Math Help for Section 3.4, Page 131
Solving
Proportions
For the proportion
$\displaystyle{a \over b} = {c \over
d}$,
the quantities a
and d are
called the extremes
of the
proportion, whereas b
and c are
called the means
of the proportion.
The following proof may help you understand why cross-multiplication
works.
| Statement: | If $\displaystyle{a \over b} = {c \over d},$ then $ad = bc.$ |
||||
| Proof: | $\displaystyle{a \over b}$ | = | $\displaystyle{c \over d}$ | Write assumption. |
|
| $\displaystyle{a \over b} \bullet b$ | = | $\displaystyle{c \over d} \bullet b$ |
Use the Multiplication Property of Equality to multiply each side by b. |
||
| $\displaystyle a$ | = | $\displaystyle{c \over d} \bullet b$ | Cancel out b on the left side. | ||
| $\displaystyle a \bullet d$ | = | $\displaystyle{c \over d} \bullet b \bullet d$ |
Use the Multiplication Property of Equality to multiply each side by d. |
||
| $a \bullet d$ | = | $c \bullet b$ | Cancel out d on the right side. | ||
| $a \bullet d$ | = | $b \bullet c$ | Use the Commutative Property of Multiplication to switch the order of b and c. |
||
| $ad$ | = | $bc$ | The statement is true. |
||
Example
5(b): Tip
Be sure to put the quanity $x-2$ into parenthesis before
you cross multiply.
Example
5: Check
You can check the answer by confirming that the quotients on each side
of the proportion are equal.
$\eqalign{{\textbf{a.}\quad\quad\quad} {{50}
\over x} =& {2 \over
{28}}&{\small\color{red}\quad\quad\text{Write original
equation.}} \cr {{50} \over {\color{red}700}}
\overset{?}{=}& {2 \over
{28}}&{\small\color{red}\quad\quad\text{Substitute 700 for }x.} \cr 0.0714… =&
0.0714…&{\small\color{red}\quad\quad\text{Solution checks. }\checkmark} \cr} $
$\eqalign{{\textbf{b.}\quad\quad} {{x – 2} \over
5} =& {4 \over 3}&{\small\color{red}\quad\quad\text{Write
original equation.}}
\cr {{{\textstyle\color{red}{{26} \over 3}} – 2} \over 5}
\overset{?}{=}& {4 \over
3}&{\small\color{red}\quad\quad\text{Substitute }{26 \over 3}\text{
for }x.} \cr
{{{\textstyle{{26} \over 3}} – {\textstyle{6 \over 3}}} \over 5}
\overset{?}{=}& {4 \over
3}&{\small\color{red}\quad\quad\text{Rewrite 2 as }{6 \over
3}.} \cr {{{\textstyle{{20}
\over 3}}} \over 5} \overset{?}{=}& {4 \over
3}&{\small\color{red}\quad\quad\text{Subtract.}} \cr
{{20} \over 3} \bullet {1 \over 5} \overset{?}{=}& {4 \over
3}&{\small\color{red}\quad\quad\text{Invert divisor and
multiply.}} \cr {{20} \over {15}}\overset{?}{=}&
{4 \over 3}&{\small\color{red}\quad\quad\text{Multiply.}} \cr
{4 \over 3} =& {4 \over
3}&{\small\color{red}\quad\quad\text{Solution
checks. }\checkmark} \cr} $