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1.3 Fractions: equivalent fractions

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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

Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
how many start and codon
Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
innocent
Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
What is specific heat capacity
Destiny Reply
Specific heat capacity is a measure of the amount of energy required to raise the temperature of a substance by one degree Celsius (or Kelvin). It is measured in Joules per kilogram per degree Celsius (J/kg°C).
AI-Robot
specific heat capacity is the amount of energy needed to raise the temperature of a substance by one degree Celsius or kelvin
ROKEEB
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Source:  OpenStax, Contemporary math applications. OpenStax CNX. Dec 15, 2014 Download for free at http://legacy.cnx.org/content/col11559/1.6
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