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Cross Multiplication with Fractions,
Mean Proportional



Cross multiplication with fractions is a process where we multiply the numerators and denominators of fractions together.

We can look at cross multiplication involving two fractions.


With the fractions   \bf{\frac{3}{4}}   and   \bf{\frac{5}{6}},

cross multiplication between them gives:     3 × 6 = 18     and     4 × 5 = 20.



Fractions with same Value

Cross multiplication turns out to produce an interesting result when done with two different fractions that are of the same overall value.

If two fractions   \frac{\mathtt{a}}{\mathtt{b}}   and   \frac{\mathtt{c}}{\mathtt{d}}   are the same overall value.

Then.   a × d  =  b × c



Examples




1.1

\bf \frac{1}{2}  =  \bf \frac{4}{8}     ,    1 × 8 = 8      2 × 4 = 8     ,     1 × 8  =  2 × 4


1.2

\bf \frac{3}{5}  =  \bf \frac{9}{15}     ,    3 × 15 = 45      5 × 9 = 45     ,     3 × 15  =  5 × 9


1.3

\bf \frac{7}{4}  =  \bf \frac{14}{8}     ,    7 × 8 = 56      4 × 14 = 56     ,     7 × 8  =  4 × 14






Cross Multiplication with Fractions Uses

This cross multiplication might not look like anything particularly useful at first glance in the first few examples with numbers.

But cross multiplication is something that really can be particularly handy when working with fractions and variables.


In a situation such as:     \frac{x}{4}  =  \frac{12}{6}

Cross multiplication can help us find the value of the variable  x.

x \times 6   =   4 \times 12       =>       6x  =  48

Then dividing each side by  6.

\frac{6x}{6}  =  \frac{48}{6}

x  =  \frac{48}{6}     ,     x  =  8



Example



2.1

Establish the value of the variable  h  below.

Solution

\bf{\frac{8}{h}}  =  \bf{\frac{36}{9}}


8 × 9   =   36 × h

  72   =   36h

\bf{\frac{72}{36}}   =   \bf{\frac{36h}{36}}       ,       2  =  h






Mean Proportional

Cross multiplication with fractions also helps us find out what is called the “Mean Proportional” of  2  numbers.


If we have a situation with two positive numbers  a  and  b  such that:

\tt{\bf{\frac{a}{m}}}  =  \tt{\bf{\frac{m}{b}}}

Then  m  is known as the mean proportional, and the value of it can be found with cross multiplication.

a × b   =   m × m

ab  =  m2      =>      \bf\sqrt{ab}  =  m



Examples




3.1
Mean proportional of  4  and  16  ?

\bf{\frac{4}{m}}  =  \bf{\frac{m}{16}}

4 × 16  =  m × m

64  =  m2      =>      \bf\sqrt{64}  =  m      =>      8  =  m



3.2
Mean proportional of  12  and  27  ?

We can really just go straight to  \bf\sqrt{ab}.

m   =   \bf\sqrt{12\times27}   =   \bf\sqrt{324}   =   18



3.3
If the Mean Proportional of two numbers  5  and  a  is  10, find the value of  a.

Solution

\bf{\frac{5}{10}}  =  \bf{\frac{10}{a}}

5 × a   =   10 × 10

5a  =  100      =>      a  =  \bf{\frac{10}{5}}      =>      a  =  20







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