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Cross Multiplying Fractions


Cross multiplying fractions in Math is a process that can often be required when solving various sums or equations that involve fractions.




Cross Multiplication with 2 Fractions

When cross multiplication deals with two fractions, the process is multiplying the numerator of one of the fractions, by the denominator of the other fraction.


Consider the fractions   \bf{\frac{2}{3}}   and   \bf{\frac{4}{7}},

cross multiplication between them results in:

2 × 7 = 14   and   4 × 3 = 12.


It’s actually the case that cross multiplication gives an interesting result when it’s performed on 2 different fractions that have the same overall value.

If two fractions  \tt{\bf{\frac{a}{b}}}   and   \tt{\bf{\frac{c}{d}}}   are the same value, then:

a × d  =  b × c



Examples    


(1.1) 

\bf{\frac{3}{4}}  =  \bf{\frac{9}{12}}     ,     3 × 12 = 36     4 × 9 = 36

( 3 × 12  =  4 × 9 )


(1.2) 

\bf{\frac{5}{2}}  =  \bf{\frac{20}{8}}     ,     5 × 8  =  40     2 × 20  =  40

( 5 × 8  =  2 × 20 )







Variables and Cross Multiplication


The process of cross multiplication may not look all that interesting or useful at first glance in examples that just deal with numbers.
However, cross multiplication becomes particularly handy when dealing with sums that involve fractions and variables.


If we have a situation such as:     \bf{\frac{t}{6}} = \bf{\frac{20}{4}}

Cross multiplication can be used to help find the value of the variable  t.

t × 4   =   6 × 20

4t  =  120

Now we can proceed by dividing each side by  4.

\bf{\frac{4t}{4}}   =   \bf{\frac{120}{4}}

=>   t  =  \bf{\frac{120}{4}}     ,     t  =  30




Example    


(2.1) 

Find the value of the variable  v  below.

\bf{\frac{16}{v}}  =  \bf{\frac{28}{7}}

Solution   

16 × 7   =   t × 28

112   =   28v

\bf{\frac{112}{28}}   =   \bf{\frac{28v}{28}}     ,     4  =  v







Cross Multiplying Fractions,
Mean Proportional


It’s the case that fraction cross multiplication also assists in finding out what is referred to as the “Mean Proportional” of numbers.


Say you had a situation with two positive numbers  g  and  h where:

\tt{\bf{\frac{g}{m}}}  =  \tt{\bf{\frac{m}{h}}}

In this situation,  m  is called the mean proportional, and the value can be found with fraction cross multiplication.

g × h   =   m × m
gh  =  m2     =>     \bf\sqrt{gh}  =  m




Examples    


(3.1) 

What is the mean proportional of  6  and  24  ?

Solution   

\bf{\frac{6}{m}}  =  \bf{\frac{m}{24}}

6 × 24  =  m × m
144  =  m2     =>     \bf\sqrt{144}  =  m    =>     12  =  m



(3.2) 

What is the mean proportional of  4  and  16  ?

Solution   

We can really just jump straight to  \bf\sqrt{ab}  for an answer.

\bf\sqrt{4\times16}   =   \bf\sqrt{64}   =   8





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