Equivalent Fractions
Equivalent Fractions have the same value, even though they may look different.
These fractions are really the same:
1
2
= 2
4
= 4
8
Some reasons why:
Look at all the blocks below. In each case the whole is the same length. But each
whole has been cut into different size parts. Notice, 1 of the halves is the same as 2
of the quarters which is the same as 3 of the sixths which is the same as 4 of the
eighths.
Look at the pizzas below. They all represent the same amount of pizza (half of the
whole). But the fraction we might use to say the amount is different each time as the
size of the pieces are different:
Look on the number lines below. Each time, the distance from 0 to 1 is the same.
Each time we point to the same point on the number line (half way between 0 and 1).
Yet the fraction name we give each time is different.
A Rule For Equivalent Fractions
The rule to remember is:
"Change the bottom using multiply or divide,
And the same to the top must be applied"
So, here is why those fractions are really the same:
× 2
× 2
1
= 2
2
4
= 4
8
× 2
× 2
Making Equivalent Fractions
You can make equivalent fractions simply by multiplying the numerator and
denominator by any number you like:
Example: Make an equivalent fraction for 4/5.
Some Solutions:
I. Let’s say we choose 3 to multiply by.
Then we get
! ×!
! ×!
!"
= !"
So 12/15 must be equivalent to 4/5.
II. Let’s say we choose 7 to multiply by.
Then we get
! ×!
! ×!
!"
= !"
So 28/35 must be equivalent to 4/5.
Making Equivalent Fractions Through Division
Provided that the number you are dividing by will go into BOTH numerator and
denominator, you can also make equivalent fractions by dividing.
Example:
Find an equivalent fraction of 30/40 by dividing.
One Solution:
Let’s divide by 2.
!" ÷!
!" ÷!
!"
= !"
So 15/20 must be equivalent to 30/40.
Making A Whole List For Equivalent Fractions
You can make a whole list of equivalent fractions using lists of multiples:
Example:
Give the first 16 equivalent fractions of 2/3.
Solution:
By starting with 2 on top and 3 on the bottom, if you write out the
multiples of 2 on the top, and multiples of 3 on the bottom, note that
you get a whole list of equivalent fractions.
!
!
!
!
!
!
!
!"
= = =
=
!"
!"
=
!"
!"
=
!"
!"
=
!"
!"
=
!"
!"
=
!"
!"
=
!!
!!
=
!"
!"
=
!"
!"
=
!"
!"
=
!"
!"
=
!"
!"
Note, 4/6 was made by multiplying the original 2 and 3 by 2.
6/9 was made by multiplying the original 2 and 3 by 3.
8/12 was made by multiplying the original 2 and 3 by 4.
10/15 was made by multiplying the original 2 and 3 by 5.
Checking for Equivalence
If one the fraction is in simplest form (see Simplifying Fractions if you are unsure
what this means), you can check for equivalence by seeing if you can “make” the
simplified fraction turn into the other fraction with the SAME multiplication to the
numerator and denominator.
Example: Is 4/5 the same as 28/35?
Solution:
Think. How could I turn the denominator form 5 into 35? Multiply by 7!
So if I multiply the numerator by 7 too, I get 4 x 7 = 28. So yes, by
multiplying BOTH the numerator and denominator by 7 I get 28/35.
The 2 fractions are equivalent.
Example: Is 2/3 the same as 16/27?
Solution:
Think. How could I turn the denominator from 3 to 27? Multiply by 9!
So if I multiply the numerator by 9, I get 2 x 9 = 18. So the equivalent
fraction of 2/3 is 18/27, NOT 16/27. So no, they are not equivalent.
Checking for Equivalence when neither is in simplified form
If neither fraction is in simplest form, the only way to test for equivalence is to
simplify both fractions and see if they simplify to the same ratio
Example: Is 18/27 the same as 20/30?
Solution:
Note that 18/27 can be simplified by dividing by 9. Counting in 9s, we
get: 9 18 27. So notice, 18/27, is 2 nines out of 3 nines. It is 2/3.
Note that 20/30 can be simplified by dividing by 10. Counting in 10s,
we get: 10 20 30. So notice, 20/30 is 2 tens out of 3 tens. Again, 2/3.
So yes, both fractions are equivalent to 2/3. They are equivalent to
each other.