While some people might be breathing deeply into a paper bag at the thought of calculating with fractions, if you understand each step and why it's necessary, it can become a piece of cake. Or 1/6 of a cake, if you like.
A rule to remember:
Whether you're adding, subtracting, multiplying or dividing fractions, you should always aim for the "tidiest" answer possible. Tell your fraction if "its bum looks big in this" and simplify it down to the smallest option (e.g. 3/6 becomes 1/2, which retains the same proportions but is less clumsy).
How to add and subtract fractions
- Check: do your denominators match?
- If not, multiply them together (and balance the numerators accordingly)
- Add or subtract using the numerators, keeping the denominator the same.
For an adding or subtracting task, you need to make "common denominators". The hardest thing about common denominators is trying to say it. Go on, recite it quickly ten times and come back to read the rest of the article when you've finished crying.
What are denominators and numerators?
A denominator is the number hanging out below the fraction line, so "common denominator" just means that you need all those numbers in the sum to be the same as each other. The number sitting on top of the fraction line is called the numerator.
When it comes to adding and subtracting fractions, it's very simple once you've checked for (or created, if necessary) a common denominator.
Step 1) Check: do your denominators match?
If you're very lucky, your denominators will already be the same, so you just add together the numerators from the top halves and keep your existing denominator under the line.
Example: 1/4 + 1/4 = 2/4
. And it's the same deal for a subtraction sum, except you subtract the numerators instead:
Example: 2/3 – 1/3 = 1/3
Step 2) If denominators do not match
If the denominators are not already the same number (like if you want to add 1/4 + 2/3), you'll need to multiply them together. The figure you get from the multiplication becomes the common denominator, and therefore forms the bottom part of each fraction in the sum.
Example: If you need to do 1/4 + 2/3, make a common denominator:
4 x 3 = 12. Now 12 goes on the bottom of each fraction.
Now you're working with the same number under each fraction. And of course, you need to adjust the numerators accordingly, or your fractions won't be equivalent. Think about it. If you paid for 1/4 of a cheesecake, you would be furious – and hungry – if you were just handed 1/12 of the cheesecake. Those aren't equivalent, so you need to adjust the numerator so it's proportionately the same size.
You do this by multiplying the numerator by the same figure you used to multiply the denominator. Look:
1/4 becomes 3/12 (both parts, top and bottom, have been multiplied by 3)
And 2/3 becomes 8/12 (both parts, top and bottom, have been multiplied by 4)
Step 3) Add or subtract your numerators
So, you've made sure that there's a common denominator, and that nobody's ripped you off with a smaller piece of cake (i.e. you adjusted your numerator accordingly). Now you're now free to do your simple sum, adding or subtracting, using the numerators. Leave your common denominator as it is; that doesn't change now. That's it!
Example: SUM: 1/4 + 2/3
Multiply your denominators together (4 x 3 = 12), and adjust the numerators proportionately: 3/12 + 8/12
Do the sum on top of the fraction = 11/12
Does my bum look big in this? Nope. This is the smallest this fraction can be.
Let's try another example.
Example: SUM: 3/4 – 1/8
Multiply your denominators together (4 x 8 = 32), and adjust the numerators proportionately: 24/32 – 4/32
Do the sum on top of the fraction = 20/32
Does my bum look big in this? Sorry, yeah it does a bit. 20/32 can be simplified down to 5/8. Much better!
Education link: Check out the fantastic math worksheets for adding fractions and subtracting fractions at DadsWorksheets.com.
How to multiply fractions
When you want to multiply fractions together, it's beautifully simple.
- Multiply the numerators together: anything sitting on top of the fraction gets multiplied together. This becomes the numerator of your answer.
- Multiply the denominators together: anything lurking below the line in the fraction gets multiplied together. This becomes the denominator in your answer.
- Does my bum look big in this? Shrink the fraction to the smallest denominator possible.
Example: 4/5 x 1/4 = 4/20 = simplified is 1/5
Education link: Check out the worksheets for multiplying fractions at DadsWorksheets.com.
How to divide fractions
[TIP] When dividing fractions, you don't ever have to do any division. Weird, right? Instead, we are going to multiply. It makes more sense if you consider that division is the opposite of multiplication, so we flip one of the fractions upside down to compensate for this.
The first thing you want to do is make it a multiplication sum. Get rid of that divide symbol and replace with an X.
But now it's a totally different sum, right? It's going to make a bigger number instead of a smaller one? Aha! Well.
Flip the second fraction upside down, putting the old denominator on TOP of the line and the numerator underneath.
Let's say you have the following fraction sum:
Keep the first fraction the same:
Change the ÷ to x:
1/2 x 3/4
And flip the second fraction upside down.
So your calculation (and you know how to multiply from the previous section) is:
1/2 ÷ 3/4
1/2 x 4/3
Does my bum look big in this? No, you look wonderful, 2/3! You're perfectly simple as you are.
Education link: Check out the worksheets for dividing fractions at DadsWorksheets.com.
Calculations complete. Should you wish to check your answers, we have a handy fractions calculator that you can use.
Please rate this article below. If you have any feedback on it, please contact me.
Before you can go on to master more advanced concepts in algebra and geometry, you need to first master all mathematical functions relating to fractions. In this article, we will review how to add, subtract, multiply, and divide two fractions as well as a fraction and an integer. We’ll also introduce complex fractions along with methods for simplifying them. Before your proceed though, make sure you fully understand the four basic mathematical operations: adding, subtracting, multiplying and dividing.
o Common denominator
o Complex fraction
o Understand how to interpret fractions that involve negative numbers
o Recognize and simplify complex fractions
Now that we have developed a solid foundation regarding what fractions are as well as some different types of fractions, we can now turn to application of the basic arithmetic operations (addition, subtraction, multiplication, and division) to fractions.
In cases that involve simple numbers, addition and subtraction of fractions is easy enough. For instance, adding one third and one third obviously gives us two thirds. Likewise, three fifths minus two fifths is one fifth. The first case is illustrated below.
But what about cases such as one half plus one third?
Notice that adding (subtracting) fractions with the same denominator is very simple-we simply add (subtract) the numerators and divide by the same denominator. We should already know that we can write equivalent fractions that have different numerators and denominators. Thus, if we simply convert one or both of the fractions that we are adding or subtracting into equivalent fractions with the same denominator, then we are able to add the fractions in the simple manner described above. Then, if necessary, we can reduce the result to lowest terms.
The challenge in adding and subtracting fractions is finding a common denominator. The most straightforward approach to finding a common denominator is to simply multiply the two existing denominators and then convert the numerators accordingly to create equivalent fractions. Although this approach is conceptually simple, it can be mathematically difficult when the denominators are large. Nevertheless, let’s try this approach for the purpose of illustration. Consider the addition mentioned above.
A common denominator is 6 (or 23), because we can multiply the numerator and denominator of by 3 to get , and we can multiply the numerator and denominator of by 2 to get . The addition is then straightforward.
Practice Problem: Calculate the result in each case.
Solution: In each case, find a common denominator and convert the terms to equivalent fractions with that denominator. One possible common denominator is given for each case. The sum (difference) of the fractions is the sum (difference) of the numerators over the common denominator. Reduce the result to lowest terms if applicable.
a. Common denominator: 21
b. Common denominator: 8
c. Common denominator: 45
Multiplication and Division
Multiplying and dividing fractions is in some ways simpler than adding and subtracting them. Let’s say we want to multiply by . Intuitively, the answer is fairly obvious: half of a half is a quarter (or one-fourth–). For instance, if you have 50 cents (half of a dollar) and you want to multiply it by a half, then you end up with 25 cents (a quarter of a dollar).
To multiply two fractions, then, simply multiply the numerators and multiply the denominators to get the product. In some cases, the product will already be in lowest terms; in others, you may need to reduce it to lowest terms. For instance, the product of and is the following:
When multiplying a fraction by an integer, note that any integer is simply a fraction with the integer as the numerator and 1 as the denominator. For instance,
Practice Problem: Calculate the following products.
Solution: In each case, the product is the product of the numerators over the product of the denominators. If one of the factors is an integer, treat it as a fraction having the integer as its numerator and 1 as its denominator. Reduce the product to lowest terms if applicable.
Now consider the case of division. Let’s say we want to divide by . Intuitively, the answer is 2–for instance, 25 cents (a quarter dollar) can go into 50 cents (a half dollar) twice.
Notice above that if we were to flip the second factor so that the numerator became the denominator and the denominator became the numerator and also changed the operation from division to multiplication, we would end up with the same result.
This is, in fact, a convenient way to divide fractions. Division by a fraction is the same as multiplication by the reciprocal of that fraction. A reciprocal is simply a “flipped” fraction. Thus, for instance, the reciprocal of is (or ).
As with multiplication of fractions, remember that an integer can also be written as a fraction. Thus, for instance, the reciprocal of 6 is . We can therefore divide fractions by integers as well as by other fractions. In addition, note that the product of a fraction and its reciprocal is always 1. Consider the example below.
In light of how we have defined division and multiplication, we can provide a more rigorous justification of our method for calculating equivalent fractions. Note that the number 1 can be written as any other number divided by itself. For instance,
Thus, the process of finding equivalent fractions is nothing more than multiplying a given fraction by 1! Consider the example below.
Practice Problem: Calculate the following quotients.
Solution: In each case, multiply the dividend by the reciprocal of the divisor. Reduce the product to lowest terms if applicable.
2. Multiply the bottom numbers (the denominators).
3. Simplify the fraction if needed.
Step 1. Multiply the top numbers:
1 2 × 2 5 = 1 × 2 = 2
Step 2. Multiply the bottom numbers:
1 2 × 2 5 = 1 × 2 2 × 5 = 2 10
2 10 = 1 5
Here you can see it with pizza .
Do you see that half of two-fifths is two-tenths?
Do you also see that two-tenths is simpler as one-fifth?
With Pen and Paper
And here is how to do it with a pen and paper (press the play button):
Step 1. Multiply the top numbers:
1 3 × 9 16 = 1 × 9 = 9
Step 2. Multiply the bottom numbers:
1 3 × 9 16 = 1 × 9 3 × 16 = 9 48
Step 3. Simplify the fraction:
9 48 = 3 16
(This time we simplified by dividing both top and bottom by 3)
♫ “Multiplying fractions: no big problem,
Top times top over bottom times bottom.
“And don’t forget to simplify,
Before it’s time to say goodbye” ♫
Fractions and Whole Numbers
What about multiplying fractions and whole numbers?
Make the whole number a fraction, by putting it over 1.
Example: 5 is also 5 1
Then continue as before.
Make 5 into 5 1 :
2 3 × 5 1
Now just go ahead as normal.
Multiply tops and bottoms:
2 3 × 5 1 = 2 × 5 3 × 1 = 10 3
The fraction is already as simple as it can be.
Answer = 10 3
Or you can just think of the whole number as being a “top” number:
Multiply tops and bottoms:
3 × 2 9 = 3 × 2 9 = 6 9
Multiplying fractions is easier than adding and subtracting them. it only has two simple steps and that’s it! In fact, multiplying fractions may be easier for you than multiplying integers, because the numbers you work with in fractions are usually smaller. The other good news is that dividing fractions is as simple as multiplying them. Use the following easy steps to multiply or divide fractions.
Step by step guide to multiply and divide fractions
- Step 1: Multiply the numerators of the fractions by each other to get the numerator of the new fraction.
- Step 2: Multiply the denominators of the fractions by each other to get the denominator of the new fraction.
- Tip: Sometimes when multiplying fractions, you have the opportunity to simplify fractions. For example, when the numerator and denominator of a fraction are both even numbers, this fraction can be simplified.When multiplying fractions, you can usually simplify the fraction by removing equal factors in the numerator and denominator of the fraction. Crossing out equal factors will make the numbers smaller for you, making it easier to work with those numbers. You also get rid of simplifying the fraction at the end of the multiplication.
How to simplify fractions during multiplication operations:
- Step 1: If the numerator of one fraction and the denominator of another fraction are the same number, change both of these numbers to (1) and crossing out the corresponding numbers.
- Step 2: When the numerator of one fraction and the denominator of another fraction are both divisible by the same number, factor that number from both (both the numerator and the denominator). In other words, divide the numerator and the denominator by that common factor.
You know that inverse division is multiplication. So to divide fractions:
- Step 1: Write the first fraction in the same way as the original
- Step 2: The division sign is converted to multiplication sign
- Step 3: The second fraction is reversed and fliped the numerator and denominator
- Step 4: Now you can slove like multiplication of fractions
Multiplying and Dividing Fractions – Example 1:
Multiply fractions. (frac<2> <5>times frac<3><4>=)
Multiply the top numbers and multiply the bottom numbers.
(frac<2> <5>times frac<3><4>= frac<2 times 3><5 times 4>=frac<6><20>) , simplify: (frac<6> <20>= frac<6 div 2><20 div 2>=frac<3><10>)
Multiplying and Dividing Fractions – Example 2:
Divide fractions. (frac<1> <2>div frac<3><5>=)
Keep first fraction, change division sign to multiplication, and flip the numerator and denominator of the second fraction. Then: (frac<1> <2>times frac<5> <3>= frac<1 times 5><2 times 3>=frac<5><6>)
Multiplying and Dividing Fractions – Example 3:
Multiply fractions. (frac<5> <6>times frac<3><4>=)
Multiply the top numbers and multiply the bottom numbers.
( frac<5><6>×frac<3><4>=frac<5×3><6×4>=frac<15><24>), simplify: (frac<15><24>=frac<15÷3><24÷3>=frac<5><8>)
Multiplying and Dividing Fractions – Example 4:
Divide fractions. (frac<1> <4>div frac<2><3>=)
Keep the first fraction, change the division sign to multiplication, and flip the numerator and denominator of the second fraction. Then: (frac<1><4>×frac<3><2>=frac<1×3><4×2>=frac<3><8 >)
Hey guys! Welcome to this video on how to cross multiply fractions.
When cross multiplying fractions, the name sort of hints at how this is actually done.
You literally multiply across. Let’s say you have two fractions that are set equal to each other. So let’s say, (frac=frac
Well, to cross multiply them, you multiply the numerator in the first fraction times the denominator in the second fraction, then you write that number down. Then you multiply the numerator of the second fraction times the number in the denominator of your first fraction, and you write that number down.
The reason we cross multiply fractions is to compare them. Cross multiplying fractions tells us if two fractions are equal or which one is greater. This is especially useful when you are working with larger fractions that you aren’t sure how to reduce.
How to Cross Multiply
Cross Multiply Fractions – Example 1
Find which of the two fractions is greatest.
So, when we cross multiply it, when we set it equal, and then cross multiply these two fractions together, we get 128. So (4times 32=128) . And when we cross multiply these two, we get (7times 26=182) . So, we know that (frac<7><32>) is greater than (frac<4><26>) because 182 is greater than 128.
We must always remember that the number that we multiplied with our numerator represents that corresponding fraction. So this number (128) is representing this fraction ((frac<4><26>)), and this number (182) is representing this fraction ((frac<7><32>)). I mention this, because it may be a little confusing to see numbers taken from two different fractions being multiplied together, but the product only representing one of the fractions and not the other. 128 goes on the left side to represent (frac<4><26>) and (7times 26=182) goes on the right side to represent this fraction right here ((frac<7><32>)).
Cross multiplying fractions helps us to see if numbers are equal, and if not, which is bigger and which is smaller. But that is not its only use. Cross multiplying fractions can help us to solve for unknown variables in fractions.
Cross Multiply Fractions – Example 2
Let’s say we have two fractions (frac<9><16>=frac
I hope that this video over cross multiplying fractions has been helpful to you.
See you guys next time!
Frequently Asked Questions
How do you cross multiply fractions?
Cross multiply fractions by multiplying the denominator of one fraction with the numerator of the other fraction and then comparing the two values. The fraction with the larger value is the larger fraction.
*Note: Be careful to always go from denominator to numerator! If you go numerator to denominator, you will get the wrong fraction as the one that is greater.
Example: Which fraction is greater: (frac<4><5>) or (frac<3><8>)?
(frac<4><5>) is greater than (frac<3><8>).
When do you cross multiply fractions?
Cross multiply fractions when you want to determine if one fraction is greater than another, or if you are looking for a missing numerator or denominator in equivalent fractions.
How do you cross multiply fractions with different denominators?
Almost all fractions being cross multiplied will have different denominators. This does not affect the process at all. Cross multiply as normal.
Why does cross multiplying fractions work?
Cross multiplying fractions to determine if one is greater than the other works because it is a shortcut for converting the fractions to a common denominator and comparing fractions.
Ex. Which is greater: (frac<7><9>) or (frac<8><12>)?
84>72, so (frac<7><9>)>(frac<8><12>)
Convert and Compare:
84>72, so (frac<7><9>>frac<8><12>)
Cross multiplying fractions to find a missing numerator or denominator for equivalent fractions works because it is a shortcut for rearranging to isolate the variable.
Ex. Solve for x: (frac<3><7>=frac<2>
Can you cross multiply when adding fractions?
No, you cannot cross multiply when adding fractions. Cross multiply only when you need to determine if one fraction is greater than another, or if you are trying to find a missing numerator or denominator in equivalent fractions.
How do you cross multiply fractions with variables?
Cross multiply fractions with variables by multiplying opposite numerators and denominators of equivalent fractions, setting the values equal to one another, and solving for the variable.
Which fraction is larger: (frac<17><29>) or (frac<12><15>)?
The two fractions are equal
Cannot be determined from given information
The fraction (frac<12><15>) is larger than (frac<17><29>). Cross multiplication can be used to answer this question. First, multiply 17 by 15. (17times15=255), so write 255 above (frac<17><29>), like this:
Next, multiply 29 by 12. (29times12=348), so write 348 above (frac<12><15>), like this:
348 is larger than 255, so (frac<12><15>) is larger than (frac<17><29>).
Which of the following expressions is correct?
Cross multiplying each fraction pair results in this:
(frac<^<72>12><13>text< >frac<5^<65>><6>rightarrowfrac<12><13>>frac<5><6>text< because >72>65)
(frac<^<21>3><8>text< >frac<4^<32>><7>rightarrowfrac<3><8>) frac<6><17>text< because >51)>(30)
Therefore, (frac<3><8>) Hide Answer
Solve for the missing variable.
Fractions (or common fractions ) are used to describe a part of a whole object. There are several notations for fractions:
Fractions like 1/4 and 2/8 that have the same value are said to be equivalent fractions . This example suggests the following method for testing if two fractions are equivalent.
For example, here is how to reduce the fraction 24/42 is its simplest equivalent fraction, namely 4/7:
Improper fractions, mixed fractions and long division
Set up the long division format, namely .
Therefore we have actually shown that .
Some special fractions
- . Any number n can be turned into a fraction by writing it over a denominator of 1.
- . Anything divided by itself equals 1. We call this a UFOO (a u seful f orm o f o ne). More on UFOOs later.
- If the numerator of a fraction is a multiple of the denominator then the fraction is equal to a whole number. An example is .
- is undefined for any numerator n . Division by zero is not allowed in mathematics.
- . A zero numerator is not a problem. This fraction equals 0.
Adding or subtracting fractions
Fractions that have the same denominator are called like fractions . If you think about this example, then the following procedure for adding or subtracting like fractions is obvious:
But what if the fractions don’t have a common denominator? The answer is that they must then be converted to equivalent fractions that do have a common denominator. The procedure is illustrated in this example:
- Find the lowest common multiple of the two denominators 24 and 30. When applied to fractions this number is called the lowest common denominator (LCD). In this example the LCD is 120.
Then simplify by reducing the new fraction to lowest terms.
To multiply a fraction by a whole number, just multiply the fraction’s numerator by the whole number to get the new numerator, like this:
Then simplify by reducing the new fraction to lowest terms.
Here is an example of why the first procedure works. Suppose that there is half a pie (the fraction 1/2) as shown on the left. Now suppose that you take 2/3 of that half pie. (The word “of” translates into the mathematical operation “multiply”.) This means that you cut the half pie into 3 equal pieces and take 2 of them. The result is 2/6 of the pie.
Reciprocals and dividing fractions
4/5 and 5/4 are reciprocals because
8 and 1/8 are reciprocals because
The key is that instead of seeing a fraction divided by a fraction, look for a single fraction whose numerator and denominator just happen to be fractions. In the first step we multiplied this fraction by a UFOO whose numerator and denominator just happen to be fractions. The UFOO was chosen so the fractions in the denominator would cancel and give 1. After another simplification that left only the final multiplication of fractions.
Example 1: A fraction divided by a fraction :
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We understand that fractions may not be easy for the first time you look at them. The reality is that many math students keep struggling with fractions simply because they failed to understand its basics.
Simply put, a fraction is nothing more nor something less than a division. The number at the top is called the numerator and the number at the bottom is called the denominator. So, when you have the fraction 9/7, this is the same as stating that 9÷7.
Now that you understand the concept of fractions, then we can move on to learn how to multiply fractions.
How To Multiply Fractions
Simply put, whenever you need to multiply fractions, you just need to follow 3 simple steps:
Step #1: Multiply the numerators (or top numbers)
Step #2: Multiply the denominators (or bottom numbers)
Step #3: When it’s needed, we need to simplify or reduce the fraction.
One of the main differences between adding fractions and multiplying fractions is that you don’t have to reduce them to the same denominator. You can simply multiply them as they are, even when they have different denominators.
Let’s check an example. Let’s say that you want to know the result of 2/6 x 9/16.
So, you will need to follow the steps that we mentioned above:
Step #1: Multiply the top numbers (numerators):
Step #2: Multiply the bottom numbers (denominators):
So, at the moment, we have:
Step #3: Simplify the fraction:
18/96 dividing by 3, you get: 6/32
If you divide the fraction by 2, you get: 3/16
Multiplying Fractions With Whole Numbers
Multiplying whole numbers and fractions may stump your students because it seems that there is only one fraction and not 2. So, it is important to keep in mind that you can transform any whole number into a fraction. For example, 4 is exactly the same as 4/1. With this in mind, we believe that you won’t ever struggle when you need to multiply a fraction by a whole number.
Let’s take a look at an example. Imagine that you want to know the result of 2 × ⁵⁄₁₃.
The first thing that you need to do is to rewrite the whole number as a fraction:
Now, you just need to follow the same steps described above:
#1: Multiply numerators: 2 × 5 = 10
#2: Multiply denominators: 1 × 13 = 13
#3: New fraction: ¹⁰⁄₁₃
Multiplying Improper Fractions
Simply put, there is no difference in terms of multiplication of proper and improper fractions. Nevertheless, when some students see an improper fraction, they simply block. So, let’s take a look at a practical example.
Multiplying numerators and denominators to get the LCD in all fraction denominators[ = frac<-1 times 16> <5 times 16>– frac<1 times 5> <16 times 5>+ frac<3 times 10> <8 times 10>]Then rewriting the equation with the equivalent fractions[ = frac<-16> <80>– frac<5> <80>+ frac<30> <80>]With like denominators we can operate on just the numerators[ = frac<-16 - 5 + 30> <80>][ = frac<9> <80>]
Add and subtract proper and improper fractions with this calculator and see the work involved in the solution.
Select the number of fractions in your equation and then input numerators and denominators in the available fields. Click the Calculate button to solve the equation and show the work.
You can add and subtract 3 fractions, 4 fractions, 5 fractions and up to 9 fractions at a time.
How to Add and Subtract Fractions
When the Denominators are the Same
When fractions have the same denominators we simply add or subtract the numerators as indicated and place the result over the common denominator. If necessary we can simplify the fraction to lowest terms or a mixed number.
When the Denominators are Unlike or Different
When fractions have unlike denominators the first step is to find equivalent fractions so that all of the denominators are the same. We find the Least Common Denominator (LCD) then rewrite all fractions in the equation as equivalent fractions using the LCD as the denominator. When all denominators are alike, simply add or subtract the numerators and place the result over the common denominator. The resulting fraction can be simplified to lowest terms or written as a mixed number.
How to Work With Negative Fractions
When an equation calls for adding a negative fraction, we can rewrite the equation as subtracting a positive fraction. Likewise, if the equation calls for subtracting a negative fraction, this is the same as adding a positive fraction and can be rewritten this way. This calculator rewrites negative fractions when it shows the work involved in finding the answer.
Simplifying Operations on Negative Numbers
Whether you are working with fractions, whole numbers or decimals, use these guidelines when adding and subtracting positive and negative numbers.
The multiplication of fractions starts with the multiplication of the given numerators, followed by the multiplication of the denominators. Then, the resultant fraction is simplified further and reduced to its lowest terms, if needed.
There is an interesting rhyme to remember the steps given above. "Multiplying fractions is no big problem; Top times top over bottom times bottom. And don’t forget to simplify, Before it’s time to say goodbye." With this, let us now look ahead to learn more about the multiplication of fractions.
|1.||Introduction to Multiplication of Fractions|
|2.||Visual Representation of Multiplication of Fractions|
|3.||Multiplication of Fractions|
Introduction to Multiplication of Fractions
The multiplication of fractions is not like the addition or subtraction of fractions, where the denominator should be the same. Here, any two fractions with different denominators can easily be multiplied. The only thing to be kept in mind is that the fractions should not be in the mixed form, they should either be proper fractions or improper fractions. Let us learn how to multiply fractions through the following steps:
- Multiply the numerators.
- Multiply the denominators.
- Reduce the resultant fraction to its lowest terms.
For example, let's multiply the following fractions: 2/3 × 4/5. We start by multiplying the numerators: 2 × 4 = 8, then, the denominators: 3 × 5 = 15. This can be written as: (2 × 4)/(3 × 5) = 8/15. Now, the product is already in its lowest terms, so we need not reduce it and we give this as the answer.
Visual Representation of Multiplication of Fractions
Now, let us see the visual representation for the multiplication of fractions. Visualizing multiplication of fractions using fractional squares is a very interesting method to understand the concept. Do you know what are fractional squares? Fractional squares are a representation of the given fraction in the form of squares where the numerator is indicated as the shaded portion. Let us see how to multiply a fraction using fractional squares. Let's multiply these two fractions: 2/3 × 1/2.
In the above figure, the fractional square on the left has 2 orange-shaded parts out of 3 equal sections. This orange area represents two-thirds of the fractional square. Similarly, the second shaded area represents half of the fraction square. By multiplying these two fraction squares, we get 2/6. This can be reduced to its simplest form and represented by one part of three squares. Thus, 2/3 × 1/2 = 2/6 = 1/3. Now that you have the insight into multiplying fractions, let us explore this topic further.
Multiplication of Fractions
Multiplication of Fractions with Whole Numbers : Multiplying fractions by whole numbers is an easy concept. Let us consider this example: 4 × 2/3. We will first represent this example using fractional squares. Four times two-thirds is represented as:
Now let us change the improper fraction obtained into a mixed fraction. 8/3 = (2frac<2><3>). Two whole and two thirds which is equal to 8/3. Finally, we get the following representation.
Multiplication of Proper Fractions : Multiplication of proper fractions is the easiest of all. For example, let us take 2/3 × 4/6. Here, 2/3 and 4/6 are proper fractions. To multiply them, we will take the following steps:
- We will multiply the numerators together: 2 × 4 = 8.
- Then multiply the denominators together: 3 × 6 = 18. This can also be written as: (2 × 4)/(3 × 6) = 8/18.
- Then, reduce the resultant fraction to its lowest terms which will be 4/9.
Multiplication of Improper Fractions : Now let us understand the multiplication of improper fractions. We already know that an improper fraction is one where the numerator is bigger than the denominator. When multiplying two improper fractions, we frequently end up with an improper fraction. For example, to multiply 3/2 × 7/5 which are two improper fractions, we need to take the following steps:
- Multiply the numerators and denominators.(3 × 7)/(2 × 5) = 21/10.
- The fraction 21/10 cannot be reduced further to its lowest terms.
- Hence, the answer is: 21/10 which can be written as (2frac<1><10>).
Multiplication of Mixed Fractions : Mixed fractions are fractions that have a whole number and a fraction, like (2frac<1><2>). When multiplying mixed fractions, we need to change the mixed fractions into an improper fraction before multiplying. For example, if the number is (2frac<2><3>), you should change this to (3 × 2 + 2)/3 = 8/3. Let's consider an example. To multiply (2frac<2><3>) and (3frac<1><4>), the following steps can be used:
- Change the given mixed fractions to improper fraction. (8/3) × (13/4)
- Multiply the numerators of the improper fractions, and then multiply the denominators. This will give 104/12.
- Now, reduce the resultant fraction to its lowest terms, which will make it 26/3.
- Further, convert the answer back to mixed fraction which will be: (8frac<2><3>)
Observe that the above example shows how mixed fractions can be represented by fractional squares. The first two blocks show the whole number 2 and the third one represents the fraction 2/3. Now that we have seen the multiplication of fractions in various forms, let us revise the steps which explain how two given fractions are multiplied.The following figure shows the steps of multiplying two mixed fractions.
Here are a few important notes which are helpful in the multiplication of fractions.
What do we use in everyday life that we don’t even realize is a math skill? Fractions! Have you ever had to convert a recipe to a smaller or bigger serving?
What about how many yards someone runs in football? They ran 20 out of the 100 yards. Yes, that’s a fraction too.
But what would you do if someone told you to sit down and work on fractions right now? You probably wouldn’t know where to start adding subtracting multiplying and dividing fractions, right?
Whether you never learned it, weren’t paying attention when you did, or just need a refresher – we’ve got your guide.
And if you read all this just to decide no, I’m not willing to do the math, we’ve got your link to a fractions calculator too!
When you’re dealing with adding subtracting multiplying and dividing fractions, there are few things you want to keep in mind.
One, it’s always better to reduce. Reducing fractions means seeing how many times the top number (numerator) goes into the denominator (bottom number).
For example, if you got the answer of 5 over 25 (5/25) and you did the math correctly, it’s still wrong. The correct answer, reduced down, is 1/5.
Why? Because 5 (the numerator) goes into the denominator (25) five times. So we reduce and get 1/5 from 5/25.
Second, when you’re doing anything with fractions that involve two sets always keep your work organized. With that, you’re ready for our how-to guide below.
The process for adding and subtracting fractions is the same, save for the mathematical function. If both bottom numbers are the same, you’re in luck and can get through the problem quickly.
For equations with the same denominator, all you have to do is add the numerators.
Example: 1/4+2/4= 3/4. If you learn better in words, that same equation looks like this.
Someone divides a pie into four slices onto different plates. (Four is the denominator). There’s one plate with one slice of pie (1/4) and one plate with two slices of pie (2/4).
If I tried to put them back in the pie dish, how much pie would I have left? Well, we put the one slice (out of the original four) in to represent 1/4.
Then we put the other two slices of the original pie in – so now there are three pieces of the original four in the pan.
To keep from having too many numbers, let’s stay with our pie example. The idea of subtracting fractions is the same- only worry about the numerators when the denominators are equal.
So, if someone took one slice out of the three slices we had in the pie tin from the previous question out, how many are left?
3/4 – 1/4 = 2/4. Is that our final answer?
No, if you caught the trick, 2/4 is too big of a number because we can reduce it down. Two goes into four twice, so the actual answer is 1/2.
We have 1/2 of the pie left, which is the same thing as 2/4.
When things get tricky is when you don’t have the same bottom numbers. When that happens, we have to make the denominators the same, by multiplying one set to match the number in the other.
Let’s say we have an equation that asks us to add 1/3 and 1/6. We can’t just add through and get 2/9. That’s incorrect. We need to make three matches the six.
Thankfully, we picked easy numbers and know that 3 goes into six twice.
So, we look at 1/3 and multiply BOTH TOP AND BOTTOM by two. Now we have 2/6 instead of 1/3.
The new equation is 2/6 + 1/6 which equals 3/6. Reduce it down to 1/2 and you get your final answer.
The exact same is true for subtraction. If we have 1/3 – 1/6 we do all the exact steps as above. We multiply 1/3 by two and get 2/6 – then, instead of adding, we subtract.
So, 2/6 – 1/6 = 1/6 and that’s as low as our fraction can go – so there’s no need to reduce.
Multiplying fractions is almost easier than adding and subtracting them because you don’t need to find a common denominator.
To multiply two fractions, you multiply across. Let’s list the steps, then do an example.
- Reduce fractions if needed
- Set up equation
- Multiply across
Example: Multiply the fractions 2/10 and 2/3
- Reduce the fraction 2/10 to 1/5.
- Now our equation is 1/5 X 2/3
- Multiply the top numbers (1 x 2 =2) and the bottom (5 x 3 = 15)
The answer you get is 2/15. Two doesn’t go into 15 evenly, so we can’t reduce.
Ok, hang on to your hat here. We promise that dividing fractions isn’t hard – but there are two extra steps you have to remember.
Instead of dividing the numbers, you’re going to multiply. Let’s talk through how a division problem becomes multiplication.
The way to remember and work through this problem is to leave, switch, flip.
If you have 1/3 divided by 2/5 you’re eventually going to get 5/6. Here’s what we did.
- Leave 1/3 alone. That’s the leave of leave, switch, flip.
- Switch the division sign to multiplication.
- FLIP the second fraction upside down. That means 2/5 becomes 5/2.
Now that you have 1/3 x 5/2 you can do the multiplication process from above. 1 x 5 = 5 and 2 x 3 = 6.
The answer is 5/6.
Mastering Adding Subtracting Multiplying and Dividing Fractions
Like so many things in math, the key to learning how to do all these functions is practice. You’re not going to remember leave, switch, flip until you’ve said it in your head and done it a few times.
If you need more equations than your workbook or teacher provides, you can always ask them for more help adding subtracting multiplying and dividing fractions worksheets.
Keep practicing and you’ll get better – we promise! Or you could just use our fraction calculator. That’s up to you and your teacher.