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Multiplying a Whole Number by a Fraction

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Multiplying a Whole Number by a Fraction

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This is "Multiplying Fractions by Whole number" by Mableton Elementary School on Vimeo, the home for high quality videos and the people who love them.

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Engage NY Review Sheets (Pick a few to try)

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Engage NY Review Sheets (Pick a few to try)

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Learning Objectives

Find the common denominator of two or more fractions

Use the common denominator to add or subtract fractions

Simplify a fraction to its lowest terms

Multiply two or more fractions

Multiply a fraction by a whole number

Find the reciprocal of a number

Divide a fraction by a whole number

Divide a fraction by a fraction

Introduction

One of the most common things people forget from arithmetic is how to add and subtract fractions. This section will remind you how to do this. You will also get to practice both adding and subtracting fractions as well as multiplying and dividing them. As you work through the rest of this course, remember to come back to this section for a quick reminder of these skills if the process becomes fuzzy.

Before we get started, here is some important terminology that will help you understand the concepts about working with fractions in this section.

product: the result of multiplication

factor: something being multiplied - for [latex]3 \cdot 2 = 6[/latex] , both 3 and 2 are factors of 6

numerator: the top part of a fraction - the numerator in the fraction [latex]\frac{2}{3}[/latex] is 2

denominator: the bottom part of a fraction - the denominator in the fraction [latex]\frac{2}{3}[/latex] is 3

Note About Instructions

Many different words are used by math textbooks and teachers to provide students with instructions on what they are to do with a given problem. For example, you may see instructions such as "Find" or "Simplify" in the example in this module. It is important to understand what these words mean so you can successfully work through the problems in this course. Here is a short list of the words you may see that can help you know how to work through the problems in this module.

Find

Perform the indicated mathematical operations which may include addition, subtraction, multiplication, division.

Simplify

1) Perform the indicated mathematical operations including addition, subtraction, multiplication, division

2) Write a mathematical statement in smallest terms so there are no other mathematical operations that can be performed-often found in problems related to fractions and the order of operations

Evaluate

Perform the indicated mathematical operations including addition, subtraction, multiplication, division

Reduce

Write a mathematical statement in smallest or lowest terms so there are no other mathematical operations that can be performed-often found in problems related to fractions or division

Adding Fractions

When you need to add or subtract fractions, you will need to first make sure that the fractions have the same denominator. The denominator tells you how many pieces the whole has been broken into, and the numerator tells you how many of those pieces you are using.

The "parts of a whole" concept can be modeled with pizzas and pizza slices. For example, imagine a pizza is cut into 4 pieces, and someone takes 1 piece. Now, [latex]\frac{1}{4}[/latex] of the pizza is gone and [latex]\frac{3}{4}[/latex] remains. Note that both of these fractions have a denominator of 4, which refers to the number of slices the whole pizza has been cut into. What if you have another pizza that had been cut into 8 equal parts and 3 of those parts were gone, leaving [latex]\frac{5}{8}[/latex]?

How can you describe the total amount of pizza that is left with one number rather than two different fractions? You need a common denominator, technically called the least common multiple. Remember that if a number is a multiple of another, you can divide them and have no remainder.

One way to find the least common multiple of two or more numbers is to first multiply each by 1, 2, 3, 4, etc. For example, find the least common multiple of 2 and 5.

First, list all the multiples of 2:

Then list all the multiples of 5:

[latex]2\cdot 1 = 2[/latex]

[latex]5\cdot 1 = 5[/latex]

[latex]2\cdot 2 = 4[/latex]

[latex]5\cdot 2 = 10[/latex]

[latex]2\cdot 3 = 6[/latex]

[latex]5\cdot 3 = 15[/latex]

[latex]2\cdot 4 = 8[/latex]

[latex]5\cdot 4 = 20[/latex]

[latex]2\cdot 5 = 10[/latex]

[latex]5\cdot 5 = 25[/latex]

The smallest multiple they have in common will be the common denominator for the two!

To add fractions with unlike denominators, first rewrite them with like denominators. Then, you know what to do! The steps are shown below.

Adding Fractions with Unlike Denominators

1. Find a common denominator.

2. Rewrite each fraction using the common denominator.

3. Now that the fractions have a common denominator, you can add the numerators.

4. Simplify by canceling out all common factors in the numerator and denominator.

Simplifying a Fraction

Often, if the answer to a problem is a fraction, you will be asked to write it in lowest terms. This is a common convention used in mathematics, similar to starting a sentence with a capital letter and ending it with a period. In this course, we will not go into great detail about methods for reducing fractions because there are many. The process of simplifying a fraction is often called reducing the fraction. We can simplify by canceling (dividing) the common factors in a fraction's numerator and denominator. We can do this because a fraction represents division.

For example, to simplify [latex]\frac{6}{9}[/latex] you can rewrite 6 and 9 using the smallest factors possible as follows:

[latex]\frac{6}{9}=\frac{2\cdot3}{3\cdot3}[/latex]

Since there is a 3 in both the numerator and denominator, and fractions can be considered division, we can divide the 3 in the top by the 3 in the bottom to reduce to 1.

[latex]\frac{6}{9}=\frac{2\cdot\cancel{3}}{3\cdot\cancel{3}}=\frac{2\cdot1}{3}=\frac{2}{3}[/latex]

Rewriting fractions with the smallest factors possible is often called prime factorization.

In the next example you are shown how to add two fractions with different denominators, then simplify the answer.

You can find a common denominator by finding the common multiples of the denominators. The least common multiple is the easiest to use.

In the following video you will see an example of how to add two fractions with different denominators.

You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.

Think About It

Add [latex] \frac{3}{4}+\frac{1}{6}+\frac{5}{8}[/latex]. Simplify the answer and write as a mixed number.

What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together.

Subtracting Fractions

When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. Below are some examples of subtracting fractions whose denominators are not alike.

The example below shows how to use multiples to find the least common multiple, which will be the least common denominator.

In the following video you will see an example of how to subtract fractions with unlike denominators.

h2>Multiply Fractions

Just as you add, subtract, multiply, and divide when working with whole numbers, you also use these operations when working with fractions. There are many times when it is necessary to multiply fractions. A model may help you understand multiplication of fractions.

When you multiply a fraction by a fraction, you are finding a "fraction of a fraction." Suppose you have [latex]\frac{3}{4}[/latex] of a candy bar and you want to find [latex]\frac{1}{2}[/latex] of the [latex]\frac{3}{4}[/latex]:

By dividing each fourth in half, you can divide the candy bar into eighths.

Then, choose half of those to get [latex]\frac{3}{8}[/latex].

In both of the above cases, to find the answer, you can multiply the numerators together and the denominators together.

Multiplying Two Fractions

[latex] \large\frac{a}{b}\cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}=\frac{\text{product of the numerators}}{\text{product of the denominators}}[/latex]

Multiplying More Than Two Fractions

[latex] \large\frac{a}{b}\cdot \frac{c}{d}\cdot \frac{e}{f}=\frac{a\cdot c\cdot e}{b\cdot d\cdot f}[/latex]

To review: if a fraction has common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

For example,

Given [latex] \frac{8}{15}[/latex], the factors of 8 are: 1, 2, 4, 8 and the factors of 15 are: 1, 3, 5, 15. [latex] \frac{8}{15}[/latex] is simplified because there are no common factors of 8 and 15.

Given [latex] \frac{10}{15}[/latex], the factors of 10 are: 1, 2, 5, 10 and the factors of 15 are: 1, 3, 5, 15. [latex] \frac{10}{15}[/latex] is not simplified because 5 is a common factor of 10 and 15.

You can simplify first, before you multiply two fractions, to make your work easier. This allows you to work with smaller numbers when you multiply.

In the following video you will see an example of how to multiply two fractions, then simplify the answer.

Think About It

Multiply [latex] \large\frac{2}{3}\cdot \frac{1}{4}\cdot\frac{3}{5}[/latex]. Simplify the answer.

What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would multiply three fractions together.

Divide Fractions

There are times when you need to use division to solve a problem. For example, if painting one coat of paint on the walls of a room requires 3 quarts of paint and you have a bucket that contains 6 quarts of paint, how many coats of paint can you paint on the walls? You divide 6 by 3 for an answer of 2 coats. There will also be times when you need to divide by a fraction. Suppose painting a closet with one coat only required [latex] \frac{1}{2}[/latex] quart of paint. How many coats could be painted with the 6 quarts of paint? To find the answer, you need to divide 6 by the fraction, [latex] \frac{1}{2}[/latex].

Before we begin dividing fractions, let's cover some important terminology.

reciprocal: two fractions are reciprocals if their product is 1 (Don't worry; we will show you examples of what this means.)

quotient: the result of division

Dividing fractions requires using the reciprocal of a number or fraction. If you multiply two numbers together and get 1 as a result, then the two numbers are reciprocals. Here are some examples of reciprocals:

[latex]\large\frac{3}{4}[/latex]

[latex]\large\frac{4}{3}[/latex]

[latex]\large\frac{3}{4}\cdot \frac{4}{3}=\frac{3\cdot 4}{4\cdot 3}=\frac{12}{12}=1[/latex]

[latex]\large\frac{1}{2}[/latex]

[latex]\large\frac{2}{1}[/latex]

[latex]\large\frac{1}{2}\cdot\frac{2}{1}=\frac{1\cdot}{2\cdot1}=\frac{2}{2}=1[/latex]

[latex] 3=\large\frac{3}{1}[/latex]

[latex]\large\frac{1}{3}[/latex]

[latex]\large\frac{3}{1}\cdot \frac{1}{3}=\frac{3\cdot 1}{1\cdot 3}=\frac{3}{3}=1[/latex]

[latex]2\large\frac{1}{3}=\large\frac{7}{3}[/latex]

[latex]\large\frac{3}{7}[/latex]

[latex]\large\frac{7}{3}\cdot\frac{3}{7}=\frac{7\cdot3}{3\cdot7}=\frac{21}{21}=1[/latex]

Sometimes we call the reciprocal the "flip" of the other number: flip [latex]\large\frac{2}{5}[/latex] to get the reciprocal [latex]\large\frac{5}{2}[/latex].

Division by Zero

You know what it means to divide by 2 or divide by 10, but what does it mean to divide a quantity by 0? Is this even possible? Can you divide 0 by a number? Consider the fraction

[latex]\large\frac{0}{8}[/latex]

We can read it as, "zero divided by eight." Since multiplication is the inverse of division, we could rewrite this as a multiplication problem.

[latex]\text{?}\cdot{8}=0[/latex].

We can infer that the unknown must be 0 since that is the only number that will give a result of 0 when it is multiplied by 8.

Now let's consider the reciprocal of [latex]\frac{0}{8}[/latex] which would be [latex]\frac{8}{0}[/latex]. If we rewrite this as a multiplication problem, we will have

[latex]\text{?}\cdot{0}=8[/latex].

This doesn't make any sense. There are no numbers that you can multiply by zero to get a result of 8. The reciprocal of [latex]\frac{8}{0}[/latex] is undefined, and in fact, all division by zero is undefined.

Caution! Division by zero is undefined and so is the reciprocal of any fraction that has a zero in the numerator. For any real number a, [latex]\large\frac{a}{0}[/latex] is undefined. Additionally, the reciprocal of [latex]\large\frac{0}{a}[/latex] will always be undefined.

Divide a Fraction by a Whole Number

When you divide by a whole number, you are multiplying by the reciprocal. In the painting example where you need 3 quarts of paint for a coat and have 6 quarts of paint, you can find the total number of coats that can be painted by dividing 6 by 3, [latex]6\div3=2[/latex]. You can also multiply 6 by the reciprocal of 3, which is [latex] \frac{1}{3}[/latex], so the multiplication problem becomes

[latex]\large\frac{6}{1}\cdot \frac{1}{3}=\frac{6}{3}=2[/latex].

Dividing is Multiplying by the Reciprocal

For all division, you can turn the operation into multiplication by using the reciprocal. Dividing is the same as multiplying by the reciprocal.

The same idea will work when the divisor (the thing being divided) is a fraction. If you have [latex] \frac{3}{4}[/latex] of a candy bar and need to divide it among 5 people, each person gets [latex] \frac{1}{5}[/latex] of the available candy:

[latex]\large\frac{1}{5}\text{ of }\frac{3}{4}=\frac{1}{5}\cdot \frac{3}{4}=\frac{3}{20}[/latex]

Each person gets [latex]\large\frac{3}{20}[/latex] of a whole candy bar.

If you have a recipe that needs to be divided in half, you can divide each ingredient by 2, or you can multiply each ingredient by [latex]\frac{1}{2}[/latex] to find the new amount.

For example, dividing by 6 is the same as multiplying by the reciprocal of 6, which is [latex]\frac{1}{6}[/latex]. Look at the diagram of two pizzas below. How can you divide what is left (the red shaded region) among 6 people fairly?

Each person gets one piece, so each person gets [latex]\arge\frac{1}{4}[/latex] of a pizza.

Dividing a fraction by a whole number is the same as multiplying by the reciprocal, so you can always use multiplication of fractions to solve division problems.

Divide a Fraction by a Fraction

Sometimes you need to solve a problem that requires dividing by a fraction. Suppose you have a pizza that is already cut into 4 slices. How many [latex]\large\frac{1}{2}[/latex] slices are there?

There are 8 slices. You can see that dividing 4 by [latex]\large\frac{1}{2}[/latex] gives the same result as multiplying 4 by 2.

What would happen if you needed to divide each slice into thirds?

You would have 12 slices, which is the same as multiplying 4 by 3.

Dividing with Fractions

1. Find the reciprocal of the number that follows the division symbol.

2. Multiply the first number (the one before the division symbol) by the reciprocal of the second number (the one after the division symbol).

Any easy way to remember how to divide fractions is the phrase "keep, change, flip." This means to KEEP the first number, CHANGE the division sign to multiplication, and then FLIP (use the reciprocal) of the second number.

When solving a division problem by multiplying by the reciprocal, remember to write all whole numbers and mixed numbers as improper fractions. The final answer should be simplified and written as a mixed number.

In the following video you will see an example of how to divide an integer by a fraction, as well as an example of how to divide a fraction by another fraction.

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Find the common denominator of two or more fractions Use the common denominator to add or subtract fractions Simplify a fraction to its lowest terms Multiply two or more fractions Multiply a fraction by a whole number Find the reciprocal of a number Divide a fraction by a whole number Divide a fraction by a fraction One of the most common things people forget from arithmetic is how to add and subtract fractions.

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Basic Math for Adults/Fractions - Wikibooks, open books for an open world

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When you divide (fractionate) something into parts, two or more, you have what is known as a common fraction. A common fraction is usually written as two numbers; a top number and a bottom number. Common fractions can also be expressed in words.

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Home Answer Booklets Curriculum Overview Curriculum Maps ESLRs Poetry Standards & Benchmarks Student Projects Student Page

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Adding and Subtracting Fractions with Like Denominators

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Adding and Subtracting Fractions with Like Denominators

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4.4: Add and Subtract Fractions with Common Denominators

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To add fractions, add the numerators and place the sum over the common denominator. To subtract fractions, subtract the numerators and place the difference over the common denominator.

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Attributed to: Copyright ©2000 - 2018, vBulletin Solutions Inc. XP Math

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XP Math - Math Games Arcade - Comparing Fractions - FREE

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Mr. Hui's XP Math features free math games aligned to Common Core State Standards: Number Sense and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability.

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Adding Fractions With Like Denominators 

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Adding Fractions With Like Denominators 

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Attributed to: Khan Academy
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So we're asked to add 3/15 plus 7/15, and then simplify the answer. So just the process when you add fractions is if they already-- well, first of all, if they're not mixed numbers, and neither of these are, and if they have the same denominator. In this example, the denominators are already the same. The denominator is 15. So if you add these two fractions, your sum is going to have the same denominator, 15, and your numerator is just going to be the sum of the numerator, so it's going to be 3 plus 7, or it's going to be equal to 10/15. Now, if we wanted to simplify this, we'd look for the greatest common factor in both the 10 and the 15, and as far as I can tell, 5 is the largest number that goes into both of them. So divide the 10 by 5 and you divide the 15 by 5, and you get-- 10 divided by 5 is 2 and 15 divided by 5 is 3. You get 2/3. Now, to understand why this works, let's draw it out. Let's split something up into 15 sections. So let me split it up into 15 sections. Let me see how well I can do this. Well, actually, even a better way, an easier way might be to draw circles. So let me do the 15 sections. So let me draw. So that is one section right over there. That is one section and then if I copy and paste it, that is a second section, and then a third section, fourth section, and then we have a fifth section. Let me copy and paste this whole thing. So that's five sections right there. Let me copy and then paste that. So that is 10 sections, and then let me do it one more time. So that is 15 sections. So you can imagine this whole thing is like a candy bar or something, and we have now split it up into 15 sections. Now, what is 3/15? Well, it's going to be 3 of the 15 sections. So 3/15 is going to be one, two, three: 3/15. Now, to that, were adding 7 of the 1/15 sections, or 7 of the sections. So we're adding 7 of those to it. So that's one, two, three, four, five, six, seven. And you see now, if you take the orange and the blue, you get one, two, three, four, five, six, seven, eight, nine, ten of the sections, or 10 of the 15 sections. And then to see why this is the same thing as 2/3, you can just split this candy bar into thirds, so each third would have five sections in it. So let's do that. One, two, three, four, five, so that is 1/3 right there. One, two, three, four, five, that is another third right there. And notice, when you do it like this, we have filled out exactly two-- one, two-- of the thirds. This is the third third, but that's not filled in. So 10/15 is the same thing as 2/3.

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Adding fractions with like denominators (video) | Khan Academy

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So we're asked to add 3/15 plus 7/15, and then simplify the answer. So just the process when you add fractions is if they already-- well, first of all, if they're not mixed numbers, and neither of these are, and if they have the same denominator.

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We can test if two fractions are equivalent by cross-multiplying their numerators and denominators. This is also called taking the cross-product:

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Comparing fractions means determining which is greater and which is less. For example:

Determine which fraction represents a greater value:

For many students (and adults), answering questions like this causes one of two reactions. Either they get extremely nervous or they go right to their trusty algorithm: finding common denominators. It doesn't have to be this way!

Relying on common denominators is often not the most efficient way to compare fractions. And, an over-reliance on this algorithm can obfuscate much of what makes fractions make sense to students. To help, I am sharing 5 strategies you and your students can use to compare fractions.

Common Denominators

Okay, let's get this one out of the way right off the bat. Finding common denominators works. It really does! And, I totally understand why so many people rely on it. You can almost turn your brain off while using this strategy because the steps are the same each time.

Let's look at the example from the top of this post. I can make common denominators by finding equivalent fractions.

\(\dfrac{7}{13}*\dfrac{11}{11}=\dfrac{77}{143}\)

\(\dfrac{5}{11}*\dfrac{13}{13}=\dfrac{65}{143}\)

Instead of comparing the original fractions, I can compare equivalent fractions that now have the same denominator.

\(\dfrac{77}{143}>\dfrac{65}{143}\)

As promised, this strategy worked. Since both fractions now have 143 as a denominator, each fraction represents partitioning the whole into the same sized pieces. Since \(\dfrac{77}{143}\) has more of the same sized parts, it is greater.

So, that strategy was successful, but took WAY more calculating than was needed. Many people would not feel comfortable doing those calculations without pencil and paper or calculator. The good news is, there is another way. Making equivalent fractions with common denominators can be a great strategy for some comparisons, but I want to share some other strategies you (and hopefully your students too) can use.

Benchmarks

A benchmark is an idea or value you know and understand well that you can compare other things to. For example, I know that my stride is almost exactly 1 yard or 3 feet. So if I want to estimate a length, I can use my stride as a benchmark by counting my steps. Similarly, I know that I can reach about 8 feet in the air without jumping or 9 feet with a small hop. I can estimate short heights by reaching up in the air and comparing to the benchmark of my reach. This is great for estimating ceiling heights which, you know, is pretty important.

For fractions, most students can easily use 1 whole and \(\dfrac{1}{2}\) as benchmarks. Most can tell, for example, that \(\dfrac{6}{10}\) is greater than \(\dfrac{1}{2}\) because \(\dfrac{5}{10}=\dfrac{1}{2}\) and \(\dfrac{6}{10}>\dfrac{5}{10}\).

When two fractions fall on either side of a benchmark, it can be easy to tell which one is larger without manipulating them in any way. Let's return to the same example from above. How do these compare to the benchmark \(\dfrac{1}{2}\) ?

\(\dfrac{7}{13}\) is greater than \(\dfrac{1}{2}\)

\(\dfrac{5}{11}\) is less than \(\dfrac{1}{2}\)

Therefore, \(\dfrac{7}{13}>\dfrac{1}{2}>\dfrac{5}{11}\)

Comparing these fractions to \(\dfrac{1}{2}\) is more efficient because it requires less calculating. More importantly though, this strategy requires students to think about the relative sizes of fractions and their locations on the number line which helps them develop deeper understanding.

Here is a diagram of this comparison on the number line. They aren't placed exactly, but each fraction's location is correct relative to \(\dfrac{1}{2}\). Because we can locate\(\dfrac{5}{11}\) to the left of \(\dfrac{7}{13}\),\(\dfrac{7}{13}\) must be greater:

Common Numerators

If common denominators allow us to compare fractions just by looking at the numerators, then common numerators should allow us to compare fractions by looking only at the denominator. For this strategy to make sense, it is easiest to first compare unit fractions.

Which of these fractions is greater? To compare these, students will often use a sharing argument. Would you rather have one piece of a cake split into 7 pieces, or one piece of a cake split into 256 pieces? The answer is obvious: \(\dfrac{1}{7}\) is larger. But, this strategy works even if the fractions are not unit fractions:

One thirteenth is greater than one nineteenth, so even if you have more than 1 piece: \(\dfrac{4}{19}

More and Larger / Fewer and Smaller

The name really says it all for this strategy. Sometimes one fraction represents larger pieces while also having a greater number of those pieces. Let's look at an example:

\(\dfrac{26}{11}\) Vs. \(\dfrac{25}{12}\)

In this case, \(\dfrac{26}{11}\) has more pieces (26 > 25) while also being made up of larger pieces (\(\dfrac{1}{11} > \dfrac{1}{12}\)). Again, no calculating is necessary in order to see that: \(\dfrac{26}{11}>\dfrac{25}{12}\)

Did you notice that we were working with improper fractions in this example? It doesn't even matter! Also, aren't you glad that you did not need to find common denominators for this one?

Missing Parts

Okay, this is the trickiest one. Sometimes you need to compare fractions that are "missing" the same amount. For example:

Both of these fractions are one part away from being equivalent to 1. So, one might say that "both fractions are missing a single piece." To tell which one is larger, I just need to see which one is missing less.

I know that \(\dfrac{1}{8} < \dfrac{1}{7}\), so that means that \(\dfrac{7}{8}\) is missing a smaller piece and is therefore closer to one.

Going back to sharing can help here too. Would you rather have \(\dfrac{1}{8}\) or \(\dfrac{1}{7}\) removed from your cake? \(\dfrac{1}{8}\) is a smaller piece, so I would much rather give that away! Therefore:

\(\dfrac{7}{8} > \dfrac{6}{7}\)

Here is a model (drawn to scale) of another comparison. Note that elevenths are larger so \(\dfrac{12}{13}\) represents a greater value.

You can even use missing parts when more than 1 part is missing. Here is an example:

\(\dfrac{93}{97}\) Vs. \(\dfrac{60}{64}\)

Using a missing parts argument, I can see that both fractions are missing 4 parts. I can compare the fractions by comparing the missing parts. I know that \(\dfrac{4}{97}\) < \(\dfrac{4}{64}\), so I am missing less from \(\dfrac{93}{97}\). Therefore, it must be greater.

Follow Up:

I want to thank Ben (thanks!), who gave me a ton of feedback to help make this post better

I often make videos for my students that can be watched for review or when students are absent. I sometimes assign them as homework to avoid wasting class time 'lecturing'. Here is a video I made on comparing fractions:

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Comparing fractions means determining which is greater and which is less. For example: Determine which fraction represents a greater value: $$\dfrac{7}{13}$$ Vs. $$\dfrac{5}{11}$$ For many students (and adults), answering questions like this causes one of two reactions. Either they get extremely nervous or they go right to their trusty algorithm: finding common denominators.