How to Divide Fractions by Fractions

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Dividing fractions by fractions can be a perplexing concept for many, but with a clear understanding of the underlying principles, it becomes much simpler to tackle. It is a fundamental skill in mathematics that allows us to solve a wide range of real-world problems, from recipes and proportions to financial calculations. This guide will break down the steps involved in dividing fractions by fractions, providing you with a step-by-step approach to understanding and mastering this essential mathematical operation. So, whether you are a student seeking help with dividing fractions or someone wanting to refresh their math skills, this guide will equip you with the knowledge and techniques needed to confidently divide fractions by fractions.

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This article was co-written by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math teacher at City University of San Francisco and previously worked in the math department of Saint Louis University. She has taught math at the elementary, middle, high, and college levels. She holds a master’s degree in education from Saint Louis University, majoring in management and supervision in education.

This article has been viewed 153,252 times.

Dividing fractions by fractions sounds complicated, but it’s actually very simple. All you need to know is to inverse fractions, multiply and simplify fractions. This article will decipher this process, and you will find dividing fractions as easy as eating candy.

Table of Contents

Steps

Practice dividing fractions by fractions

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Start with a sample example. Calculate 2/3 ÷ 3/7 . This question asks us how many 3/7 units out of 2/3 units. Don’t worry; It sounds complicated and difficult to understand, but it’s not difficult at all!

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Change the division sign to the multiplication sign. The new formula will be: 2/3 * __ (we will fill in the blanks in a later step).

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Find the inverse of the second fraction. That is, we will be upside down 3/7, then the numerator (3) will be “pushed” down, and the denominator (7) will be “pulled” up. The reciprocal of 3/7 is 7/3. We will fill in the blanks with this new fraction in the previous step:

• 2/3 * 7/3 = __

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Multiply two fractions. First we multiply the two numerators together: 2 * 7 = 14. 14 is the numerator (number above) of the result. Then we multiply the two denominators: 3 * 3 = 9. 9 is the denominator (bottom number) of the result. So we have: 2/3 * 7/3 = 14/9.

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Simplify fractions. In this case, since the numerator is larger than the denominator, our fraction is larger than 1, and we can split the fraction into a mixed number. (A mixed number consists of an integer and a fraction, like 1 2/3. ^{[1] X Research Source} )

• First, divide 14 by 9. 14 divided by 9 gives 1 remainder 5, so we have a mixed number: 1 5/9 (“one-fifth”).
• This is the final answer! We can see that the fraction cannot be reduced further because the numerator is not divisible by the denominator (5 is not divisible by 9) and the numerator is a prime number, i.e. a positive integer that is only divisible by 1. and itself. ^{[2] X Research Source}

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Let’s try another example! Calculate 4/5 ÷ 2/6 = . First, replace the division sign with a multiplication sign ( 4/5 * __ = ), then find the inverse of 2/6 to get 6/2. So we have 4/5 * 6/2 =__ . Next multiply the numerators by 4 * 6 = 24 , multiply the denominators together by 5 * 2 = 10 . Here we have 4/5 * 6/2 = 24/10. Now we will reduce the fraction. Since the numerator is larger than the denominator, we need to convert this fraction to a mixed number.

• Divide the numerator by the denominator ( 24/10 = 2 remainder 4 ).
• So we have 2 4/10 . However, we can still reduce this mixed number.
• We see that 4 and 10 are even numbers, so we can divide both numbers by 2, so we reduce 4/10 to 2/5.
• Since the numerator (2) is a prime number that is not divisible by the denominator (5), we cannot reduce it any further. The end result is: 2 2/5 .

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Simplify fractions. You’ve probably learned a lot about reducing fractions before you’ve learned to divide fractions, but if you need to learn from scratch or review how to reduce fractions, you can easily find other articles on network. ^{[3] X Research Sources}

Understand how to divide fractions by fractions

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Understand what fractional division really is. Question 2 ÷ 1/2 really wants to know “In 2 units, how many half units are there?” The correct answer is 4, because each base unit (1) will consist of 2 halves of units (because 1/2 +1/2 = 1/2 * 2 = 1), so with 2 units we will have: 2 halves/1 unit * 2 units = 4 halves.

• Think about it another way, take a glass of water as an example, ask: If you have two cups of water, how many half a cup of water do you have? You can pour half a cup twice to fill a glass of water, which means you add the two halves together, so when you have two cups: 2 halves/1 cup * 2 cups = 4 halves.
• When the divisor is between 0 and 1, the result is always greater than the original value of the divisor! This is always true whether the number being divided is an integer or a fraction.

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Division is the inverse of multiplication. So, dividing by a fraction is equivalent to multiplying by the reciprocal of that fraction. The inverse of a fraction is the inversion of the numerator and denominator of the original fraction. ^{[4] X Research Source} Next we will divide the fraction by the fraction by finding the inverse of the second fraction and multiplying this inverse by the first fraction. However, first you need to understand the inverse:

• The reciprocal of 3/4 is 4/3.
• The reciprocal of 7/5 is 5/7.
• The reciprocal of 1/2 is 2/1, which is also 2.

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Remember the steps after dividing a fraction by a fraction. The steps to divide fractions by fractions include:

• Temporarily do not consider the first fraction.
• Convert the division sign in the calculation to the multiplication sign.
• Find the inverse of the second fraction. That is, reverse the positions of the numerator and denominator.
• Multiply the numerator (number above) of two fractions together to get the numerator of the calculation. ^{[5] X Research Sources}
• Multiply the denominator (bottom number) of the two fractions to get the denominator of the result.
• Minimize the resulting fraction.

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Practice the steps above with the calculation 1/3 ÷ 2/5. First, we ignore the first fraction, then replace the division sign with the multiplication sign:

• 1/3 ÷ 2/5 = will become:
• 1/3 * __ =
• Then we invert the second fraction (2/5) to get its reciprocal of 5/2:
• 1/3 * 5/2 =
• Now multiplying the two numerators of the first fraction and the reciprocal of the second fraction together, we get 1*5 = 5.
• 1/3 * 5/2 = 5/
• Similarly, multiplying the two denominators together, we get 3*2 = 6.
• So we have: 1/3 * 5/2 = 5/6
• This is a simple fraction so it is also the final result of the calculation.

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We can summarize the above steps according to the following frog poem: “Dividing fractions/by fractions, is not a/problem, first is/divide by multiply, then inverse/second, multiply by two multiply/multiply by two denominators, and reduce/and that’s it.” Original: “Dividing fractions, as easy as pie, Flip the second fraction, then multiply. And don’t forget to simplify, Before it’s time to say goodbye.” ^{[6] X Research Sources}

• Another way to help you remember what to do with each part of the calculation is: “ Ignore me (first fraction), Change me (division), Invert me (second fraction))

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This article was co-written by Grace Imson, MA. Grace Imson is a math teacher with over 40 years of teaching experience. Grace is currently a math teacher at City University of San Francisco and previously worked in the math department of Saint Louis University. She has taught math at the elementary, middle, high, and college levels. She holds a master’s degree in education from Saint Louis University, majoring in management and supervision in education.

This article has been viewed 153,252 times.

Dividing fractions by fractions sounds complicated, but it’s actually very simple. All you need to know is to inverse fractions, multiply and simplify fractions. This article will decipher this process, and you will find dividing fractions as easy as eating candy.

In conclusion, dividing fractions by fractions may initially appear confusing and tiresome, but with a solid understanding of the underlying principles, it becomes an easily manageable process. By following the steps outlined in this guide – inverting the second fraction and multiplying the fractions – individuals can successfully divide fractions by fractions. The key is to remember that dividing fractions is equivalent to multiplying by the reciprocal, and simplifying the resulting fraction is crucial. With practice, individuals can become proficient in dividing fractions by fractions and confidently tackle more complex mathematical problems involving fractions.

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