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By the end of this section, you will be able to:
  • Find equivalent fractions
  • Simplify fractions
  • Multiply fractions
  • Divide fractions
  • Simplify expressions written with a fraction bar
  • Translate phrases to expressions with fractions

A more thorough introduction to the topics covered in this section can be found in the Prealgebra chapter, Fractions .

Find equivalent fractions

Fractions are a way to represent parts of a whole. The fraction 1 3 means that one whole has been divided into 3 equal parts and each part is one of the three equal parts. See [link] . The fraction 2 3 represents two of three equal parts. In the fraction 2 3 , the 2 is called the numerator    and the 3 is called the denominator    .

Two circles are shown, each divided into three equal pieces by lines. The left hand circle is labeled “one third” in each section. Each section is shaded. The circle on the right is shaded in two of its three sections.
The circle on the left has been divided into 3 equal parts. Each part is 1 3 of the 3 equal parts. In the circle on the right, 2 3 of the circle is shaded (2 of the 3 equal parts).
Doing the Manipulative Mathematics activity “Model Fractions” will help you develop a better understanding of fractions, their numerators and denominators.


A fraction is written a b , where b 0 and

  • a is the numerator and b is the denominator .

A fraction represents parts of a whole. The denominator b is the number of equal parts the whole has been divided into, and the numerator a indicates how many parts are included.

If a whole pie has been cut into 6 pieces and we eat all 6 pieces, we ate 6 6 pieces, or, in other words, one whole pie.

A circle is shown and is divided into six section. All sections are shaded.

So 6 6 = 1 . This leads us to the property of one that tells us that any number, except zero, divided by itself is 1.

Property of one

a a = 1 ( a 0 )

Any number, except zero, divided by itself is one.

Doing the Manipulative Mathematics activity “Fractions Equivalent to One” will help you develop a better understanding of fractions that are equivalent to one.

If a pie was cut in 6 pieces and we ate all 6, we ate 6 6 pieces, or, in other words, one whole pie. If the pie was cut into 8 pieces and we ate all 8, we ate 8 8 pieces, or one whole pie. We ate the same amount—one whole pie.

The fractions 6 6 and 8 8 have the same value, 1, and so they are called equivalent fractions. Equivalent fractions are fractions that have the same value.

Let’s think of pizzas this time. [link] shows two images: a single pizza on the left, cut into two equal pieces, and a second pizza of the same size, cut into eight pieces on the right. This is a way to show that 1 2 is equivalent to 4 8 . In other words, they are equivalent fractions    .

A circle is shown that is divided into eight equal wedges by lines. The left side of the circle is a pizza with four sections making up the pizza slices. The right side has four shaded sections. Below the diagram is the fraction four eighths.
Since the same amount is of each pizza is shaded, we see that 1 2 is equivalent to 4 8 . They are equivalent fractions.

Equivalent fractions

Equivalent fractions are fractions that have the same value.

How can we use mathematics to change 1 2 into 4 8 ? How could we take a pizza that is cut into 2 pieces and cut it into 8 pieces? We could cut each of the 2 larger pieces into 4 smaller pieces! The whole pizza would then be cut into 8 pieces instead of just 2. Mathematically, what we’ve described could be written like this as 1 · 4 2 · 4 = 4 8 . See [link] .

A circle is shown and is divided in half by a vertical black line. It is further divided into eighths by the addition of dotted red lines.
Cutting each half of the pizza into 4 pieces, gives us pizza cut into 8 pieces: 1 · 4 2 · 4 = 4 8 .

This model leads to the following property:

Questions & Answers

find the solution to the following functions, check your solutions by substitution. f(x)=x^2-17x+72
Carlos Reply
Aziza is solving this equation-2(1+x)=4x+10
Sechabe Reply
No. 3^32 -1 has exactly two divisors greater than 75 and less than 85 what is their product?
x^2+7x-19=0 has Two solutions A and B give your answer to 3 decimal places
Adedamola Reply
please the answer to the example exercise
Patricia Reply
3. When Jenna spent 10 minutes on the elliptical trainer and then did circuit training for20 minutes, her fitness app says she burned 278 calories. When she spent 20 minutes onthe elliptical trainer and 30 minutes circuit training she burned 473 calories. How manycalories does she burn for each minute on the elliptical trainer? How many calories doesshe burn for each minute of circuit training?
Edwin Reply
John left his house in Irvine at 8:35 am to drive to a meeting in Los Angeles, 45 miles away. He arrived at the meeting at 9:50. At 3:30 pm, he left the meeting and drove home. He arrived home at 5:18.
DaYoungan Reply
p-2/3=5/6 how do I solve it with explanation pls
Adedamola Reply
don't understand answer
is y=7/5 a solution of 5y+3=10y-4
Adedamola Reply
don't understand answer
Lucinda has a pocketful of dimes and quarters with a value of $6.20. The number of dimes is 18 more than 3 times the number of quarters. How many dimes and how many quarters does Lucinda have?
Rhonda Reply
Find an equation for the line that passes through the point P ( 0 , − 4 ) and has a slope 8/9 .
Gabriel Reply
is that a negative 4 or positive 4?
y = mx + b
if negative -4, then -4=8/9(0) + b
if positive 4, then 4=b
then plug in y=8/9x - 4 or y=8/9x+4
Macario is making 12 pounds of nut mixture with macadamia nuts and almonds. macadamia nuts cost $9 per pound and almonds cost $5.25 per pound. how many pounds of macadamia nuts and how many pounds of almonds should macario use for the mixture to cost $6.50 per pound to make?
Cherry Reply
Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350 % mark - up
Makaila Reply
the sum of two Numbers is 19 and their difference is 15
Abdulai Reply
2, 17
Felecia left her home to visit her daughter, driving 45mph. Her husband waited for the dog sitter to arrive and left home 20 minutes, or 13 hour later. He drove 55mph to catch up to Felecia. How long before he reaches her?
Rafi Reply
hola saben como aser un valor de la expresión
Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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