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Reciprocal and Division of Fractions are two different terms in solving fractions. Reciprocal is when the numerator and the denominator of the fractions are interchanged.
- Division is the normal division done using different steps for a fraction.
- A fraction represents a part of the whole.
- The reciprocal of number is opposite of fraction.
- It means if determine the reciprocal of unit fraction then it is equivalent to whole number.
- On the other hand division is a process of sharing and put numbers into equal sections.
- It is process of multiplying fraction with the reciprocal of fraction.
- For eg, if we say 3/4 of the Pizza had cheese, it means that 3 out 4 parts of the Pizza had cheese on it.
- Offering return gift to guest coming in a birthday party is real life example of reciprocal.
Key Terms: Reciprocal and Division of Fractions, Reciprocal, Division, Multiplication Inverse, Fractions, Numerator, Denominator, Proper Fraction, Improper Fraction, Mixed Fractions, Like Fractions, Unlike Fractions
Reciprocal of Fractions
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Reciprocal of fractions is a process of turning the fraction upside down. In other words, the positions of the numerator and the denominator are interchanged.
- The numerator of the fraction includes the number in the upper part of the fraction.
- The denominator occupies the lower part.
- Reciprocal of mixed fractions is determined by converting mixed numbers into improper fractions.
- It is also known as the multiplicative inverse of each other.
- The process is the same for fractions with exponents.
- Feedback mechanism for a service is a common example of the reciprocal of fractions.
- It can be represented as:
Fraction: Numerator/ Denominator
Reciprocal: Denominator / Numerator
Example of Reciprocal of FractionsExample 1: Consider the fraction: ¾? Ans: Here 3 is the numerator and 4 becomes the denominator. A reciprocal is written as 4/3 Example 2: Consider the fraction: 6/7? Ans: Here 6 is the numerator and 7 becomes the denominator. A reciprocal is written as 7/6 |
Reciprocal of fractions
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Division of Fractions
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The division is a process of sharing an item equally among a number of participants. For solving a division of a fraction with another fraction, the second fraction’s reciprocal is taken and multiplied with the first.
- The process of division is similar to the process of multiplication of fractions.
- Multiplication happens between the numerator and denominator of one fraction with the numerator and denominator of the other.
- It consists of fractions within a fraction.
- Division of fractions is related to the reciprocal fractions.
- The division changes to multiplication as the second fraction reciprocates.
Thus division of fraction follows these steps:
- Divide
- Reciprocate
- Multiply
- Reduce the fraction to its simplest form (if applicable)
Example of Division of FractionsExample: 2/7 is divided by 4/9 While dividing, we can observe in the first step that in division of fraction, it has a fraction in its numerator and denominator. Having a fraction within a fraction, makes the fraction in the denominator reciprocate itself.
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Division of fractions
Rules in Division of Fractions
Product of multiplying two fractions is equivalent to ratio of product of numerators to product of denominators. Similar method is followed when a fraction is divided by a whole number and when a whole number is divided by a fraction.
- There are three rules followed in division of fractions which are as follows:
Division of the Whole Number by a Fraction
In this whole number is multiplied by the numerator of the fraction. It can be explained with the example which is as follows:
| Example: 24 ÷ 4/7 Solution: 24 ÷ 4/7 = 24/1 × 7/4
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Division of Fraction by a Whole Number
In this whole number is multiplied by the denominator of the fraction. It can be explained with the example which is as follows:
| Example: Divide 7/3 by 3 Solution: We need to simplify, 7/3 ÷ 3 The reciprocal of 3 is 1/3. Now writing the given expression into multiplication form, 7/3 × 1/3 = 7 /9 Therefore, 7/3 ÷ 3 = 7/9 |
Division of a Fraction by another Fraction
Division of a fraction by another fraction is equivalent to product of numerators to product of denominators. It can be explained with the example which is as follows:
| Example 3: 8/3 ÷ 8/3 Solution: 8/3 ÷ 8/3 The reciprocal of second term 8/3 is 3/8. Now multiply the first term with the reciprocal of the second term. 8/3 × 3/8 = 8/8 = 1 |
Types of Fractions
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The steps of division of fractions might slightly increase based on the type of fractions. There are three types of fractions.
Proper Fraction
A proper fraction is a fraction which has a smaller numerator than its denominator. In these type of fraction both numerator and denominator are positive.
Proper Fraction = Numerator < Denominator
Example of Proper FractionExample: 3/5, 1/ 2 and 5 / 9 |
Improper Fraction
An improper fraction is a fraction in which the value of numerator is greater than its denominator.
Improper Fraction = Numerator > Denominator
Example of Improper FractionExample: 6/5, 3/2 and 23/19 |
Mixed Fraction
A fraction is represented with its quotient and the remainder is a mixed fraction. Mixed fraction is written with a combination of a fraction as well as a whole number.
Example of Mixed FractionExample: Convert the mixed fraction 12/7 into improper fraction First multiply the denominator with the number beside it (1 × 7 = 7), now to this value add the numerator (2 + 7 = 9). Thus while writing the improper fraction the denominator remains unchanged, only the numerator is written after solving. The improper fraction of 12/7 is 97. |
Things to Remember
- Reciprocal and division of fractions are interrelated with each other.
- You will always get one as an answer when you multiply a fraction and its reciprocal.
- Any whole number will always have ‘1’ as its denominator, even if not mentioned.
- While solving mixed fractions, start division by first converting the mixed fraction into an improper fraction.
- For comparison, both the fractions should share a common denominator: convert them into like fractions.
Read More:
| Class 7 Maths Related Concept | ||
|---|---|---|
| Additive Inverse | Descending Order | Decimal to Fraction Formula |
| Rationalize the Denominator | Number Lines | Addition and Subtraction of Fractions |
Sample Questions
Ques. Find the value of 3/32 ÷ 15/64. (2 marks)
Ans. To divide 3/32 ÷ 15/32, we will be using the steps of the division of fractions. The first step is to keep the first fraction as it is. Then change the division sign to multiplication sign and at last, flip the second fraction to its reciprocal.
- This implies 3/32 × 64/15.
- After simplifying, we get (3 × 64) / (32 × 15) = 2/5.
Ques. Divide 5 by 3/2 and convert it into mixed fraction. (2 marks)
Ans. Any whole number will always have ‘1’ as its denominator, thus
5/1 x 3/2=5/1 x2/3=10/3
For converting 10/3 into mixed fraction,
| 3 | |
| 3 | 10 |
| 9 | |
| 1 |
10/3 can also be written as 31/3.
Ques. Out of the work given to Piya and Monu, Piya completed 5/7 of the total work while Monu completed 2/5 of the whole work. Compare and conclude on who completed more work. (3 marks)
Ans. Piya completed 5/8 of the total work
Monu completed 2/5 of the total work
Converting both the fractions into like fractions, 5/7 = 25/35 and 2/5 = 14/35
[while converting fractions into like fractions one needs to multiply both the denominators in such a way that they become equal. Here 7 × 5 = 35 thus both the numerator and the denominator of 5/7 is multiplied by 5 to give the answer 25/ 35. 5 × 7 = 35 thus both the numerator and the denominator of 25 is multiplied by 7 to give the answer 14/35.]
Now comparing the two like fractions,
25/35 > 14/35 [because 25 > 14]
Thus,
57 > 25
Thus, Pooja completed more work than Mark.
Ques. Divide 2/5 with 8/5. (2 marks)
Ans.
2/5x8/5=2/5x5/8
=2/8
=1/4
Ques. Divide the following. (2 marks)
- 1/2 ÷ 3/4
- 1/4 ÷ 5/6
Ans. The process is as follows:
- 12 ÷ 34 = 1/2x3/4=1/2x4/3=6
- 1/4 ÷ 5/6 = 1/4x5/6=1/4x6/5=8/45
Ques. Find the value of 7/32 ÷ 14/32. (2 marks)
Ans. To divide 7/32 ÷ 14/32, we will be using the steps of the division of fractions. The first step is to keep the first fraction as it is. Then change the division sign to multiplication sign and at last, flip the second fraction to its reciprocal.
- This implies 7/32 × 14/32.
- After simplifying, we get (7 × 32) / (32 × 14) = 1/2.
Ques. What is the value of the reciprocal of the sum of fractions 4/5 and 5/6. (2 marks)
Ans. The given fractions are 4/5 and 5/6. The addition of unlike fractions is done by taking the LCM of the denominators.
- The LCM of 5 and 6 is 6.
- So, 4/5 + 5/6 = (24+25)/30
- 49/30.
- Therefore, the reciprocal of fraction 49/30 is 30/49.
Ques. Use the steps of dividing fractions with whole numbers to find the value of 8/7 ÷ 7. (2 marks)
Ans. To divide a fraction with a whole number, we multiply the given whole number with the denominator of the fraction. Here,
- 8/7 ÷ 7
- 8/7 × 1/7
- 8/49.
Ques. If the reciprocal of x is 6/11, find the value of x + 6. (2 marks)
Ans. Given that the reciprocal of x is 6/11. It means that x is 11/5. Now, x + 6 = 11/6 + 6.
⇒ 11/6 + 6/1
⇒ (11+36)/6
⇒ 47/6
Ques. What is the value of the reciprocal of the sum of fractions 4/5 and 15/16. (2 marks)
Ans. The given fractions are 4/5 and 15/16. The addition of unlike fractions is done by taking the LCM of the denominators.
- The LCM of 5 and 16 is 6.
- So, 4/5 + 15/16 = (64+75)/80
- 139/80.
- Therefore, the reciprocal of fraction 139/80 is 80/139.
Ques. Use the steps of dividing fractions with whole numbers to find the value of 18/17 ÷ 17. (2 marks)
Ans. To divide a fraction with a whole number, we multiply the given whole number with the denominator of the fraction. Here,
- 18/17 ÷ 17
- 18/17 × 1/17
- 18/289.
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