1.2 BEDMAS and fractions

Reliable calculations require an agreed order for interpreting operations and a consistent way to work with parts of a whole. BEDMAS records the order of operations:

brackets,
exponents,
division and multiplication from left to right,
addition and subtraction from left to right.

Fraction. A quotient \(a/b\) in which \(a\) is the numerator, \(b\) is the denominator, and \(b\ne0\).

Fractions follow the same BEDMAS rules. To multiply fractions, multiply their numerators and denominators. To divide, multiply by the reciprocal. To add or subtract, first use a common denominator.

For nonzero denominators,

\[\frac ab\frac cd=\frac{ac}{bd},\qquad \frac ab\div\frac cd=\frac ab\frac dc,\]

and

\[\frac ab+\frac cb=\frac{a+c}{b}.\]

When denominators differ, use the least common denominator rather than adding denominators. A fraction bar also acts as a grouping symbol. Everything in its numerator is evaluated as one group and everything in its denominator as another. In an engineering calculation, keep several digits during intermediate work and round only the final result to a precision justified by the data.

Worked Example: Comparing two fractional operating periods

A pump operates for \(3/8\) of an hour at one setting and \(5/12\) of an hour at another. The total operating time is

\[\frac38+\frac5{12}=\frac9{24}+\frac{10}{24}=\frac{19}{24}\ \text{h}.\]

Try It. A backwash cycle uses \(2/15\) h for draining and \(1/10\) h for rinsing. Find the combined time.

Worked Example: Combining fractions of capacity

If two operating periods process \(1/6\) and \(4/9\) of a tank capacity, then

\[\frac16+\frac49=\frac3{18}+\frac8{18}=\frac{11}{18}.\]

Try It. Combine \(1/4\) and \(2/3\) of a tank capacity.

1.2.1 Complex numerical fractions

A complex fraction has one or more fractions in its numerator or denominator. Simplify the numerator and denominator separately, or multiply both by the least common denominator of all smaller fractions. For example,

\[ \frac{\frac16+\frac49}{\frac24-\frac2{10}} =\frac{\frac{11}{18}}{\frac3{10}} =\frac{11}{18}\cdot\frac{10}{3} =\frac{55}{27}. \]

The equality at every stage makes the order visible. Entering an entire complex fraction into a calculator without grouping parentheses can produce a different expression.

Worked Example: Simplifying a complex fraction

Simplify

\[ \frac{\dfrac{3}{4}+\dfrac{1}{6}} {\dfrac{5}{8}-\dfrac{1}{4}}. \]

Approach

First, add the fractions in the numerator and subtract the fractions in the denominator. Then divide the resulting fractions by multiplying by the reciprocal.

Worked Solution

Simplify the numerator:

\[ \frac{3}{4}+\frac{1}{6} = \frac{9}{12}+\frac{2}{12} = \frac{11}{12}. \]

Simplify the denominator:

\[ \frac{5}{8}-\frac{1}{4} = \frac{5}{8}-\frac{2}{8} = \frac{3}{8}. \]

The original complex fraction becomes

\[ \frac{\dfrac{3}{4}+\dfrac{1}{6}} {\dfrac{5}{8}-\dfrac{1}{4}} = \frac{\dfrac{11}{12}}{\dfrac{3}{8}} = \frac{11}{12}\cdot\frac{8}{3} = \frac{22}{9}. \]

Therefore,

\[ \frac{\dfrac{3}{4}+\dfrac{1}{6}} {\dfrac{5}{8}-\dfrac{1}{4}} = \frac{22}{9}. \]

Interpretation

The main fraction bar represents division. Once the numerator and denominator have each been simplified, divide by multiplying by the reciprocal of the denominator.

Try It. A dimensionless treatment-performance index is calculated using

\[ I= \frac{\dfrac{5}{6}-\dfrac{1}{4}} {\dfrac{2}{3}+\dfrac{1}{9}}. \]

Simplify the complex fraction and determine the value of \(I\).

1.2.2 Units, conversion factors, and precision

Units may be treated as algebraic factors. A conversion factor is a ratio equal to 1, such as \(1000\ \text{L}/1\ \text{m}^3\). Arrange it so the unwanted unit cancels. For example,

\[ 2.4\ \text{m}^3\left(\frac{1000\ \text{L}}{1\ \text{m}^3}\right)=2400\ \text{L}. \]

This cancellation shows why multiplying by 1000 is appropriate. If the units do not cancel as intended, the conversion factor is upside down or the relationship is incomplete. A rate such as litres per minute is a fraction, so multiplying by minutes produces litres.

Exact numbers, such as 1000 L in 1 m³, do not limit precision. Measured values do. Retain guard digits during a calculation and round the final result once. A reported answer should not imply more precision than the measurements support. Estimation provides a quick reasonableness check: a basin roughly 20 m by 8 m by 3 m should have a volume near 480 m³, not 48 m³ or 4800 m³.

Worked Example: Calculating a rectangular basin volume

A clearwell is 18.0 m long, 7.5 m wide, and filled to a depth of 3.2 m. Using \(V=LWD\),

\[ V = (18.0)(7.5)(3.2) = 432\ \text{m}^3. \]

The numerical result is 432, and its unit is cubic metres because three lengths were multiplied.

Try It. Find the volume of a basin measuring 12 m by 6.5 m by 2.0 m.

Worked Example: Converting a daily flow to litres

A small facility treats \(0.85\ \text{m}^3\) of water per day. Since \(1\ \text{m}^3=1000\ \text{L}\),

\[0.85\ \text{m}^3\left(\frac{1000\ \text{L}}{1\ \text{m}^3}\right)=850\ \text{L}.\]

Try It. Convert \(3.6\ \text{m}^3\) to litres.

Applied Problem: Estimating stored water. A rectangular equalization basin is 14.0 m long and 6.0 m wide. The water depth is \(2\tfrac14\) m. Find the stored volume in cubic metres and litres.

1.2.3 Practice Problems

Evaluate \(18-3(4+1)\).

Check Your Work
\(18-15=3\).
Evaluate \(6+2^3(5-2)\).

Check Your Work
\(6+8(3)=30\).
Calculate \(3/8+5/12\).

Check Your Work
\(9/24+10/24=19/24\).
Calculate \((7/10)\div(14/15)\).

Check Your Work
\((7/10)(15/14)=3/4\).
A tank is \(9.0\) m by \(4.0\) m by \(2.5\) m. Find its volume.

Check Your Work
\(V=(9.0)(4.0)(2.5)=90\ \text{m}^3\).
A filter is online for \(5/12\) of a day and in standby for \(1/8\) of a day. What fraction of the day do these periods occupy?

Check Your Work
\(5/12+1/8=10/24+3/24=13/24\) of a day.
Convert \(4.75\ \text{m}^3\) of water to litres.

Check Your Work
\(4.75(1000)=4750\) L.
A chemical feed pump delivers 18 L/min for 35 min. Find the delivered volume.

Check Your Work
\((18\ \text{L/min})(35\ \text{min})=630\) L.