Fractions represent a part of an integer and are very useful for making measurements or calculating values with precision. The concept of a fraction or fractional number can be difficult to understand, as it is characterized by specific terminology and precise rules for being applied and used within equations. When you understand all the parts that make up a fraction, you can practice solving mathematical problems in which you will have to add or subtract them. Once you master the process of adding and subtracting fractions, you can go a step further by trying to multiply and divide with fractional numbers.
Steps
Method 1 of 3: Understanding what Fractions are
Step 1. Identify the numerator and denominator
The value at the top of the fraction is known as the numerator and represents the part of the whole value expressed by the fraction itself. The value at the bottom of the fraction represents the denominator and indicates the number of parts that represent the whole. If the numerator is smaller than the denominator, it is called a "proper" fraction. If the numerator is larger than the denominator it is called an "improper" fraction.
- For example, examining the fraction ½, one senses that the number 1 is the numerator, while the number 2 is the denominator.
- Fractions can also be reported on a single line as follows 4/5. In this case the number to the left of the fraction line is the numerator, while the number to the right will always be the denominator.
Step 2. Remember that if you multiply the numerator and denominator by the same number you will get a fraction equivalent to the original one, ie of equal value
Equivalent fractions represent the same value as the original, but use different numerators and denominators from the latter. If you want to calculate a fraction equivalent to the one you are looking at, simply multiply the numerator and denominator by the same number and report the result as a fraction.
- For example, if you want to find an equivalent fraction of 3/5, you have to multiply both the numerator and the denominator by 2 to get the new fraction 6/10.
- Using a real example, if you have two identical slices of pizza, by cutting one in half you will still have a quantity of pizza equal to that of the slice still intact.
Step 3. Simplify a fraction by dividing the numerator and denominator by a common multiple
In many cases you will be required to simplify a fraction to a minimum. If the fraction you are studying has a very large number in both the numerator and denominator, look for a multiple that is common to both. Now divide both the numerator and denominator by the number you have identified to simplify the fraction into a form that is easier to read and understand.
For example, the fraction 2/8 has the numerator and denominator that are divisible by 2. By dividing both values by the number 2, you get the simplified fraction 1/4
Step 4. Convert an improper fraction to a mixed number
Improper fractions have the characteristic of having the numerator greater than the denominator. To simplify an improper fraction, divide the numerator by the denominator to identify the integer part and the fractional part (the remainder of the division) indicated by the fraction itself. As a result it reports the whole part followed by a new fraction in which the remainder represents the numerator while the denominator will remain the same as that of the starting fraction.
For example, if you need to simplify the improper fraction 7/3, start by dividing 7 by 3 to get 2 with the remainder of 1. The mixed number you end up with is 2 ⅓
Advise:
if the numerator and denominator are the same, the fraction always represents the number 1.
Step 5. Return a mixed number as a fraction if you need to use it in an equation
When you need to use a mixed number in an equation, it will be much easier to report it as an improper fraction for calculations. To convert a mixed number to an improper fraction, multiply the integer part by the denominator, then add the result to the numerator.
For instance. To convert the mixed number 5 ¾ into the corresponding improper fraction, start by multiplying 5 by 4 to get 5 x 4 = 20. Now add the value 20 to the numerator of the fraction to get the final result 23/4
Method 2 of 3: Adding and Subtracting Fractions
Step 1. Just add or subtract the numerators if the denominator of the fractions is the same
If all the denominators of the fractions involved are identical, then you can perform the calculations simply by adding or subtracting the numerators from each other. Rewrite the equation so that there is only one denominator and the numerators that are added or subtracted from each other are enclosed in parentheses. Perform calculations to the numerator of the fraction and simplify the final result if necessary.
- For example, if you have to solve the following calculation 3/5 + 1/5, rewrite the equation as (3 + 1) / 5 and perform the calculations resulting in 4/5.
- If you have to solve the following calculation 5/6 - 2/6, rewrite the starting expression as (5-2) / 6 and perform the calculations resulting in 3/6. In this case both the numerator and the denominator are divisible by the number 3, so simplifying the result you will get the final fraction 1/2.
- If there are mixed numbers in the equation, remember to transform them into the equivalent improper fractions before performing the calculations. For example, if you have to do the following calculation 2 ⅓ + 1 ⅓, start by transforming both mixed numbers into improper fractions, resulting in the following expression 7/3 + 4/3. Now rewrite the equation in this way (7 + 4) / 3 and perform the calculations resulting in the fraction 11/3. Now convert the improper fraction into a mixed number, resulting in 3 ⅔.
Warning:
never add or subtract denominators. The denominators of the fractions simply represent the number of parts indicating the unit or the whole, while the numerators represent the parts indicated by the fraction.
Step 2. Find a common multiple if the denominators of the fractions under consideration are different
In most cases you will have to face problems where the denominators of the fractions are different from each other. In this case you will first have to identify a common denominator, otherwise the calculations you will perform will be incorrect. Make a list of the multiples of each denominator until you find one that is in common with all the fractions you are studying. If you can't find a common multiple for all denominators, multiply them and use the product you get.
- For example, if you need to perform the following calculation 1/6 + 2/4, start by creating the list of multiples of the numbers 6 and 4.
- Multiples of 6: 0, 6, 12, 18 …
- Multiples of 4: 0, 4, 8, 12, 16 …
- The least common multiple of 6 and 4 is the number 12.
Step 3. Calculate the equivalent fractions based on the least common multiple to make sure that the denominators are all equal
Multiply the numerator and denominator of the first fraction by the correct multiple, so that the denominator of the new fraction is equal to the least common multiple you found in the previous step. At this point, do the same process with the second fraction of the equation, so that also in this case the denominator is equal to the least common multiple you have identified.
- Continuing with the previous example, 1/6 + 2/4, multiply the numerator and denominator of the first fraction (1/6) by 2 to get 2/12, then multiply the numerator and denominator of the second fraction (2/4) for 3 to get 6/12.
- Rewrite the starting equation as follows 2/12 + 6/12.
Step 4. Then perform the calculations as you normally would
Once you have found a common denominator for all the fractions, you can add or subtract the numerators according to your needs as you normally would. If you can, reduce the final fraction to its lowest terms.
- Continuing with the previous example, you rewrite the starting equation, 2/12 +6/12, in this way (2 + 6) / 12, obtaining as a final result 8/12.
- Simplify the final fraction by dividing the numerator and denominator by 4 to get ⅔.
Method 3 of 3: Multiply and Divide Fractions
Step 1. Multiply the numerators and denominators together separately
When you need to multiply two fractions to calculate the product of two fractions. Start by multiplying the two numerators together and return the result to the numerator of the final fraction, then multiply the two denominators and return the product to the denominator of the final fraction. At this point, simplify the result you have obtained to a minimum.
- For example, if you have to do the following calculation 4/5 x ½, multiplying the numerators will give you 4 x 1 = 4.
- Multiplying the denominators you get 5 x 2 = 10.
- The final result of the multiplication is therefore 4/10. You can simplify it by dividing both the numerator and the denominator by 2 to get 2/5.
- Now try the following calculation: 2 ½ x 3 ½ = 5/2 x 7/2 = (5 x 7) / (2 x 2) = 35/4 = 8 ¾.
Step 2. If you need to divide fractions, start by calculating the reciprocal of the second fraction, ie invert the numerator with the denominator
When dealing with this type of problem with fractional numbers, you need to calculate the inverse of the second fraction, also known as reciprocal. To calculate the reciprocal of a fraction simply invert the numerator with the denominator.
- For example, the reciprocal of 3/8 is 8/3.
- To calculate the reciprocal of a mixed number, start by converting it to the equivalent improper fraction. For example, convert the mixed number 2 ⅓ to the fraction 7/3, then calculate the reciprocal which is 3/7.
Step 3. To perform fraction division, you actually multiply the first number by the reciprocal of the second
Then start by transforming the original problem into a multiplication of fractions, remembering to use the reciprocal of the second fraction. Multiply the numerators together, then calculate the product of the denominators and you will get the final result you were looking for. Minimize the fraction you got if you can.
- For example, if you have to perform the following calculation 3/8 ÷ 4/5, start by calculating the reciprocal of the fraction 4/5 which is 5/4.
- At this point, reset the starting problem as if it were a multiplication using the reciprocal of the second fraction: 3/8 x 5/4.
- Multiply the numerators to get the numerator of the final fraction: 3 x 5 = 15.
- Now multiply the denominators to get 8 x 4 = 32.
- Report the final result as a fraction 15/32.
Advice
- Always simplify the final fraction to the smallest terms, so that it is easier to read and understand.
- Some calculators allow you to perform calculations with fractional numbers. If you have trouble doing the calculations by hand, help yourself with these types of tools.
- Remember that, in the case of addition and subtraction, the denominators must never be added or subtracted from each other.
