four operations: addition, subtraction, multiplication and division. It shows the solution step by step: it finds the LCM, brings the fractions to a common denominator, adds the numerators, reduces by the GCD and takes out the whole part. A zero denominator and a division by zero are blocked with an error message. What it does not do is draw graphs, explain what a fraction means in words, or check whether you have understood the topic.