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B1.04: Section 3

Example 1

Solve $\frac{14}{3}=\frac{8}{x}$. Discussion: Sometimes students learn to solve problems like this one by “cross-multiplying.” That is correct. However, most math teachers prefer to think of solving this by multiplying both sides by the same thing—in this case the product of the two denominators. The result is the same. Both methods are shown below.

 Solution Method: Cross-multiply: Alternate Solution Method: Multiply by common denominator Check: \begin{align}&\frac{14}{3}=\frac{8}{x}\\&14x=3\cdot8\\&14x=24\\&\frac{14x}{14}=\frac{24}{14}\\&x=1.7143\\\end{align} [latex-display]\begin{align}&\frac{14}{3}=\frac{8}{x}\\&3\cdot{x}\cdot\frac{14}{3}=3\cdot{x}\cdot\frac{8}{x}\\&\frac{3}{3}\cdot{x}\cdot14=\frac{x}{x}\cdot3\cdot8\\&1\cdot{x}\cdot14=1\cdot3\cdot8\\\end{align}[/latex-display] \begin{align}&14x=24\\&\frac{14x}{14}=\frac{24}{14}\\&x=1.7143\\\end{align} [latex-display]\begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{14}{3}=\frac{8}{x}\\&\,\,\,\,\,\,\,\,\,\,\,\,\frac{14}{3}\,\,\,?=?\,\,\,\frac{8}{1.7143}\\&4.66667\,\,\,?=?\,\,\,4.66663\\\end{align}[/latex-display] Here, the two sides aren’t exactly equal.   But that isn’t surprising, because we rounded the final answer to the original problem, so we don’t expect the two sides here to be exactly equal.
Notice that the quotient, which is the final answer, didn’t come out even, so I had to round off. I chose to round off to four decimal places. We’ll talk more specifically about how exactly answers should be given later in the course. For now, when you have to round in the final computation, always keep at least three decimal places. That will ensure that your checking will produce answers close enough that you can recognize them as being essentially the same, so your checking is useful.

Example 2

Solve $\frac{7}{33}=\frac{x}{5}$ Discussion: Here the variable isn’t in the denominator, but this illustrates that the basic principle of multiplying both sides by the same non-zero expression works here too.

 Solution Method: Cross-multiply: Alternate Solution Method: Multiply by common denominator Check: \begin{align}&\frac{7}{33}=\frac{x}{5}\\&33x=7\cdot5\\&33x=35\\&\frac{33x}{33}=\frac{35}{33}\\&x=1.0606\\\end{align} \begin{align}&\frac{7}{33}=\frac{x}{5}\\&33\cdot5\cdot\frac{7}{33}=33\cdot5\cdot\frac{x}{5}\\&\frac{33}{33}\cdot5\cdot7=\frac{5}{5}\cdot33\cdot{x}\\&1\cdot{x}\cdot35=1\cdot33\cdot{x}\\&33x=35\\&\frac{33x}{33}=\frac{35}{33}\\&x=1.0606\\\end{align} \begin{align}&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{7}{33}=\frac{x}{5}\\&\,\,\,\,\,\,\,\,\,\,\,\,\frac{7}{33}\,\,\,?=?\,\,\,\frac{1.0606}{5}\\&\,\,\,0.21212\,\,\,\,=\,\,\,0.21212\\\end{align}

Example 3

Solve $\frac{12}{x}=6$

 Solution Method: Cross-multiply: Solution Method: Multiply by common denominator Check: \begin{align}&\frac{12}{x}=\frac{6}{1}\\&6x=12\\&x={}^{12}\!\!\diagup\!\!{}_{6}\;=2\\\end{align} \begin{align}&\frac{12}{x}=6\\&x\cdot\frac{12}{x}=x\cdot6\\&\frac{x}{x}\cdot12=6x\\&12=6x\\&x={}^{12}\!\!\diagup\!\!{}_{6}\;=2\\\end{align} \begin{align}&\frac{12}{x}=6\\&\,\frac{12}{2}\,\,=\,6\\\end{align}

Example 4

Find a formula for h (that is, solve for h.) $\frac{h}{36}=\frac{m}{k}$.

 Solution Method: Cross-multiply: Alternate Solution Method: Multiply by common denominator Check: \begin{align}&\frac{h}{36}=\frac{m}{k}\\&hk=36m\\&\frac{hk}{k}=\frac{36m}{k}\\&h=\frac{36m}{k}\end{align} \begin{align}&\frac{h}{36}=\frac{m}{k}\\&36k\frac{h}{36}=36k\frac{m}{k}\\&hk=36m\\&\frac{hk}{k}=\frac{36m}{k}\\&h=\frac{36m}{k}\end{align} [latex-display]\frac{h}{36}=\frac{m}{k}[/latex-display] Substitute \begin{align}&\frac{\frac{36m}{k}}{36}=\\&\frac{36\cdot{m}}{k}\div36=\\&\frac{36\cdot{m}}{k}\cdot\frac{1}{36}=\\&=\frac{m}{k}\end{align}

Example 5

Find a formula for d (that is, solve for d.) $\frac{a}{0.37}=\frac{r}{d}$.

 Solution Method: Cross-multiply: Alternate Solution Method: Multiply by common denominator Check: \begin{align}&\frac{a}{0.37}=\frac{r}{d}\\&0.37r={a}\cdot{d}\\&d\frac{0.37r}{a}=\frac{a\cdot{d}}{a}\\&\frac{0.37r}{a}=d\\&d=\frac{0.37r}{a}\end{align} \begin{align}&\frac{a}{0.37}=\frac{r}{d}\\&0.37d\cdot\frac{a}{0.37}=0.37d\cdot\frac{r}{d}\\&d\cdot{a}=0.37r\\&\frac{d\cdot{a}}{a}=\frac{0.37r}{a}\\&d=\frac{0.37r}{a}\end{align} [latex-display]\frac{a}{0.37}=\frac{r}{d}[/latex-display] Substitute for d on the right. \begin{align}&\frac{r}{d}=\frac{r}{\frac{0.37r}{a}}\\&=r\div\frac{0.37r}{a}\\&=\frac{r}{1}\div\frac{0.37r}{a}\\&=\frac{r}{1}\cdot\frac{a}{0.37r}\\&=\frac{ra}{0.37r}\\&=\frac{a}{0.37}\end{align}