FRACTIONS LESSON

Equivalent and Comparing Fractions

Equivalent Fractions

Equivalent fractions are equal in value. When we reduce them to lowest terms, they are identical. For instance, if you were holding 5 dimes, you wouldn't say I have 5 tenths of a dollar. You'd probably say I have a dollar because you know that 5 tenths equals or is equivalent to half of a dollar.

Example:

When we cut a pizza into two pieces, each piece is one-half of the pizza.
If we cut it into 4 pieces, then it takes 2 such pieces or 2 quarters, to make half of a pizza,
and 4 such pieces or quarters to get back the whole pie.
So, we see that is exactly equal or equivalent to , and 1 is equal to .

 

A whole pizza is equivalent to 2 halves of a pizza or 4 quarters of a pizza.

 

Think money! You know 2 quarters ( 's) make a half-dollar and 4 quarters ( 's) make a buck.

To create a fraction that is equivalent to a given fraction, we multiply both its numerator and denominator by the same number so we're multiplying by one (1) and we're not changing the value, just the form.

Remember there's no prejudice in math!!! -- what you do to the top, you do to the bottom!

So, to change into we would multiply top and bottom by 2 like this:

 

We have twice as many pieces but they're half as big as before.

 

Example: Fill in the blank space to create an equivalent fraction:

 


Since we multiplied 7 (the denominator) by 6 to get 42, we do the same to the 5.

The solution is

 

But, 6 over 6 = 1, so we haven't changed the value of the original fraction.
We just multiplied it by one to change its form, not its value.

 

Equivalent Fractions Exercise #1:

Fill in the blanks to create an equivalent fraction.

a)

b)

c)

d)

e)

(solutions)


 

Reducing Fractions to Lowest Terms

Just as we can get equivalent fractions when we multiply by one, we can do the same when we divide by one. In this case we reduce the fraction to lowest terms.
So an answer of 15/20 would reduce to 3/4, since

 

In the same way,

 

Equivalent Fractions Exercise #2:

Now we divide top and bottom by the same value
Factor these fractions as in the examples, then Reduce them to lowest terms.

 

a)

b)

c)

d)

e)

(solutions)


 

Comparing the Size of Fractions

John and Don, a pair of quarrelsome twins, were arguing one day. Seems their Mom asked them to choose between two pieces of chocolate candy. One piece was three quarters of a whole candy bar; the other was five eighths of a bar. Since they both loved candy, they had to figure out which piece was bigger. John said 5 eighths must be bigger than 3 quarters because 5 is bigger than 3. But Don said that 4 was smaller than 8 so 3 big pieces might be bigger than 5 smaller ones.

They decided to figure it out! They wrote the two fractions like this: , and asked

 

themselves which symbol: < (less than), > (greater than), or = (equal to) made sense between them. Then Don got a brilliant idea. He decided to turn the 3-quarter-sized-pieces into
6-eighth-sized-pieces by cutting each one of them in half like we did before with the pizza remember? This, of course, doubled the number of pieces to create 6 pieces from the 3 but now each one was only one-eighth of a candy bar no longer a quarter. Both twins quickly concluded that
6-eighths is more candy than 5-eighths so they chose the bigger piece the one that was 3 quarters of the bar.

What our twins did was find a common denominator for the two fractions.
They cut the two pieces of candy into the same sized pieces.
They found that

 

When fractions have the same denominators they are easy to compare.
We look at their numerators and compare them.

When fractions have the same denominators, the biggest
fraction is the one with the biggest numerator
.

Ex: In the same way, , because 2 < 6.

 

When fractions have the same numerators they are also easy to compare. We just have to know into how many pieces the whole pie, candy bar, whatever was cut.
As we just learned,
eighth-sized-pieces are half the size of quarter-sized-pieces, and so on.

When the numerators are the same, the biggest fraction
is the one with the smallest denominator
.

Ex: . In the same way, , because 7 > 6.

 

Equivalent Fractions Exercise #3:

Change these into equivalent fractions. Write < (less than), > (greater than), or = (equal to)
between them to indicate the relation.

a)

b)

c)

d)

e)

(solutions)

 

Solutions for Exercises

 


Exercise #1:

a)

b)

c)

d)

e)

 

 

Exercise #2:

Factor these fractions then Reduce to lowest terms.

a)

b)

c)

d)

e)

.

 

Exercise #3:

a)

b)

c)

d)

e)

 

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