1. Home
  2. Contemporary math applications
  3. Exam 1
  4. Fractions
  5. Fractions: equivalent fractions

1.3 Fractions: equivalent fractions

<< Chapter < Page
  Contemporary math applications     Page 1 / 3
Chapter >> Page >
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses equivalent fractions, reducing fractions to lowest terms, and raising fractions to higher terms. By the end of the module students should be able to recognize equivalent fractions, reduce a fraction to lowest terms and be able to raise a fraction to higher terms.

Section overview

  • Equivalent Fractions
  • Reducing Fractions to Lowest Terms
  • Raising Fractions to Higher Terms

Equivalent fractions

Let's examine the following two diagrams.

A rectangle divided equally into three parts, each marked one-third. The left two parts are shaded. To the right of the box is the caption, two-thirds of the whole is shaded. Below this is a rectangle equally divided into six part, with the leftmost four part shaded. to the right of this rectangle is the caption, four-sixths of the whole is shaded.

Notice that both 2 3 size 12{ { {2} over {3} } } {} and 4 6 size 12{ { {4} over {6} } } {} represent the same part of the whole, that is, they represent the same number.

Equivalent fractions

Fractions that have the same value are called equivalent fractions . Equiva­lent fractions may look different, but they are still the same point on the number line.

There is an interesting property that equivalent fractions satisfy.

two-thirds and four-sixths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

A test for equivalent fractions using the cross product

These pairs of products are called cross products .

Is two time six equal to three times four? Yes.

If the cross products are equal, the fractions are equivalent. If the cross products are not equal, the fractions are not equivalent.

Thus, 2 3 size 12{ { {2} over {3} } } {} and 4 6 size 12{ { {4} over {6} } } {} are equivalent, that is, 2 3 = 4 6 size 12{ { {2} over {3} } = { {4} over {6} } } {} .

Sample set a

Determine if the following pairs of fractions are equivalent.

3 4 and 6 8 size 12{ { {3} over {4} } `"and " { {6} over {8} } } {} . Test for equality of the cross products.

three-fourths and six-eigths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

Is three times eight equal to six times four? yes. The cross products are equals.

The fractions 3 4 and 6 8 are equivalent, so 3 4 = 6 8 .

3 8 and 9 16 size 12{ { {3} over {8} } " and " { {9} over {"16"} } } {} . Test for equality of the cross products.

Three-eights and nine-sixteenths, with an arrow from each denominator pointing to the numerator of the opposite fraction.

is three times sixteen equal to nine times eight? No. forty-eight does not equal seventy-two. The cross products are not equal.

The fractions 3 8 size 12{ { {3} over {8} } } {} and 9 16 size 12{ { {9} over {"16"} } } {} are not equivalent.

Practice set a

Determine if the pairs of fractions are equivalent.

1 2 size 12{ { {1} over {2} } } {} , 3 6 size 12{ { {3} over {6} } } {}

Six equals six. , yes

4 5 size 12{ { {4} over {5} } } {} , 12 15 size 12{ { {"12"} over {"15"} } } {}

Sixty equals sixty. , yes

2 3 size 12{ { {2} over {3} } } {} , 8 15 size 12{ { {8} over {"15"} } } {}

30 ≠ 24 , no

1 8 size 12{ { {1} over {8} } } {} , 5 40 size 12{ { {5} over {"40"} } } {}

Forty equals forty. , yes

3 12 size 12{ { {3} over {"12"} } } {} , 1 4 size 12{ { {1} over {4} } } {}

Twelve equals twelve. , yes

Reducing fractions to lowest terms

It is often very useful to conver t one fraction to an equivalent fraction that has reduced values in the numerator and denominator. We can suggest a method for doing so by considering the equivalent fractions 9 15 size 12{ { {9} over {"15"} } } {} and 3 5 size 12{ { {3} over {5} } } {} . First, divide both the numerator and denominator of 9 15 size 12{ { {9} over {"15"} } } {} by 3. The fractions 9 15 size 12{ { {9} over {"15"} } } {} and 3 5 size 12{ { {3} over {5} } } {} are equivalent.

(Can you prove this?) So, 9 15 = 3 5 size 12{ { {9} over {"15"} } = { {3} over {5} } } {} . We wish to convert 9 15 size 12{ { {9} over {"15"} } } {} to 3 5 size 12{ { {3} over {5} } } {} . Now divide the numerator and denominator of 9 15 size 12{ { {9} over {"15"} } } {} by 3, and see what happens.

9 ÷ 3 15 ÷ 3 = 3 5 size 12{ { {9 div 3} over {"15" div 3} } = { {3} over {5} } } {}

The fraction 9 15 size 12{ { {9} over {"15"} } } {} is converted to 3 5 size 12{ { {3} over {5} } } {} .

A natural question is "Why did we choose to divide by 3?" Notice that

9 15 = 3 ⋅ 3 5 ⋅ 3 size 12{ { {9} over {"15"} } = { {3 cdot 3} over {5 cdot 3} } } {}

We can see that the factor 3 is common to both the numerator and denominator.

Reducing a fraction

From these observations we can suggest the following method for converting one fraction to an equivalent fraction that has reduced values in the numerator and denominator. The method is called reducing a fraction .

A fraction can be reduced by dividing both the numerator and denominator by the same nonzero whole number.

Nine-twelfths is equal to nine divided by three, over nine divided by three, which is equal to three-fourths. Sixteen thirtieths is equal to sixteen divided by two, over thirty divided by 2, which is equal to eight-fifteenths. Notice that three over three and two over two are both equal to 1.

Consider the collection of equivalent fractions

5 20 size 12{ { {5} over {"20"} } } {} , 4 16 size 12{ { {4} over {"16"} } } {} , 3 12 size 12{ { {3} over {"12"} } } {} , 2 8 size 12{ { {2} over {8} } } {} , 1 4 size 12{ { {1} over {4} } } {}

Reduced to lowest terms

Notice that each of the first four fractions can be reduced to the last fraction, 1 4 size 12{ { {1} over {4} } } {} , by dividing both the numerator and denominator by, respectively, 5, 4, 3, and 2. When a fraction is converted to the fraction that has the smallest numerator and denomi­nator in its collection of equivalent fractions, it is said to be reduced to lowest terms . The fractions 1 4 size 12{ { {1} over {4} } } {} , 3 8 size 12{ { {3} over {8} } } {} , 2 5 size 12{ { {2} over {5} } } {} , and 7 10 size 12{ { {7} over {"10"} } } {} are all reduced to lowest terms.

Questions & Answers

differentiate between demand and supply giving examples
Lambiv Reply
differentiated between demand and supply using examples
Lambiv
what is labour ?
Lambiv
how will I do?
Venny Reply
how is the graph works?I don't fully understand
Rezat Reply
information
Eliyee
devaluation
Eliyee
t
WARKISA
hi guys good evening to all
Lambiv
multiple choice question
Aster Reply
appreciation
Eliyee
explain perfect market
Lindiwe Reply
In economics, a perfect market refers to a theoretical construct where all participants have perfect information, goods are homogenous, there are no barriers to entry or exit, and prices are determined solely by supply and demand. It's an idealized model used for analysis,
Ezea
What is ceteris paribus?
Shukri Reply
other things being equal
AI-Robot
When MP₁ becomes negative, TP start to decline. Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of lab
Kelo
Extuples Suppose that the short-run production function of certain cut-flower firm is given by: Q=4KL-0.6K2 - 0.112 • Where is quantity of cut flower produced, I is labour input and K is fixed capital input (K-5). Determine the average product of labour (APL) and marginal product of labour (MPL)
Kelo
yes,thank you
Shukri
Can I ask you other question?
Shukri
what is monopoly mean?
Habtamu Reply
What is different between quantity demand and demand?
Shukri Reply
Quantity demanded refers to the specific amount of a good or service that consumers are willing and able to purchase at a give price and within a specific time period. Demand, on the other hand, is a broader concept that encompasses the entire relationship between price and quantity demanded
Ezea
ok
Shukri
how do you save a country economic situation when it's falling apart
Lilia Reply
what is the difference between economic growth and development
Fiker Reply
Economic growth as an increase in the production and consumption of goods and services within an economy.but Economic development as a broader concept that encompasses not only economic growth but also social & human well being.
Shukri
production function means
Jabir
What do you think is more important to focus on when considering inequality ?
Abdisa Reply
any question about economics?
Awais Reply
sir...I just want to ask one question... Define the term contract curve? if you are free please help me to find this answer 🙏
Asui
it is a curve that we get after connecting the pareto optimal combinations of two consumers after their mutually beneficial trade offs
Awais
thank you so much 👍 sir
Asui
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities, where neither p
Cornelius
In economics, the contract curve refers to the set of points in an Edgeworth box diagram where both parties involved in a trade cannot be made better off without making one of them worse off. It represents the Pareto efficient allocations of goods between two individuals or entities,
Cornelius
Suppose a consumer consuming two commodities X and Y has The following utility function u=X0.4 Y0.6. If the price of the X and Y are 2 and 3 respectively and income Constraint is birr 50. A,Calculate quantities of x and y which maximize utility. B,Calculate value of Lagrange multiplier. C,Calculate quantities of X and Y consumed with a given price. D,alculate optimum level of output .
Feyisa Reply
Answer
Feyisa
c
Jabir
the market for lemon has 10 potential consumers, each having an individual demand curve p=101-10Qi, where p is price in dollar's per cup and Qi is the number of cups demanded per week by the i th consumer.Find the market demand curve using algebra. Draw an individual demand curve and the market dema
Gsbwnw Reply
suppose the production function is given by ( L, K)=L¼K¾.assuming capital is fixed find APL and MPL. consider the following short run production function:Q=6L²-0.4L³ a) find the value of L that maximizes output b)find the value of L that maximizes marginal product
Abdureman
types of unemployment
Yomi Reply
What is the difference between perfect competition and monopolistic competition?
Mohammed
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Contemporary math applications' conversation and receive update notifications?

Ask