Fractions and Rational Numbers
Maya ate $\frac{2}{4}$ of a pizza. Jayden ate $\frac{3}{6}$ of an identical pizza. Who ate more? And what would a fraction like $-\frac{3}{4}$ even mean? By the end of this page you will be able to answer both, and to place every fraction, positive or negative, exactly where it belongs.
Fractions as parts of a whole
A fraction $\frac{a}{b}$ describes equal parts of a whole. The denominator $b$ tells how many equal parts the whole is cut into. The numerator $a$ tells how many of those parts we are talking about.
A fraction is also a division: $\frac{3}{8} = 3 \div 8$.
The word equal matters. Cutting a pizza into 8 pieces of different sizes does not give eighths.
At a dinner with friends, you have ordered a family pizza that comes divided into $8$ equal slices. If you have eaten $3$ slices, identify the terms of the fraction that…
At a dinner with friends, you have ordered a family pizza that comes divided into $8$ equal slices. If you have eaten $3$ slices, identify the terms of the fraction that represents the part of the pizza you have consumed.
- a) What is the numerator and what does it indicate in this context?
- b) What is the denominator and what does it indicate in this context?
Look at the following geometric figures and write the fraction that represents the colored part in each case. Then, indicate which of them is a proper fraction and which…
Look at the following geometric figures and write the fraction that represents the colored part in each case. Then, indicate which of them is a proper fraction and which is a fraction equal to one.
Proper fractions, improper fractions, and mixed numbers
Compare the numerator with the denominator.
| Type | Rule | Value | Example |
|---|---|---|---|
| Proper fraction | numerator < denominator | less than 1 | $\frac{5}{8}$ |
| Equal to one | numerator = denominator | exactly 1 | $\frac{6}{6}$ |
| Improper fraction | numerator > denominator | greater than 1 | $\frac{11}{4}$ |
(Some books also call $\frac{6}{6}$ improper, since its numerator is not smaller than its denominator.)
An improper fraction can be written as a mixed number: a whole number plus a proper fraction. Divide the numerator by the denominator. The quotient is the whole part, and the remainder goes over the same denominator.
Classify the following fractions into proper or improper. Remember that to differentiate them you must look at the relationship between the numerator and the…
Classify the following fractions into proper or improper. Remember that to differentiate them you must look at the relationship between the numerator and the denominator:
$$\frac{3}{5}, \quad \frac{7}{4}, \quad \frac{11}{12}, \quad \frac{9}{2}, \quad \frac{5}{5}, \quad \frac{1}{8}$$After an athletics training session, you have drunk $\frac{7}{3}$ of a gallon of water. Express this amount as a mixed number to find out how many full gallons and what…
After an athletics training session, you have drunk $\frac{7}{3}$ of a gallon of water. Express this amount as a mixed number to find out how many full gallons and what fraction of the next one you have consumed.
A fraction of a quantity
To find $\frac{a}{b}$ of an amount, divide by the denominator, then multiply by the numerator. Dividing by $b$ finds the size of one equal part. Multiplying by $a$ collects $a$ of them.
When you need a fraction of a fraction, "of" means multiply. Multiply the numerators and multiply the denominators: $\frac{a}{b} \text{ of } \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$. Why? If a piece that is $\frac{1}{5}$ of the whole is cut into 2 equal pieces, each is $\frac{1}{2}$ of $\frac{1}{5}$, which is $\frac{1}{10}$ of the whole.
If you know how much a fraction of the whole is worth, you can go backward to find the whole: divide by the numerator to get one part, then multiply by the denominator.
Your cell phone battery has a total capacity of $4500$ mAh. If the screen shows that you have exactly $\frac{2}{9}$ of the total charge left, how many mAh of battery do…
Your cell phone battery has a total capacity of $4500$ mAh. If the screen shows that you have exactly $\frac{2}{9}$ of the total charge left, how many mAh of battery do you have available at this moment?
A well-known "streamer" has $420\,000$ followers on her main platform. When analyzing her statistics, she observes that $\frac{3}{7}$ of her followers connect from the…
A well-known "streamer" has $420\,000$ followers on her main platform. When analyzing her statistics, she observes that $\frac{3}{7}$ of her followers connect from the United States, while $\frac{2}{5}$ do so from Latin America.
- a) Calculate the number of followers who connect from the United States.
- b) Calculate the number of followers who connect from Latin America.
- c) What fraction of the total followers do those who do not connect from either of these two regions represent?
A content creator on a streaming* platform starts their live stream with a certain number of viewers. During the first hour, they lose $\frac{2}{9}$ of their followers.…
A content creator on a streaming platform starts their live stream with a certain number of viewers.
-
During the first hour, they lose $\frac{2}{9}$ of their followers.
-
In the second hour, $\frac{1}{4}$ of those remaining leave.
-
At the end of the third hour, $45$ new people join, which means they now have $\frac{3}{4}$ of the initial amount of viewers.
Calculate the total fraction of viewers who left relative to the start and determine how many viewers they had when starting the live stream.
Fractions of an area
Many problems show a figure and ask what fraction is shaded. The rule is the same as before: count the equal pieces in the whole, then count the shaded ones.
If the pieces in the picture are not equal, cut the figure into the smallest equal pieces that fit.
When you cut one piece into smaller equal pieces, each new piece is a fraction of a piece. If a piece that is $\frac{1}{5}$ of the whole is cut into 2 equal parts, each part is $\frac{1}{10}$ of the whole. If the whole has a known area, find a fraction of it exactly as in the last section.
In a data center, the storage panel is divided into identical modules. Look at the following graphical representation where the colored squares indicate the space…
In a data center, the storage panel is divided into identical modules. Look at the following graphical representation where the colored squares indicate the space occupied by a social network's database.
- a) Write the fraction that represents the occupied space and the fraction that represents the free space.
- b) Is it a proper or improper fraction? Justify your answer by comparing the numerator and the denominator.
Look at the following geometric figure representing a square solar panel of side $L$. The panel is divided into different sections to maximize its efficiency. Calculate…
Look at the following geometric figure representing a square solar panel of side $L$. The panel is divided into different sections to maximize its efficiency.
Calculate what fraction of the total area of the square the shaded region A represents. If the total area is $1$ square foot, express the area of region B as an irreducible fraction.
A graphic designer is creating a logo based on a regular hexagon. They have divided the hexagon by joining its opposite vertices to form six equilateral triangles.…
A graphic designer is creating a logo based on a regular hexagon. They have divided the hexagon by joining its opposite vertices to form six equilateral triangles. Subsequently, they have divided one of those triangles into four smaller triangles by joining the midpoints of its sides.
If we shade one of those tiny triangles (as shown in orange), what fraction of the total area of the hexagon does it represent? Express the result as an irreducible fraction.
Equivalent fractions and simplest form
Different fractions can name the same amount. $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$ are equivalent fractions.
To make an equivalent fraction, multiply (or divide) the numerator and denominator by the same nonzero number. You are really multiplying by $\frac{3}{3}$ or $\frac{5}{5}$, which equals 1, so the value does not change. In the picture it means cutting every piece into smaller equal pieces. Some problems call this amplifying a fraction.
A fraction is in simplest form (also called lowest terms) when the numerator and denominator have no common factor except 1. To get there, divide both by their GCF (greatest common factor). Some problems say irreducible fraction. It means the same thing.
Checking by cross-multiplication. Two fractions are equivalent exactly when their cross products are equal: $\frac{a}{b} = \frac{c}{d}$ when $a \cdot d = b \cdot c$. The reason is that both fractions rewritten over the common denominator $b \cdot d$ have numerators $a \cdot d$ and $b \cdot c$.
Finding a missing number. To solve $\frac{6}{8} = \frac{x}{20}$, you can simplify first, $\frac{6}{8} = \frac{3}{4}$. Since $4 \times 5 = 20$, multiply the top by 5 too, so $x = 15$. Or cross-multiply: $8x = 6 \times 20 = 120$, so $x = 15$. Cross-multiplying always works, even when the scale factor is not a whole number.
Back to the start: Maya ate $\frac{2}{4} = \frac{1}{2}$ of her pizza and Jayden ate $\frac{3}{6} = \frac{1}{2}$ of his. They ate exactly the same amount.
A video game takes up $\frac{12}{16}$ of the space on a memory card. Your friend tells you that this value is equivalent to $\frac{3}{4}$ of the card. Check if the two…
A video game takes up $\frac{12}{16}$ of the space on a memory card. Your friend tells you that this value is equivalent to $\frac{3}{4}$ of the card. Check if the two values are equivalent fractions using the cross-multiplication method.
Simplify the following fractions until you obtain their irreducible fraction: a) $\frac{15}{20}$ b) $\frac{24}{36}$ c) $\frac{10}{100}$
Simplify the following fractions until you obtain their irreducible fraction:
- a) $\frac{15}{20}$
- b) $\frac{24}{36}$
- c) $\frac{10}{100}$
To prepare a lilac paint mixture, an artist uses $\frac{4}{10}$ of a gallon of blue color for every gallon of white paint. If the artist wants to keep exactly the same…
To prepare a lilac paint mixture, an artist uses $\frac{4}{10}$ of a gallon of blue color for every gallon of white paint. If the artist wants to keep exactly the same shade but needs to make a larger quantity:
- a) Find the value of $x$ in the following equality so that the fractions are equivalent: $\frac{4}{10} = \frac{x}{25}$.
- b) Write three fractions equivalent to $\frac{4}{10}$ by amplification and one by simplification.
Comparing and ordering fractions
Which fraction is bigger? Use the easiest tool that works.
-
Same denominator: the larger numerator wins. $\frac{5}{9} > \frac{4}{9}$.
-
Same numerator: the smaller denominator wins, because the pieces are bigger. $\frac{7}{8} > \frac{7}{12}$.
-
Common denominator: rewrite both fractions with the LCM (least common multiple) of the denominators, then compare numerators.
-
Cross products: for $\frac{a}{b}$ and $\frac{c}{d}$ with positive denominators, compare $a \cdot d$ with $b \cdot c$. The larger product is on the side of the larger fraction.
-
Simplify first: to compare proportions like $\frac{36}{54}$ and $\frac{12}{18}$, reduce both to simplest form. If the simplest forms match, the proportions are equal.
You can also turn each fraction into a decimal by dividing, which is handy for checking that a fraction lies between two others.
What happens if you add the same number to the top and bottom? Take $\frac{3}{7}$ and add 2 to both: $\frac{5}{9}$. Cross products: $3 \times 9 = 27$ and $7 \times 5 = 35$, so $\frac{3}{7} < \frac{5}{9}$. The fraction grew. Now try an improper fraction, $\frac{7}{3}$, adding 2: $\frac{9}{5}$. Cross products: $7 \times 5 = 35$ and $3 \times 9 = 27$, so $\frac{7}{3} > \frac{9}{5}$. This time the fraction shrank. Always test with cross products instead of guessing.
In a school basketball championship, the "Tigers" team has scored $48$ baskets out of $72$ attempts, while the "Falcons" have scored $36$ baskets out of $54$ attempts.…
In a school basketball championship, the "Tigers" team has scored $48$ baskets out of $72$ attempts, while the "Falcons" have scored $36$ baskets out of $54$ attempts.
- a) Write the success fraction for each team.
- b) Reduce both fractions to their irreducible form using the Greatest Common Factor (GCF) method.
- c) In light of the results, which of the two teams has a better success rate?
Three friends are downloading the same update file for a video game. After 10 minutes, the progress of each one is as follows: Marcus: $\frac{5}{6}$ of the total. Lucy:…
Three friends are downloading the same update file for a video game. After 10 minutes, the progress of each one is as follows:
-
Marcus: $\frac{5}{6}$ of the total.
-
Lucy: $\frac{3}{4}$ of the total.
-
Sam: $\frac{11}{12}$ of the total.
- a) Reduce the three fractions to a common denominator using the least common multiple (LCM).
- b) Order the friends from least to most progress using the $<$ or $>$ symbols.
Analyze the following property of fractions: "If we have a positive proper fraction $\frac{n}{d}$ (where $n < d$) and we add the same natural number $k$ to both the…
Analyze the following property of fractions: "If we have a positive proper fraction $\frac{n}{d}$ (where $n < d$) and we add the same natural number $k$ to both the numerator and the denominator, we obtain a new fraction $\frac{n+k}{d+k}$".
Prove, using the fraction $\frac{2}{5}$ as an example and adding $k=3$, whether the new fraction is greater than, less than or equal to the original one. Would this conclusion hold if the original fraction were improper ($\frac{5}{2}$)?
Rational numbers
So far every fraction has been positive. Now we allow negatives too.
Every integer is rational: $5 = \frac{5}{1}$ and $-2 = \frac{-4}{2}$. Some fractions are integers in disguise: $\frac{15}{3} = 5$. So the number sets fit inside each other.
Here is the vocabulary. The natural numbers are the counting numbers $1, 2, 3, \dots$. Put 0 in with them and you get the whole numbers $0, 1, 2, 3, \dots$. Add the negatives and you get the integers. So $0$ is a whole number and an integer, but it is not a natural number. (Some books count 0 as natural. In the US, we start counting at 1.) To classify a fraction, simplify it first and see what number you get.
Classify the following expressions, indicating whether they represent a natural number, an integer (but not a natural number) or a non-integer rational number: a)…
Classify the following expressions, indicating whether they represent a natural number, an integer (but not a natural number) or a non-integer rational number:
- a) $\frac{15}{3}$
- b) $-\frac{12}{4}$
- c) $\frac{7}{5}$
- d) $\frac{0}{8}$
- e) $-\frac{1}{2}$
- f) $\frac{24}{6}$
Negative fractions: opposite and reciprocal
A fraction can be negative, and the negative sign can sit in three equivalent places.
Why does flipping work? Because $\frac{a}{b} \times \frac{b}{a} = \frac{ab}{ab} = 1$. And the opposite works because $\frac{a}{b} + \left(-\frac{a}{b}\right) = 0$.
Write the opposite and the reciprocal of each of the following fractions. Pay attention to the sign: a) $\frac{2}{5}$ b) $-\frac{3}{7}$ c) $4$ (remember that an integer…
Write the opposite and the reciprocal of each of the following fractions. Pay attention to the sign:
- a) $\frac{2}{5}$
- b) $-\frac{3}{7}$
- c) $4$ (remember that an integer can also be written as a fraction)
Rational numbers on the number line
To place $\frac{a}{b}$ on a number line, cut each unit into $b$ equal gaps and count $a$ gaps from 0. For negative numbers, count to the left. Mixed numbers help: $-1\tfrac{1}{2}$ is halfway between $-1$ and $-2$.
Two facts you will use all the time:
-
Order: the farther right, the greater. So $-\frac{1}{2} > -\frac{3}{4}$.
-
Absolute value $|x|$ is the distance from 0, so it is never negative. $\left|-\frac{5}{4}\right| = \frac{5}{4}$.
To find the point halfway between two numbers, average them. To split a segment from $A$ to $B$ into $n$ equal parts, each part has length $\frac{B - A}{n}$. Add that step again and again, starting at $A$.
On a drawing, distances are proportional to the numbers, so you can turn a length in inches into a number-line distance with a proportion.
Look at the position of the highlighted point on the following number line: a) Write the improper fraction that represents the orange point. b) Express this fraction as…
Look at the position of the highlighted point on the following number line:
- a) Write the improper fraction that represents the orange point.
- b) Express this fraction as a mixed number.
- c) If we divided each unit into 8 equal parts instead of 4, what numerator would the equivalent fraction to the previous one have?
In a laboratory experiment, the variation in the temperature of a fluid with respect to the room temperature is recorded. The following values in degrees Fahrenheit are…
In a laboratory experiment, the variation in the temperature of a fluid with respect to the room temperature is recorded. The following values in degrees Fahrenheit are obtained:
$$-\frac{3}{2}, \quad \frac{1}{4}, \quad -\frac{5}{4}, \quad \frac{7}{8}, \quad -1$$- a) Represent these numbers approximately on a number line.
- b) Order the values from smallest to largest (from coldest to warmest).
- c) Indicate which of these values has the greatest absolute value.
On the number line, point A corresponds to the fraction $\frac{1}{6}$ and point B corresponds to $\frac{1}{2}$. If we divide the line segment AB into three equal parts,…
On the number line, point A corresponds to the fraction $\frac{1}{6}$ and point B corresponds to $\frac{1}{2}$.
If we divide the line segment AB into three equal parts, what fractions correspond to the division points? Also place the origin ($0$) on the line and explain what graphical distance (in inches or grid units) there would be between $0$ and point A if the distance between A and B is $12$ in.
Strategies for challenge problems
Harder problems combine the ideas above. Three strategies cover most of them.
1. Unit fractions: subtract what you know, then test. A unit fraction has numerator 1. Suppose $\frac{1}{2} + \frac{1}{x} + \frac{1}{y} = \frac{5}{6}$. Subtract the known part: $\frac{5}{6} - \frac{1}{2} = \frac{1}{3}$. Now find two different unit fractions that add to $\frac{1}{3}$. Try $\frac{1}{4}$: $\frac{1}{3} - \frac{1}{4} = \frac{1}{12}$, which is a unit fraction. So $x = 4$ and $y = 12$.
2. Between two fractions. The midpoint is the average: add the two fractions and divide by 2. The midpoint of $\frac{1}{4}$ and $\frac{1}{3}$ is $\frac{1}{2}\left(\frac{7}{12}\right) = \frac{7}{24}$. For any value written in simplest form, every equivalent fraction is a multiple of its numerator and denominator. So $\frac{7}{24}$ can be written as $\frac{14}{48}$ or $\frac{21}{72}$, but never with a denominator that is not a multiple of 24. Fractions that are merely somewhere between two numbers are different: there are infinitely many, and you can find them by rewriting both fractions over a larger common denominator.
3. Constraint searches. When a fraction must satisfy several conditions, list the candidates (for example, each allowed denominator), and test one condition at a time with decimals or cross products. Cross out the ones that fail.
In an archaeological excavation, a papyrus has been found describing a distribution of land using unit fractions (those whose numerator is 1). The scribe noted that a…
In an archaeological excavation, a papyrus has been found describing a distribution of land using unit fractions (those whose numerator is 1). The scribe noted that a farmer's plot is the sum of three different unit fractions and that the total adds up to $\frac{13}{12}$ of the basic unit of measurement.
If we know that the smallest denominator of these fractions is $2$, find the other two denominators so that the equality is correct.
Two friends, Lucy and Jacob, are comparing their progress in an obstacle course. Lucy has completed $\frac{5}{7}$ of the course and Jacob $\frac{6}{7}$. A third runner,…
Two friends, Lucy and Jacob, are comparing their progress in an obstacle course. Lucy has completed $\frac{5}{7}$ of the course and Jacob $\frac{6}{7}$. A third runner, Matthew, is located somewhere between the two (not necessarily at the midpoint).
Find a fraction that represents Matthew's position such that its denominator is a prime number greater than $10$. Justify whether there is more than one possibility.
Find an irreducible fraction $\frac{a}{b}$ that satisfies the following three conditions simultaneously: 1. It is greater than $\frac{3}{5}$ but less than $\frac{2}{3}$.…
Find an irreducible fraction $\frac{a}{b}$ that satisfies the following three conditions simultaneously:
-
It is greater than $\frac{3}{5}$ but less than $\frac{2}{3}$.
-
The sum of its numerator and its denominator is a multiple of $11$.
-
The denominator $b$ is an even number less than $20$.
Justify the steps followed to find it.