U Teach Your Child Fractions 5: Equivlent Fractions

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Note: in this note a / b is used to represent the fraction with numerator a and denominator b.

For the child encountering fractions for the first time, keep two things in mind. First, fractions are unlike any numbers he has met. For, it takes two numbers to represent a fraction, and each number conveys different information. The numbers in his previous experiences conveyed its information with one number. Second, one fraction has an infinite number of representations. His previous experiences featurd numbers that had only one representation. This new terrain, fractions, often excites the quick learner but others adjust to it over time.

Good examples bring excitement to this new terrain. So draw pictures and use the fraction terminology freely, in particular, point out the relationship between the number of cells in the part and the total number of cells in the whole.

A helpful clue that will enhance your child’s success in learning fractions is to teach her how to select the appropriate representation of a fraction when she has to solve a problem. For example, when adding fractions, addition is greatly simplified when the appropriate representation (often referred to as common denominator) is used.

The goals of this note are to:
•    Name the representations of a given fraction
•    Determine when two fractions represent the same fraction
•    Generate other fractions that represent a given fraction

•    Assign a name to the family of representations of a given fraction.

Different representations of the same fraction are called equivalent fractions.

How do you determine when fractions are equivalent?

Given the fractions a / b and c / d.

The fraction c / d is equivalent to a / b  when ad=bc.

In words, the numerator of the first fraction times the denominator of the second fraction is equal to the denominator of the first fraction times the numerator of the second.

In symbols, a times d  =  b times c.

We write the equivalence of  a / b and  c / d by the expression a / b = c / d.


2 / 3 is equivalent to 4 / 6 because 2 * 6 = 3 * 4.  So 2 / 3 = 4 / 6.

4 / 7 is equivalent to 12 / 21 because 4 * 21 = 7 * 12.  So 4 / 7 = 12 / 21.

Why should you study equivalent fractions?

Equivalent fractions have the same mathematically information as the given fraction, but they make computing with fractions simpler.

How can I generate equivalent fractions of a given fraction?

Answer: Multiply the numerator and denominator of the given fraction by the same number.

The chosen number cannot be ZERO.

Also dividing the numerator and denominator of a given fraction by the same number also generates an equivalent fractions. Again the chosen number cannot be ZERO.

A fraction has many representations. So which name should one use to identify it?  

We use the representation that is in the simplest form. This will be further clarified in future tutorial.

In the above example, the simplest form is 5/8.


Give four equivalent fractions to each of these fractions:  3/7,  4/11,  2/5,  60/100.

Remember: Zero cannot be used when forming equivalent fractions. All other numbers are permitted.

When a given fraction is given, let’s demonstrate how to get the equivalent fraction that has one in the numerator.

So we ask, what is the equivalent fraction of 3/4 that has one i the numerator. Divide the numerator and the denominator by 3. See the following example.

Click the link for more on equivalent fractions.


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