Confused between various fraction types? If yes, then check here for the important type of fraction i.e., equivalent fraction. Definition of Equivalent Fractions is here. Check rules, methods, and formulae of Equivalent Fractions. Refer step by step procedure to know the problems of equal fractions. Follow the important points and example problems to know in-depth of equivalent problems. Check the below sections to know the detailed description of equivalent fractions and their rules.
What are Equivalent Fractions?
Equivalent Fractions or equal fractions are the fractions that have different numerators and denominators but gives the same value. For example, the value of both the fractions \(\frac { 4 }{ 8 } \) and \(\frac { 3 }{ 6} \) is equal to \(\frac { 1 }{ 2 } \). Hence, both the values are the same they are equivalent in nature. This equivalent fraction represents a similar proportion of the whole.
To define the equivalent fractions, suppose that \(\frac { a }{ b } \) and \(\frac { c }{ d } \) are 2 fractions. After simplification of the given fractions, both results in equal fractions suppose e/f which are equal to each other.
Why do fractions have the same values in spite of having a different number?
For the above question, the answer is the denominator and numerator are not co-prime numbers. This fraction has a common multiple that gives the same value therefore they have a common multiple, which on division gives exactly the same value.
Example:
\(\frac { 2 }{ 4 } \)= \(\frac { 1 }{ 2 } \)
= \(\frac { 4 }{ 8 } \)
In the above given example, it is clearly shown that the fractions have different denominators and numerators.
Dividing both denominator and numerator by their common factor, we have:
= \(\frac { 2 }{ 2 } \) ÷ \(\frac { 4 }{ 2 } \)
= \(\frac { 1 }{ 2 } \)
In a similar way, on simplifying \(\frac { 4 }{ 8 } \) we get
= \(\frac { 4 }{ 4} \) ÷ \(\frac { 8 }{ 4 } \)
= \(\frac { 1 }{ 2 } \)
Therefore both the fractions have an equal value \(\frac { 1 }{ 2 } \)
How to Find Equal Fractions?
Equal fractions are actually similar because when we divide or multiply both the denominator and numerator by the same number, the fraction value doesn’t change. On simplifying the value of the equivalent fractions, the value will be the same.
Example:
Simplify the fraction \(\frac { 1 }{ 5 } \)
On multiplying denominator and numerator with 2, the result will be = \(\frac { 2 }{ 10 } \)
On multiplying denominator and numerator with 3, the result will be = \(\frac { 3 }{ 15 } \)
Multiplying denominator and numerator with 4, the result will be = \(\frac { 4 }{ 20 } \)
Thus, we can conclude from the above simplification as,
\(\frac { 1 }{ 5 } \) = \(\frac { 2 }{ 10 } \) = \(\frac { 3}{ 15 } \) = \(\frac { 4 }{ 20 } \)
We can only divide or multiply by similar numbers to get an equal or equivalent fraction and not subtraction or addition. Simplification has to be done where both the denominator and numerator should be whole numbers.
Key Points to Remember
- Equal Fractions or equivalent fractions will look different, but they both have the same values.
- You can easily divide or multiply to find an equivalent fraction.
- The functions of addition and subtraction do not work for similar fractions or equivalent fractions.
- If you divide or multiply with the top part of the fraction, you must also do the same for the denominator.
- Use the rule of cross multiplication, to determine if both the fractions are equivalent.
How to Determine Whether Two Fractions are Equivalent?
Simplifying the given fractions is the only step to find whether the fractions are equivalent or not. Equivalent numbers simplification can be done for a point where both the denominator and numerator should be the whole number. There are some methods to identify that the given fractions are equal. Some of them are:
Step 1: Make the denominators same
Step 2: Find the decimal form of both the fraction values.
Step 3:Â Apply the cross multiplication method.
Step 4: Visualise the method of fractions.
Example 1:
Show that the fractions below are equivalent fractions
\(\frac {3}{ 7 } \), \(\frac { 12 }{ 28 } \), \(\frac { 18 }{ 42 } \), \(\frac { 27 }{ 63 } \)?
Solution:
The trick of solving the fractions is to select any of the four fractions and also using some arithmetic equations, transform one fraction into another three fractions. For this example, I would like to pick the smallest fraction which is \(\frac { 3 }{ 7 } \).
Step 1:
Converting \(\frac { 3 }{ 7 } \) into \(\frac { 12 }{ 28 } \) to prove they are equivalent fractions
To convert into an equivalent fraction, we have to multiply the fraction with \(\frac { 4 }{4 } \)
Therefore, the fraction can be written as \(\frac { 3 }{ 7 } \) * \(\frac { 4 }{ 4 } \)
= \(\frac { 12 }{ 28 } \)
Thus, the fractions are equivalent.
Step 2:
Converting \(\frac { 3 }{ 7 } \) into \(\frac { 18 }{ 42 } \) to prove they are equivalent fractions
To convert into an equivalent fraction, we have to multiply the fraction with \(\frac { 6 }{ 6 } \)
Therefore, the fraction can be written as
\(\frac { 3 }{ 7} \) * \(\frac { 6 }{6 } \)
= \(\frac { 18 }{ 42 } \)
Step 3:
Converting \(\frac { 3 }{ 7 } \) into \(\frac { 27 }{ 63 } \) to prove they are equivalent fractions
To convert into an equivalent fraction, we have to multiply the fraction with \(\frac { 9 }{ 9 } \)
Therefore, the fraction can be written as
\(\frac { 3 }{ 7 } \) * \(\frac { 9 }{ 9 } \)
= \(\frac { 27 }{ 63 } \)
Example 2:
Check whether \(\frac { 2 }{ 5 } \) is equivalent to \(\frac { 4 }{ 10 } \)?
Solution:
To find that two fractions \(\frac { 2 }{ 5 } \) and \(\frac { 4 }{ 10 } \) are equivalent, we have to apply the cross multiplication
To convert into equivalent fractions, we have to multiply the fraction \(\frac { 4 }{ 10 } \) with \(\frac { 2 }{ 2 } \)
On multiplying the fraction with \(\frac { 2 }{ 2 } \), we get \(\frac { 2 }{ 5 } \)
Therefore the fraction \(\frac { 4 }{ 10 } \) is equivalent to \(\frac { 2 }{ 5 } \).
Thus, these are called equal or equivalent fractions.
Example 3:Â
Mr.Lee is planting a vegetable garden. The garden will have no more than 16 equal sections. \(\frac { 3 }{ 4 } \) of the garden will have tomatoes. What fraction could represent the part of the garden that will have tomatoes?
Solution:
We need to find other fractions that are equivalent to \(\frac { 3 }{ 4 } \)
We can make a table of those fractions
We can use multiplication to find the equivalent fractions
Multiply the fraction with \(\frac { 2 }{ 2 } \), \(\frac { 3 }{ 3 } \), \(\frac { 4 }{ 4 } \)
\(\frac { 3 }{ 2 } \) * \(\frac { 4 }{ 2 } \) = \(\frac { 6 }{ 8 } \)
\(\frac { 3 }{ 3 } \) * \(\frac { 4 }{ 3 } \) = \(\frac { 9 }{ 12 } \)
\(\frac { 3 }{ 4 } \) * \(\frac { 4 }{ 4 } \) = \(\frac { 12 }{ 16 } \)
Each numerator represents the part of Mr.Lee’s garden that has tomatoes.
Each denominator represents how many parts there are in all his garden.
As the denominator becomes a greater number, the size of the parts becomes smaller.
Example 4:
Sopia is making bracelets with beads. Each bracelet has 4 beads and \(\frac { 3 }{ 4 } \) of the beads are red. If Sopia makes 5 bracelets, how many red beads does she need?
Solution:
As given in the question,
No of beads each bracelet has = 4
No of beads that are red = \(\frac { 3 }{ 4 } \)
To make 5 bracelets, the no of red beads she needs
5 * \(\frac { 3 }{ 4 } \) * 4 = 15
Therefore, 15 red beads are needed for Sopia to make 5 bracelets.
Example 5:
Sopia is making necklaces. The largest necklace will have 24 beads. Another necklace may contain fewer beads but will have at least 12 beads. In every necklace, half of the beads are red, \(\frac { 1 }{ 3 } \) is green and \(\frac { 1 }{ 6 } \) are yellow. What combinations of beads represent all the possible necklaces that Sopia can make?
Solution:
The common multiplies of 2, 3 and 6 are 12,18,24
Total beads in necklace = 12
Fraction of red beads = 6
Fraction of green beads = 4
Fraction of yellow beads = 2
Total beads in 2nd necklace = 18
Fraction of red beads = 9
Fraction of green beads = 6
Fraction of yellow beads = 3
Total beads in 3rd necklace = 24
Fraction of red beads = 12
Fraction of green beads = 8
Fraction of yellow beads = 4
We find equivalent fractions using common multiple as denominator
Half of them are red – \(\frac { 1 }{ 2 } \) * \(\frac { 6 }{6 } \) = \(\frac { 6 }{ 12 } \),
\(\frac { 1 }{ 3} \) – \(\frac { 1 }{ 3 } \)* \(\frac { 4 }{ 4 } \) = \(\frac { 4 }{ 12 } \),
\(\frac { 1 }{ 2 } \) are yellow – \(\frac { 1 }{ 2 } \) *\(\frac { 6 }{ 2 } \) = \(\frac { 2 }{ 12 } \)
Half of them are red – \(\frac { 1 }{ 2 } \) * \(\frac { 9 }{9 } \) = \(\frac { 9 }{ 18 } \)
\(\frac { 1 }{ 3 } \) are green – \(\frac { 1 }{ 3 } \)* \(\frac { 6 }{ 6 } \) = \(\frac { 6 }{ 18 } \)
\(\frac { 1 }{ 2 } \) are yellow – \(\frac { 1 }{ 6 } \) * \(\frac { 3 }{ 3 } \) = \(\frac { 3 }{ 18 } \)
Half of them are red – \(\frac { 1 }{ 2 } \) * \(\frac { 12 }{ 12 } \) = \(\frac { 12 }{ 24 } \)
\(\frac { 1 }{ 3 } \) are green – \(\frac { 1 }{ 3 } \) * \(\frac { 8 }{ 8 } \) = \(\frac { 8 }{ 24 } \)
\(\frac { 1 }{ 6 } \) are yellow – \(\frac { 1 }{ 6 } \)* \(\frac { 4 }{4 } \) = \(\frac { 4 }{ 24 } \)