Articles

1.5: Equivalent Fractions


  • Equivalent Fractions
  • Reducing Fractions To Lowest Terms
  • Raising Fractions To Higher Terms

Definition: Equivalent Fractions

Fractions that have the same value are called equivalent fractions

For example, (dfrac{2}{3}) and (dfrac{4}{6}) represent the same part of a whole quantity and are therefore equivalent. Several more collections of equivalent fractions are listed below:

(dfrac{15}{25}, dfrac{12}{20}, dfrac{3}{5})

(dfrac{1}{3}, dfrac{2}{6}, dfrac{3}{9}, dfrac{4}{12})

(dfrac{7}{6}, dfrac{14}{12}, dfrac{21}{18}, dfrac{28}{24}, dfrac{35}{30})

Reducing Fractions to Lowest Terms

Reduced to Lowest Terms

It is often useful to convert one fraction to an equivalent fraction that has reduced values in the numerator and denominator. When a fraction is converted to an equivalent fraction that has the smallest numerator and denominator in the collection of equivalent fractions, it is said to be reduced to lowest terms. The conversion process is called reducing a fraction.

We can reduce a fraction to lowest terms by

  1. Expressing the numerator and denominator as a product of prime numbers. (Find the prime factorization of the numerator and denominator. See Section 1.3 for this technique.)
  2. Divide the numerator and denominator by all common factors. (This technique is commonly called “cancelling.”)

Sample Set A:

Example (PageIndex{1})


(egin{aligned}
&dfrac{6}{18}=dfrac{2 cdot 3}{2 cdot 3 cdot 3}
&=dfrac{ ot{2} cdot ot{3}}{ ot{2} cdot ot{3} cdot 3} quad 2 ext { and } 3 ext { are common factors. }
&=dfrac{1}{3}
end{aligned}
)

Example (PageIndex{2})

(
egin{aligned}
dfrac{16}{20} &=dfrac{2 cdot 2 cdot 2 cdot 2}{2 cdot 2 cdot 5}
&=dfrac{ ot{2} cdot ot{2} cdot 2 cdot 2}{ ot{2} cdot ot{2} cdot 5} quad 2 ext { is the only common factor. }
&=dfrac{4}{5}
end{aligned}
)

Example (PageIndex{3})

(
egin{aligned}
&dfrac{56}{70}=dfrac{2 cdot 4 cdot 7}{2 cdot 5 cdot 7}
&=dfrac{ ot{2} cdot 4 cdot ot{7}}{ ot{2} cdot 5 cdot ot{7}} quad 2 ext { and } 7 ext { are common factors. }
&=dfrac{4}{5}
end{aligned}
)

Example (PageIndex{4})

(
dfrac{8}{15}=dfrac{2 cdot 2 cdot 2}{3 cdot 5}
) There are no common factors.

Thus, (dfrac{8}{15}) is reduced to lowest terms.

Raising a Fraction to Higher Terms

Equally important as reducing fractions is raising fractions to higher terms. Raising a fraction to higher terms is the process of constructing an equivalent fraction that has higher values in the numerator and denominator. The higher, equivalent fraction is constructed by multiplying the original fraction by 1.

Notice that (dfrac{3}{5}) and (dfrac{9}{15}) are equivalent, that is (dfrac{3}{5}) = (dfrac{9}{15}). Also,

(
egin{array}{l}
dfrac{3}{5} cdot 1=dfrac{3}{5} cdot dfrac{3}{3}=dfrac{3 cdot 3}{5 cdot 3}=dfrac{9}{15}
1=dfrac{3}{3}
end{array}
)

This observation helps us suggest the following method for raising a fraction to higher terms.

Raising a Fraction to Higher Terms

A fraction can be raised to higher terms by multiplying both the numerator and denominator by the same nonzero number.

For example, (dfrac{3}{4}) can be raised to (dfrac{24}{32}) by multiplying both the numerator and denominator by 8, that is, multiplying by 1 in the form (dfrac{8}{8}).

(
dfrac{3}{4}=dfrac{3 cdot 8}{4 cdot 8}=dfrac{24}{32}
)

How did we know to choose 8 as the proper factor? Since we wish to convert 4 to 32 by multiplying it by some number, we know that 4 must be a factor of 32. This means that 4 divides into 32. In fact, (32 div 4=8). We divided the original denominator into the new, specified denominator to obtain the proper factor for the multiplication.

Sample Set B

Determine the missing numerator or denominator.

Example (PageIndex{5})

(dfrac{3}{7}=dfrac{?}{35} . quad ext{Divide the original denominator, } 7, ext{ into the new denominator }35)

(35 div 7=5)

( ext{Multiply the original numerator by } 5.)

(dfrac{3}{7}=dfrac{3 cdot 5}{7 cdot 5}=dfrac{15}{35})

Example (PageIndex{6})

(dfrac{5}{6}=dfrac{45}{?} . quad ext{Divide the original denominator, } 5, ext{ into the new denominator }45)

(45 div 5=9)

( ext{Multiply the original numerator by } 9.)

(dfrac{5}{6}=dfrac{5 cdot 9}{6 cdot 9}=dfrac{45}{54})

Exercise (PageIndex{1})

(dfrac{6}{8})

Answer

(dfrac{3}{4})

Exercise (PageIndex{2})

(dfrac{5}{10})

Exercise (PageIndex{3})

(dfrac{6}{14})

Answer

(dfrac{3}{7})

Exercise (PageIndex{4})

(dfrac{4}{14})

Exercise (PageIndex{5})

(dfrac{18}{12})

Answer

(dfrac{3}{2})

Exercise (PageIndex{6})

(dfrac{3}{2})

Exercise (PageIndex{7})

(dfrac{20}{8})

Exercise (PageIndex{8})

(dfrac{10}{6})

Answer

(dfrac{5}{3})

Exercise (PageIndex{9})

(dfrac{14}{4})

Exercise (PageIndex{10})

(dfrac{10}{12})

Answer

(dfrac{5}{6})

Exercise (PageIndex{11})

(dfrac{32}{28})

Exercise (PageIndex{12})

(dfrac{36}{10})

Answer

(dfrac{18}{5})

Exercise (PageIndex{13})

(dfrac{26}{60})

Exercise (PageIndex{14})

(dfrac{12}{18})

Answer

(dfrac{2}{3})

Exercise (PageIndex{15})

(dfrac{18}{27})

Exercise (PageIndex{16})

(dfrac{18}{24})

Answer

(dfrac{3}{4})

Exercise (PageIndex{17})

(dfrac{32}{40})

Exercise (PageIndex{18})

(dfrac{11}{22})

Answer

(dfrac{1}{2})

Exercise (PageIndex{19})

(dfrac{17}{51})

Exercise (PageIndex{20})

(dfrac{27}{81})

Answer

(dfrac{1}{3})

Exercise (PageIndex{21})

(dfrac{16}{42})

Exercise (PageIndex{22})

(dfrac{6}{8})

Answer

(dfrac{3}{4})

Exercise (PageIndex{23})

(dfrac{39}{13})

Answer

3

Exercise (PageIndex{24})

(dfrac{44}{11})

Exercise (PageIndex{25})

(dfrac{121}{132})

Answer

(dfrac{11}{12})

Exercise (PageIndex{26})

(dfrac{30}{105})

Exercise (PageIndex{27})

(dfrac{108}{76})

Answer

(dfrac{29}{19})

For the following problems, determine the missing numerator or denominator.

Exercise (PageIndex{28})

(
dfrac{1}{3}=dfrac{?}{12}
)

Exercise (PageIndex{29})

(
dfrac{1}{5}=dfrac{?}{30}
)

Answer

6

Exercise (PageIndex{30})

(
dfrac{3}{3}=dfrac{?}{9}
)

Exercise (PageIndex{31})

(
dfrac{3}{4}=dfrac{?}{16}
)

Answer

12

Exercise (PageIndex{32})

(
dfrac{5}{6}=dfrac{?}{18}
)

Exercise (PageIndex{1})

(
dfrac{4}{5}=dfrac{?}{25}
)

Answer

20

Exercise (PageIndex{1})

(
dfrac{1}{2}=dfrac{4}{?}
)

Exercise (PageIndex{1})

(
dfrac{9}{25}=dfrac{27}{?}
)

Answer

75

Exercise (PageIndex{1})

(
dfrac{3}{2}=dfrac{18}{?}
)

Exercise (PageIndex{1})

(
dfrac{5}{3}=dfrac{80}{?}
)

Answer

48


Learn Fractions with Cuisenaire Rods

When two fractions are equal in overall quantity or value they are called equivalent fractions. We can say that two fractions are considered equivalent when it can be demonstrated that each fraction can be used to represent the same amount of a given object.

To Demonstrate an Equivalent Fraction:

  • Equivalent fractions can best be shown by trains (rods lined up end-to-end) of the same length.
  • In comparing any set of trains showing equivalent fractions, the train with the smallest number of rods represents the fraction in its lowest terms .
  • There may be several groups of equivalent fractions for each unit.

Let's say the brown rod which represents 8 cm. is the unit (meaning it is equal to 1). We can show the following 3 equivalent fraction groups.

In the above example, the two fractions shown are equivalent to each other. This is evidenced by their equal length. In the equivalent fraction group 1, we used the white and red rods because both can be evenly divided into 8. We know there are no other fractions that belong in this equivalent group because there are no other rods equal in length to 1 red rod and 2 white rods.

The numerator of each fraction is the number of rods used in the fraction. The denominator of each fraction is the number of rods that would be used if the train was equal in length to the unit. For example, in the first fraction of the equivalent fraction group 1, the numerator of 2/8 is the number of white rods used in the fraction (2) and the denominator of 2/8 is the number of white rods that would be needed to equal the unit in length (8). In the second equivalent fraction, the numerator of 1/4 is the number of red rods used in the fraction (1) and the denominator is the number of red rods (4) that would be required to equal the unit length.

The following is another equivalent fraction group representing the unit 8.

In equivalent group 2 we use the purple rods in addition to the white and red rods because the purple rods can also be divided into 8 evenly. As you can see the numerators of all 3 fractions are equal in length and the denominators are also equal in length.

As stated above, the train with the smallest number of rods represents the fraction in its lowest terms. The fraction 1/2 is the fraction in its lowest terms.

As in equivalent group 1, group 3 also only uses white and red rods because no other rods which equal 6 in length and divide evenly into 8.

The fraction 3/4 is the fraction in its lowest terms.

In the example above we see that there are 3 groups of equivalent fractions representing the unit 8. In order to determine these equivalent fractions we created trains that must be equal in length and their multiples must equal the unit which is 8 in this case.

In this example let's use the orange rod (which represents 10 cm.) as the unit. We can show the following 4 equivalent fraction groups.

In review we will use rods that are equal in length and that divide evenly into 10.

The fraction 1/5 is the fraction in its lowest terms.

The fraction 1/2 is the fraction in its lowest terms.

The fraction 3/5 is the fraction in its lowest terms.

The fraction 4/5 is the fraction in its lowest terms.

In the example above we see that there are 4 groups of equivalent fractions. As in the first example, we created trains equal in length whose multiples equal the unit which is 10 in this case. Some students may want to insert a dark green rod in the equivalent group 3 or a brown rod in the equivlent group 4, however neither of these are possible because neither 6 nor 8 divide evenly into 10 so their is no fractional equivalent for this rod when the orange rod is the unit.

1. Which equivalent fractions (with values less than 1) can be represented using a dark green rod?

2. Which equivalent fractions (with values less than 1) can be represented using a purple rod?

3. Assuming the blue rod equals 1 whole, what fractional value would be assigned to the light green and white rod?


Equivalent Fractions Worksheets

Equivalent fraction worksheets contain step-by-step solving process, identifying missing numbers, finding the value of the variables, completing the chain of equivalent fractions, writing equivalent fractions represented by pie models and fraction bars and representing the visual graphics in fractions. Explore some of these worksheets for free.

Each printable worksheet has ten problems finding equivalent fractions by undergoing a step-by-step process.

Find the missing number that makes the equivalent fractions. Easy level has 2, 3, 4 and 5 as factors. Moderate level has factors between 1 and 11. Difficult level contains factors in the range 2-25.

Find the value of the variables that makes the equivalent fractions. You may use cross-multiplication method to find the values.

Each question has 8 equivalent fractions. Using the first fraction, complete the chain of equivalent fractions.

Equivalent Fraction Pattern

Each question has a bunch of equivalent fractions. Find the missing equivalent fraction by identifying the pattern followed by both the numerators and denominators.

Hey! I give you the clue. Can you find what fraction am I? Type-2 have fractions expressed in word forms.

Identifying Equivalent Fractions

Each pdf worksheet has two sections. First section contains six multiple choices. Second section contains questions on writing your own equivalent fractions.

Equivalent or Not Equivalent

Insert the correct symbol to show the pair of pizza fractions are equivalent or not equivalent.

Writing Equivalent fractions: Pie Model

Each question in this batch of pdf worksheets has a pair of equivalent pie models. Express the pizza models in equivalent fractions.

Writing Equivalent fractions: Bar Model

Look at the equivalent fraction bars and write the fractions represented by them.

Representing Equivalent fractions: Pie Model

Shade the pie wedges to represent the equivalent fractions in these printable pdfs. Ask the children to color the visual pie graphics (not compulsory).

Representing Equivalent fractions: Bar Model

Color (or shade) the fraction bar to represent each fraction. See how they are equivalent.


What you need for Equivalent Fractions BUMP:

One thing I love about BUMP games is that they require very little prep, and this one is the same. To get started:

  1. Print out the game board and fraction cards of your choice so you can play over and over
  2. Cut out the game cards that have fractions on them.
  3. Preferably place the fraction cards in learning cubes. If you don’t have a learning cube you can tape the fraction cards together to make your own die.
  4. Grab game markers. You will need two sets of game markers that are two different colors. Use unifix cubes, counter chips, coins, beads, blocks, etc.
  5. Sit down with a fourth – sixth grader and get ready for some equivalent fraction practice!

Description Equivalent Fractions

The first step in making equivalent fractions is determining the denominator (bottom number). First you look whether you can make the smallest denominator the same as the largest denominator.

Example sum 1: 1 3 + 1 6

To make these two fractions equivalent, you need to make sure that both fractions have the same denominator.
In this example, we can easily change 1 3 -> 2 6 by multiplying both the numerator and the denominator by 2.

This way we get two equivalent fractions, namely: 2 6 + 1 6 .​

Example sum 2: 2 3 + 1 5

In this example it isn't possible to make the smallest denominator the same as the largest denominator in one go.
That's why we try to multiply the largest denominator by 2 and then see whether that can be divided by the smallest denominator. If that doesn't work, we try multiplying by 3, 4, etc.

The largest denominator is 5. 5 x 2 = 10 Now we look whether 10 can be divided by 3. No, impossible. Now we try 5 x 3 = 15. 15 can be divided by 3.

Now we need to make both denominators 15. It's important to multiply both the numerator and the denominator by the same number. To change 5 -> 15 we multiply by 3
1 5 becomes 3 15 .


Part 2: Lesson

Instructional Strategies and Activities

Warm-Up

Learners will begin by reviewing fractions and shading fraction ‘pies’ to compare fractions of the same denomination.

Introduction

Compare ½ pies and ⅓ pies from the warm up examples. Not all fractions have the same denominator and this makes it harder to compare fractions. In order to compare fractions accurately, we need to match fractions with the same denominators. This process can be shown by making a couple of changes to how we split the fractions.

Presentation / Modeling / Demonstration

Create equivalent fractions by multiplying the slices.

¼ can be multiplied by 3/3 to create 3/12.

This allows us to compare it to ⅙ by multiplying the fraction by 2/2 to create 2/12.

We have not changed the portion of the rectangle shaded, we have just divided it into the same number of segments to make it easier to compare. If we look at the two fractions, we now see that 3/12 is more than 2/12.

2/12 is ‘equivalent’ to ⅙ and 3/12 is ‘equivalent’ to ¼. Equivalent fractions are equal in value to the original fraction.

This means that ⅙ is less than ¼.

Guided Practice

Guided Practice Worksheet

Learners will complete the worksheet using similar examples as above by finding equivalent fractions and comparing fractions with the same denominator.

Before we do some shopping of our own, let’s check and make sure we have our inventory straight.

1.Name an equivalent fraction for 3/4.

Next – Choose a number that is a multiple of 4. We will choose 12.

What do you need to multiply 4 by to get the answer of 12? ____

Multiply the numerator by the same number. The numerator of ¾ is the number 3.

Shade your answer in the boxes below.

Now compare your shaded group here with ¾ above. They should be the same amount. They are equivalent.

2.Find an equivalent fraction for 1/6. Draw diagrams below if you need them.

3.Find an equivalent fraction for 1/3. Draw diagrams below if you need them.

4.Find an equivalent fraction for 1/5. Draw diagrams below if you need them.

5.Compare the fractions 1/3 and 1/5. Change them both to equivalent fractions with the same denominator. Which fraction is a larger amount? Draw diagrams below if you need them.

Evaluation

Learners will complete quiz with 80% or better success.

Name two equivalent fractions for the fractions given below.

Compare the fractions below by using the symbols (, or =).

Application

Learners will complete the worksheet comparing sale items by the fraction of the discount. Learners will write “Deal” across problems that are a deal and draw a large ‘x’ over problems that are not a deal.

Look at the objects below and compare sale prices. Write “Deal” across the fraction that is the largest discount. Write an ‘x’ over the fraction that is smaller.

Store A – 1/4 off Store B – 1/3 off

Store A – 1/5 off Store B – 1/2 off

Store A – 1/6 off Store B – 2/5 off

Store A – 3/8 off Store B – 2/5 off

Store A – 2/3 off Store B – 5/6 off

Key Terms and Concepts

Equivalent fractions - fractions that have the same value but do not have the same numerator and denominator.


Equivalent Fractions Online Quiz

Following quiz provides Multiple Choice Questions (MCQs) related to Equivalent Fractions. You will have to read all the given answers and click over the correct answer. If you are not sure about the answer then you can check the answer using Show Answer button. You can use Next Quiz button to check new set of questions in the quiz.

Q 1 - Fill in the blank to make the fractions equivalent.

Answer : D

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 7/13 by 5 to get.

Q 2 - Fill in the blank to make the fractions equivalent.

Answer : C

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 3/8 by 3 to get.

Q 3 - Fill in the blank to make the fractions equivalent.

Answer : B

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 7/8 by 6 to get.

Q 4 - Fill in the blank to make the fractions equivalent.

Answer : A

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 5/7 by 8 to get.

Q 5 - Fill in the blank to make the fractions equivalent.

Answer : C

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 4/9 by 8 to get.

Q 6 - Fill in the blank to make the fractions equivalent.

Answer : B

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 5/12 by 5 to get.

Q 7 - Fill in the blank to make the fractions equivalent.

Answer : A

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 2/5 by 11 to get.

Q 8 - Fill in the blank to make the fractions equivalent.

Answer : D

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 4/11 by 7 to get.

Q 9 - Fill in the blank to make the fractions equivalent.

Answer : A

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 4/13 by 6 to get.

Q 10 - Fill in the blank to make the fractions equivalent.

Answer : C

Explanation

Since these are equivalent fractions multiply both numerator and denominator of 7/14 by 5 to get.


Cross-Multiplication Rule

Step 1: To check if and are equivalent fractions, set them equal to each other.

Step 2: Perform the cross-multiplication procedure. The diagram below should help.

  • Multiply the left numerator to the right denominator. Write it as ad .
  • Then write the equal symbol ( = ).
  • Finally, multiply the left denominator to the right numerator. Write it as bc .

Step 3: If ad = bc is a true statement then and are equivalent fractions. Otherwise, if adbc then the two fractions are not equivalent.

Examples of How to Apply the Cross-Multiplication Rule to Verify if the Two Given Fractions are Equivalent

Example 5: Are the fractions below equivalent?

The cross products are equal. This means that they are equivalent fractions.

Example 6: Are the fractions below equivalent?

These two fractions may seem to be totally different in value. But the cross multiplication rule should reveal their equivalency.


How to Find Equivalent Decimal for 1/4?

The below workout with step by step calculation shows how to find the equivalent decimal for fraction number 1/4 manually.
step 1 Address input parameters & values.
Input parameters & values:
The fraction number = 1/4

step 2 Write it as a decimal
1/4 = 0.25
0.25 is the decimal representation for 1/4

For Percentage Conversion :
step 1 To represent 0.25 in percentage, write 0.25 as a fraction
Fraction = 0.25/1

step 2 multiply 100 to both numerator & denominator
(0.25 x 100)/(1 x 100) = 25/100


1.5: Equivalent Fractions

Look at the sets of numbers below: One set is equivalent and one set is not. Which set IS the equivalent fractions?

If I had 6/8 of a cookie and you had 4/10 of a cookie, would we have equivalent fractions?

6/9

Which fraction is not equal to the others? 6/8, 3/4, 1/10, 12/16

Andy bought a pack of 8 pencils. He gave 4 away. Which fraction shows how many of his pencils he gave away?

3/5 = 6/10


Watch the video: Equivalent Linear Systems. Page 46. Question #5; (October 2021).