5.10: Fundamentals 10 - Fractions–Basics
- Page ID
- 148797
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)
( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\id}{\mathrm{id}}\)
\( \newcommand{\Span}{\mathrm{span}}\)
\( \newcommand{\kernel}{\mathrm{null}\,}\)
\( \newcommand{\range}{\mathrm{range}\,}\)
\( \newcommand{\RealPart}{\mathrm{Re}}\)
\( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)
\( \newcommand{\Argument}{\mathrm{Arg}}\)
\( \newcommand{\norm}[1]{\| #1 \|}\)
\( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)
\( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)
\( \newcommand{\vectorA}[1]{\vec{#1}} % arrow\)
\( \newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow\)
\( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vectorC}[1]{\textbf{#1}} \)
\( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)
\( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)
\( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)
\( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)
\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)LEARNING GOALS
By the end of this lesson, you should understand
- Fraction terminology: numerator, denominator, fraction bar
- The meaning of a fraction
By the end of this lesson, you should be able to
- Convert from division calculation to a fraction, and vice versa
- Estimate with unit fractions
- Determine if two fractions are equivalent
- Simplify a fraction
FUNDAMENTALS OF THE LESSON
A division occurs when one number is divided by another. We can represent division using a fraction. For example, ten divided by five can be written as \(\dfrac{10}{5}\). Ten is called the numerator and five is called the denominator. The horizontal line is called the fraction bar.
1. Identify the numerator or denominator in the following fractions:
A. \(\dfrac{3}{4}\) Denominator_______________
B. \(\dfrac{5}{6}\) Numerator _________________
C. \(\dfrac{4}{12}\) Denominator________________ Numerator ____________
2. Write the following division calculations as a fraction:
A. 12 divided by 7
B. Six divided by two
C. 27 divided by 15
3. Write the following fractions out in words:
A. \(\dfrac{9}{3}\) ________________________________________
B. \(\dfrac{4}{8}\) _________________________________________
C. \(\dfrac{4}{5}\) _________________________________________
Fractions are of two types: proper and improper. A fraction is called proper if its numerator is less than its denominator. Proper fractions have values less than 1. A fraction is improper if its numerator is greater than or equal to its denominator. These fractions are often written as mixed numbers. A mixed number contains an integer and a proper fraction. For example, 3/2 can be written as 1 \(\frac{1}{2}\), which means 1 + \(\frac{1}{2}\).
Improper fractions can be converted to mixed numbers by dividing the numerator by the denominator.
Example: Convert 11/5 to a mixed number.
Solution: Divide 11 by 5 using long division. The fraction 11/5 represents the division: \(11\div 5\).
The resulting mixed number is 2 \(\frac{1}{5}\). In the mixed number, the quotient (2) is the whole number, the remainder (1) is the numerator in the fraction, and the divisor (5) is the denominator in the fraction.
4 Write the following improper fractions as a mixed number:
A. 4/3 ____________________
B. 12/7 ____________________
C. 19/10 __________________
Division Difficulty
Rosa works in an elementary school. During her work day, Rosa had to do several calculations. She had to divide six homework problems into two sets, 24 cupcakes between 12 children, and assign 12 children into groups of four. She had no problem with any of those calculations. The next day she had four pizzas to divide amongst 12 children. She found that division to be more challenging.
5. Can you explain why the pizza problem seems more difficult?
6. Write Rosa’s problems as division problems (using the fraction bar) and solve them. The first division is shown below:
Six problems divided by two: \(\dfrac{6}{2}\), so 3 problems in each set.
A. 24 cupcakes divided between 12 children:
B. 12 children assigned to groups of four:
C. Four pizzas divided by 12:
NEXT STEPS
Unit Fractions
Joe read that one-fifth of all Americans live in rural areas. He wanted to estimate how many people in his home state of South Carolina live in rural areas. One fifth is an example of a unit fraction. Examples of unit fractions are \(\dfrac{1}{2}, \dfrac{1}{3}, \dfrac{1}{4}\), where the numerator is one.
7. Here are some pictures to illustrate unit fractions. Write the fraction underneath. The first one is done for you.
Unit fractions can be used to determine a portion of an amount. To find a unit fraction of a number, we divide. For example, ½ of an amount is the amount divided by 2. Half of $800, is $800/2, which is $400. To find the unit fraction of a number, we divide the number by the denominator of the unit fraction.
8. For each expression below, determine the corresponding division:
A. \(\dfrac{1}{3}\) of a number = the number divided by ______
B. \(\dfrac{1}{4}\) of a number = the number divided by ______
C. \(\dfrac{1}{5}\) of a number = the number divided by ______
9. Find unit fractions of the numbers below:
A. \(\dfrac{1}{3}\) of 15 = _______
B. \(\dfrac{1}{4}\) of 28 = _______
C. \(\dfrac{1}{5}\) of 45 = _______
In the previous examples, the fractions simplified to whole numbers. \(\dfrac{1}{3}\) of 15 = \(\dfrac{15}{3}\) = 5. Since 15 is a multiple of 5, when we divide 15 by 5, we get 3, a whole number. What about when the numbers do not divide exactly. For example, what is \(\frac{1}{3}\) of 20? This is equal to 20 divided by 3. We can estimate this to be about 7, since \(\frac{20}{3}\) is close to \(\frac{21}{3}\), and \(\frac{21}{3}\) equals 7.
10. For each expression below, determine the corresponding fraction, and estimate the value.
A. \(\dfrac{1}{4}\) of 9: It’s about ________________. Exact fraction:______
B. \(\dfrac{1}{6}\) of 17: It’s about ________________. Exact fraction:______
C. \(\dfrac{1}{10}\) of 98: It’s about ________________. Exact fraction:______
11. Let’s return to the problem Joe was investigating. There are about 5 million people in South Carolina. \(\dfrac{1}{5}\) of people in South Carolina live in a rural area.
A. About how many people in South Carolina live in rural areas?
B. Joe also read that in New York state, there are 19,800,000 people. What is one fifth of the people in New York State? First estimate, then calculate.
TRY THESE
Non-Unit Fractions
There are several ways to think of a fraction that does not have one as its numerator. Let’s look at two thirds (2/3):
We can think of this fraction and represent it in two ways. As shown in image (1) below, we could divide a quantity into three equal parts, and take two of the three parts. Or, as shown in image (2) below, we could take two quantities and find one-third of each of them.
Since the whole quantities are the same in both images, these images represent the same amount.
12. Draw similar diagrams to represent each fraction below. There may be more than one correct answer.
A. \(\dfrac{2}{4}\)
B. \(\dfrac{3}{5}\)
Finding a Fraction of an Amount
When we find a non-unit fraction of a number, we must multiply and divide. We can do so in either order. For example, to find of \(\frac{2}{3}\) an amount, we can find \(\frac{1}{3}\) of the amount, and then multiply by two. Or, we can multiply the amount by 2, and then find \(\frac{1}{3}\) of the result.
Example: Find \(\frac{2}{3}\) of 9 .
Solution: We can find this amount in two ways. Both methods are shown below.
Method 1: Find \(\frac{1}{3}\) of 9 (divide 9 by 3), and then multiply by 2. This is shown below.
\(\frac{2}{3}\) of 9 = (\(\frac{1}{3}\) of 9) x 2 = \(\frac{9}{3}\) x 2 = 3 x 2 = 6
Method 2: Multiply 9 by 2, and then find ⅓ of the result (divide by 3). This is shown below.
\(\frac{2}{3}\) of 9 = (9 x 2) x \(\frac{1}{3}\) = 18 x \(\frac{1}{3}\) = \(\frac{18}{3}\) = 6
13. Find the fractions of the following amounts. Use both methods introduced in the prior example:
A. \(\dfrac{3}{4}\) of 16
B. \(\dfrac{2}{5}\) of 25
C. \(\dfrac{3}{10}\) of 100
NEXT STEPS
Equivalent Fractions
Jenna has 25 chairs in her classroom. Five of them need fixing. One of her students says that this means \(\dfrac{5}{25}\) of the chairs need mending. Another student says \(\dfrac{1}{5}\) of the chairs need mending. Who is right?
|
Diagram 1 (showing 5 of 25 chairs broken) |
Diagram 2 (showing 5 groups of chairs with 1 of 5 broken) |
*The gray boxes represent broken chairs, and the white boxes represent unbroken chairs.
You can see that both students are right. \(\dfrac{5}{25}\) and \(\dfrac{1}{5}\) are equivalent fractions. Equivalent fractions have the same value even though they have different numerators and denominators.
Two fractions are equivalent if you can get from one to the other by multiplying or dividing the top and bottom of the fraction by the same number. For example, the fractions \(\frac{1}{3}\) and \(\frac{3}{9}\) are equivalent since you can multiply the top and bottom of \(\frac{1}{3}\) by 3 to get \(\frac{3}{9}\) or, you can divide the top and bottom of \(\frac{3}{9}\) by 3 to get \(\frac{1}{3}\).
14. Decide whether the following fractions are equivalent or not. If the fractions are equivalent, determine the number that you can multiply one of the fractions by to get the other fraction, or determine the number you can divide one of the fractions by to get the other fraction.
A. \(\dfrac{3}{6}, \dfrac{1}{2}\)
B. \(\dfrac{3}{9}, \dfrac{2}{3}\)
C. \(\dfrac{16}{12}, \dfrac{4}{3}\)
D. \(\dfrac{12}{30}, \dfrac{2}{5}\)
E. \(\dfrac{24}{45}, \dfrac{3}{5}\)
Simplifying Fractions
We can simplify a fraction by reducing it to lowest terms. This means that we find an equivalent fraction that cannot be reduced any further. To simplify a fraction, we divide the top and bottom by the same number.
FURTHER APPLICATIONS
15. Joe found out that there are only about one million people living in rural areas in New York State. He wanted to know what fraction that is of the New York State population, which is about 20 million. Complete the following sentences:
A. _____________ people in New York State live in rural areas out of _____________________ total people in New York State.
B. If we simplify the fraction, we find that __________ person out of every __________ people in New York lives in a rural area.
In the above problem, twenty million is called the reference value. It is the total number (of people in this case) in which you are interested. One million is called the comparison value. It is the number that you are comparing to your reference value.
If one person out of every twenty people in New York State lives in a rural area, then \(\dfrac{1}{20}\) is the fraction of people in New York State who live in rural areas.
16. A. There are 25 students in Rosa’s class. Ten of them are in the after school program. What fraction of Rosa’s class is in the after school program? Simplify the fraction.
B. Eighteen teachers in the school are over thirty-five years of age. There are 30 teachers in the school. What fraction of the teachers in the school are over thirty-five years of age?
C. There are about 5,000 students in the school district. About 500 of them are below grade level in reading. What fraction of the students in the school district are below grade level in reading?
Questions: Fractions–Basics
- Determine which pairs of fractions below are equivalent fractions.
A. \(\dfrac{1}{4}, \dfrac{5}{28}\)
B. \(\dfrac{2}{6}, \dfrac{3}{9}\)
C. \(\dfrac{4}{12}, \dfrac{5}{21}\)
D. \(\dfrac{3}{4}, \dfrac{12}{16}\)
- Keisha has made 9 pizzas. She is having a party for 27 people.
A. Are there enough pizzas for at least one-half pizza for each person? Explain.
B. How many pizzas are there for each person?
- Raul runs a pet shop. He has 18 gallons of salt water for some of his fish. If he divides it equally among his 45 tanks for these fish, how much water will each tank get?
4 Find the following:
A. \(\dfrac{1}{12}\) of 108
B. \(\dfrac{1}{10}\) of 30
C. \(\dfrac{1}{7}\) of 1190
5. Estimate the following. For each expression, determine the whole number that is closest to the actual result:
A. \(\dfrac{1}{6}\) of 31
B. \(\dfrac{1}{8}\) of 23
C. \(\dfrac{1}{50}\) of 346
6. Find the following:
A. \(\dfrac{2}{6}\) of 42
B. \(\dfrac{5}{9}\) of 54
C. \(\dfrac{2}{100}\) of 800
7. There are 30 days in April. Last year it rained on \(\dfrac{1}{2}\) of them. How many days did it rain?
8. There are 30 days in June. Last year it was sunny on \(\dfrac{3}{5}\) of the days in June. How many days was it sunny?