Section Topics

 Fraction, Fraction with Units, Fractions &  Ratios; and Proportionality forwards & backwards.

Section Pages

Fraction How-TOs
1 What is a Fraction
2  Fraction Multiplication I
3 Fraction Multiplication II
4 Fraction  Multiplication III
5 Equivalent Fractions
6 - Products Algebraically
7. Mixed Numbers Etc.,
8. Fraction Comparison, Etc
9  Fraction Addition I
10. Fraction Addition II
11. Add, Subtract, Compare Similarities
12. Fraction Addition III
13  Fraction Multiplication IV
14.  Fraction Division & Reciprocals
15. Division Formulas Justified
16. Fraction Webvideos

Would you like to show yourself or others how to be  algebra power users? 

Efficient ways to Multiply Fractions

[Play Video] 2-3 minutes. Multiplying Fractions with  cancellation of  common factors done first (recommended) or not, with more simplification to be done later.

After reading this page, 

1. calculate a few products of fractions with and with the cancellation methods described below for "efficient" multiplication, or more precisely efficient or easier simplification after (cross) cancellation of common factors. 
2.  Find examples in which non-cancellation of factors makes the numerators and denominator multiples of ten.  For such examples, is it easier to cancel the common factors (powers of 2 and 5) before calculating the product?   
3. If one of the fractions  in a product can be simplified (reduced), should you do so to make the simplification of the product easier?

In general, we may multiply  fractions  as follows:.   

In the resulting fraction, the the numerator (top) is a product of the numerators of the factors and  the denominator (bottom) is a product of the denominator of the factors.  The foregoing describes the first method for multiplying fractions. After that, we would simplify the resulting fraction by canceling common factors in the products numerator and denominator. The rule here is multiply first and cancel second.  But this order can be changed.  Cancellation first   leads to smaller numbers and a quicker way (usually) to get the simplified form of the product. 

Example:  

Now instead of compute the products of the numerators and denominators (and then factoring the products to cancel common factors), we take advantage of the situation that the original numerators and denominators provide factors of the product numerators, and factor further to locate common factors that will cancel. Cancelled factors are crossed-out. 

Here we kept the original numerators and denominators and then factored them in a way that would help simplification (lowering terms) in the product fraction. So we cancelled the 25 and 11 after factorization. Then after no further factors could be cancelled, computed the decimal representation of the product numerator and denominator in reduced form.

 Here is the above product computation revisited with in place cancellation - the same calculation with a cosmetic change.

The first way we did the cancellation (that is,  multiplying the fractions together and then factoring to reduce) provides justification for the cancellation of common factors in the original fractions before multiplication is done. 

Algebraic Viewpoint/Description

 for reading as part of algebra skill development - optional reading for now

Algebraic Shorthand Description 

This description is rather complicated, it can be ignored. None the less, the challenge is to understand what is says or suggests, good luck. Understanding is a sign (not a guarantee) of algebra mastery. 

Euclid's Algorithm, Prime Decomposition factorization, and Rules for recognizing multiples of  whole numbers 2, 3, 5, 9, 10, 25, all provide methods to identify and cancel common factors.  These methods were presented briefly or not in the previous lesson.

 [ Back ] [ Up ] [ Next ] [Top of this Page]  
www.whyslopes.com?

Road Safety Message  Do not walk on a road with your back to the traffic - rule of thumb
Please report by email,  errors in mathematics or grammar or terminology to site author
If a mathematics topic you need is not covered in site pages,  report that as well. Topics in most demand
will be covered first in site growth.  

All trademarks and copyrights on this page are owned by their respective owners.
Copyright to comments & contributions are owned by the Poster. 
The Rest © 1995 onward by site author,   Alan Selby,  All Rights Reserved.