Fractions Part 1 Activity

Section 5.3 Fractions Part 1 Activity

Subsection 5.3.1 Fraction Arithmetic

Activity 5.3.1.

Add these fractions.

\begin{equation*} \frac{5}{12} + \frac{3}{9} \end{equation*}

We can both the numerator and denominator of each fraction by the denominator of the other fraction to make a common denominator.

\begin{align*} \frac{5}{12} + \frac{3}{9} \amp = \frac{5 \times 9}{12 \times 9} + \frac{3 \times 12}{9 \times 12} \\ \amp = \frac{45}{108} + \frac{36}{108}\\ \intertext{Then we can add the numerators.}\\ \amp = \frac{45 + 36}{108} = \frac{81}{108}\\ \intertext{We can reduce this fraction by dividing numberator and denominator by 27.}\\ \amp = \frac{3}{4} \end{align*}

We could also have noticed that \(9\) and \(12\) have a common factor of \(3\text{,}\) meaning that there is a smaller common denominator of \(36\text{.}\) Here is the calculation using that common denomninator instead.

\begin{align*} \frac{5}{12} + \frac{3}{9} \amp = \frac{5 \times 3}{12 \times 3} + \frac{3 \times 4}{9 \times 4} \\ \amp = \frac{15}{36} + \frac{12}{36}\\ \amp = \frac{15 + 12}{3r} = \frac{27}{36}\\ \intertext{We can reduce this fraction by dividing numberator and denominator by 9.}\\ \amp = \frac{3}{4} \end{align*}

After reducing to lowest terms, we see that this is the same result.

Activity 5.3.2.

Subtract these fractions.

\begin{equation*} \frac{8}{5} - \frac{4}{7} \end{equation*}

Solution

There are no common factors between \(5\) and \(7\text{,}\) so the smallest reasonable common denominator is their product, \(35\text{.}\) We multiply the numerator and denmoinator of the first fraction by \(7\) and likewise for the second fraction by \(5\text{.}\)

\begin{align*} \frac{8}{5} - \frac{4}{7} \amp = \frac{8 \times 7}{5 \times 7} - \frac{4 \times 5}{7 \times 5}\\ \amp = \frac{56}{35} - \frac{20}{35}\\ \amp = \frac{56- 20}{35} = \frac{36}{35} \end{align*}

This result is already in lowest terms, so we are finished.

Activity 5.3.3.

Multiply these fractions.

\begin{equation*} \left( \frac{3}{11} \right) \left( \frac{-4}{7} \right) \end{equation*}

Solution

To multiply fractions, we simply mulitply numerators and mulitply denominators.

\begin{equation*} \left( \frac{3}{11} \right) \left( \frac{-4}{7} \right) = \frac{3 \times (-4)}{11 \times 7} = \frac{-12}{77} \end{equation*}

This is already in lowest terms. The negative can be written out front, since multiplying either numerator or denominator by \((-1)\) is the same as making the whole fraction negative.

\begin{equation*} = - \frac{12}{77} \end{equation*}

Activity 5.3.4.

Divide these fractions.

\begin{equation*} \frac{\frac{3}{5}}{\frac{10}{3}} \end{equation*}

Solution

To divide fraction, we multiply by the reciprocal.

\begin{equation*} \frac{\frac{3}{5}}{\frac{10}{3}} = \frac{3}{5} \frac{3}{10} \end{equation*}

Then we just multiply numerator and denominator seperately.

\begin{equation*} \frac{3}{5} \frac{3}{10} = \frac{3 \times 3}{5 \times 10} = \frac{9}{50} \end{equation*}

This is in lowest terms, so we are finished.

Subsection 5.3.2 Fractions and Percentages

Activity 5.3.5.

Write this fraction as a percentage.

\begin{equation*} \frac{3}{20} \end{equation*}

Solution

As a decimal (calculate by long division or by calculator/computer), this is \(0.15\text{.}\) Multiplying by \(100\) means that this is 15%.

Activity 5.3.6.

Write this fraction as a percentage.

\begin{equation*} \frac{7}{53} \end{equation*}

Solution

As a decimal (calculate by long division or by calculator/computer), this is approximatly \(0.132\text{.}\) Multiplying by \(100\) means that this is approximately 13.2%.

Activity 5.3.7.

Write this percentage as a fraction in lowest terms.

\begin{equation*} 48\% \end{equation*}

Solution

Percentages are fractions of \(100\text{,}\) so we start with this percentage over \(100\)

\begin{equation*} \frac{48}{100} \end{equation*}

Then we try to reduce. \(4\) is a common factor, so we divide numerator and denominator by \(4\)

\begin{equation*} \frac{12}{25} \end{equation*}

Activity 5.3.8.

Calculate the percentage change if a population increases from \(1403\) members to \(1784\) members.

Solution

Percentage change is calculate following the description in Subsection 5.2.4.

\begin{equation*} \frac{1784 - 1403}{1403} \times 100 = \frac{381}{1403} \times 100 \doteq (0.272) \times 100 \doteq 27.2 \% \end{equation*}

Activity 5.3.9.

Calculate the percentage change if a population decreases from \(17,809\) members to \(12,512\) members.

Solution

Percentage change is calculate following the description in Subsection 5.2.4. Note that the second value is first in the numerator, which will lead to a negative value, which is expected for this decrease.

\begin{equation*} \frac{12512 - 17809}{17809} \times 100 = \frac{-5297}{17809} \times 100 \doteq (-0.297) \times 100 \doteq -29.7\% \end{equation*}

Subsection 5.3.3 Conceptual Questions

Activity 5.3.10.

What is a fraction? Why is it also notation for division?
Why does addition and subtraction of fractions need common denominator?
Why doesn't multiplication or division of fraction also need common denominator?
How do we manage multiply representation of numbers?
What is a percentage?