# 4 ways to calculate with fractions

Operating on fractions is not very complicated, but you have to be very careful about what you are doing. If the product of fractions is simple, their addition (or subtraction) is a bit trickier. As for the division of two fractions, it is necessary to think carefully before embarking on any calculation whatsoever. It's just a matter of practice!

## Steps

### Method 1 of 4: know the two elements of a fraction

#### Step 1. Know what a fraction looks like

It is an expression with 2 numbers separated by a so-called “fraction” line.

#### Step 2. Locate the numerator

It is the value above the fraction mark that represents how many parts of a whole are taken into account.

• In fraction 15 { displaystyle { frac {1} {5}}}

, 1 est le numérateur.

#### Step 3. Locate the denominator

It is the value under the fraction line that indicates how many parts a whole has been divided into.

• In fraction 15 { displaystyle { frac {1} {5}}}

, 5 est le dénominateur, ce qui signifie que le tout de départ est divisé en 5 parties égales.

#### Step 4. See if the fraction is incorrect

By this term is meant a fraction whose numerator is greater than the denominator. In the opposite case, the fraction is said to be “clean”.

• For example, 34 { displaystyle { frac {3} {4}}}

est une fraction propre, tandis que 53{displaystyle {frac {5}{3}}}

est impropre.

• Un entier suivi d'une fraction est appelé « nombre fractionnaire », à l'image de 112{displaystyle 1{frac {1}{2}}}
• , c'est l'autre écriture d'une fraction impropre.

### Méthode 2 sur 4: Additionner et soustraire des fractions

#### Step 1. Find fractions with the same denominator

To add or subtract fractions, you need to make sure they have the same denominator. This is a sine qua non.

#### Step 2. Find a common denominator

With two fractions having different denominators, find a common denominator. One method is to multiply the first fraction by the denominator of the second, and the second fraction by the denominator of the first.

• Calculate the following sum 13 + 25 { displaystyle { frac {1} {3}} + { frac {2} {5}}}

. Pour trouver le dénominateur commun, multipliez le 1 et le 3 de la première fraction par 5, puis le 2 et le 5 de la seconde par 3, ce qui vous donne: 515+615{displaystyle {frac {5}{15}}+{frac {6}{15}}}

. Voilà ! Vous pouvez les additionner.

#### Step 3. Operate on the numerators

Now that you have 2 fractions with the same denominator, you can add or subtract the numerators. The result obtained is placed on a fraction line under which you keep the common denominator.

• Example: 36−26 = 16 { displaystyle { frac {3} {6}} - { frac {2} {6}} = { frac {1} {6}}}

• Les dénominateurs ni ne s'ajoutent ni se soustraient.

#### Step 4. Optionally simplify the fraction

A result must always be given in the form of an irreducible fraction, that is to say simplified. By adding for example 832 { displaystyle { frac {8} {32}}}

et 1232{displaystyle {frac {12}{32}}}

, vous obtenez 2032{displaystyle {frac {20}{32}}}

. Ces deux nombres sont divisibles par 4: 2032=58{displaystyle {frac {20}{32}}={frac {5}{8}}}

### Méthode 3 sur 4: Multiplier des fractions

#### Step 1. Convert mixed numbers to improper fractions

It is often easier to work only with fractions. So, if you need to multiply a fraction with an integer or mixed number, turn these into improper fractions.

• You have to calculate 25 × 7 { displaystyle { frac {2} {5}} times 7}

. Convertissez 7 en une fraction en la ramenant sur 1, soit 71{displaystyle {frac {7}{1}}}

. Faites le calcul: 2×75×1=145{displaystyle {frac {2\times 7}{5\times 1}}={frac {14}{5}}}

• Si vous aviez eu à multiplier la fraction, non par 7, mais par 113{displaystyle 1{frac {1}{3}}}
• , il aurait fallu convertir ce nombre en une fraction impropre, soit, 43{displaystyle {frac {4}{3}}}

#### Step 2. Multiply the numerators and denominators between them

It is much simpler than the addition: you multiply the numerators between them and the denominators of the same… while keeping the line of fraction.

• You must find the result of 13 { displaystyle { frac {1} {3}}}

par 34{displaystyle {frac {3}{4}}}

, Multipliez 1 par 3 (soit 3), multipliez 3 par 4 (soit 12). Votre réponse est: 13×34=312{displaystyle {frac {1}{3}}\times {frac {3}{4}}={frac {3}{12}}}

#### Step 3. Optionally simplify the fraction

A result must always be given in the form of an irreducible fraction, that is to say simplified. Find the greatest common divisor (GCD) of the numerator and denominator, then divide the latter by that divisor.

• Our answer was 312 { displaystyle { frac {3} {12}}}

. Or, le numérateur et le dénominateur sont tous deux divisibles par 3, ce qui donne comme résultat: 14{displaystyle {frac {1}{4}}}

, fraction désormais irréductible.

### Méthode 4 sur 4: Diviser des fractions

#### Step 1. Reverse the second fraction

In fact, dividing one fraction by another is the same as multiplying the first fraction by the inverse of the second. Here you don't have to worry about the denominators.

• You have to find the result of 84 ÷ 12 { displaystyle { frac {8} {4}} div { frac {1} {2}}}

. Vous commencez par prendre l'inverse de 12{displaystyle {frac {1}{2}}}

, soit 21=2{displaystyle {frac {2}{1}}=2}

#### Step 2. Multiply the numerators and denominators between them

Since it's a multiplication, you multiply the fractions line by line and you keep the fraction line between the two.

• So you have to multiply 54 { displaystyle { frac {5} {4}}}

par 21{displaystyle {frac {2}{1}}}

, ce qui vous donne 104{displaystyle {frac {10}{4}}}

#### Step 3. Optionally simplify the fraction

A result must always be given in the form of an irreducible fraction, that is to say simplified… if there is a common divisor for the two numbers!

• 104 { displaystyle { frac {10} {4}}}

peut être simplifiée par 2, ce qui donne comme résultat: 52{displaystyle {frac {5}{2}}}

, fraction désormais irréductible.

• si l’on vous le demande et si la fraction simplifiée est impropre, vous pouvez la convertir en un nombre fractionnaire: 52=42+12=212{displaystyle {frac {5}{2}}={frac {4}{2}}+{frac {1}{2}}=2{frac {1}{2}}}

## conseils

• il peut arriver que vous ayez à opérer avec des fractions un peu compliquées (fractions sur des fractions entre parenthèses…), ne vous laissez pas impressionner !
• soyez très vigilant(e) quand vous écrivez vos fractions, faites attention à ne rien inverser.