The Set of Rational Numbers
Maya and Ethan order the same size pizza. Maya eats $\frac{3}{6}$ of hers and Ethan eats $\frac{4}{8}$ of his. Did they eat the same amount? And is a number like $5$ or $-2$ also a fraction in disguise? In this section you will meet the whole family of numbers that can be written as fractions, and learn to simplify, compare, and graph them.
The Rational Numbers
You already know two smaller sets of numbers. The natural numbers $\mathbb{N}$ are the counting numbers $1, 2, 3, \dots$ The integers $\mathbb{Z}$ are $\dots, -2, -1, 0, 1, 2, \dots$ Every natural number is an integer, and every integer is rational. That gives $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q}$.
Why is every integer rational? Because you can give it a denominator of $1$: $5 = \frac{5}{1}$ and $-2 = \frac{-2}{1}$. Some fractions are integers in disguise: $\frac{15}{3} = 5$. Decimals that end, such as $0.6 = \frac{6}{10}$, are rational too.
To classify a number, first simplify it or turn it into a fraction, then ask which sets it fits in.
Copy the table into your notebook and mark with an "X" the sets to which each number belongs (a number can belong to several sets): | Number | Natural ($\mathbb{N}$) |…
Copy the table into your notebook and mark with an "X" the sets to which each number belongs (a number can belong to several sets):
| Number | Natural ($\mathbb{N}$) | Integer ($\mathbb{Z}$) | Rational ($\mathbb{Q}$) |
|---|---|---|---|
| $-5$ | |||
| $\frac{8}{4}$ | |||
| $0,75$ | |||
| $12$ |
Classify the following numbers, indicating which set or sets they belong to (Natural numbers $\mathbb{N}$, Integers $\mathbb{Z}$ or Rational numbers $\mathbb{Q}$): a)…
Classify the following numbers, indicating which set or sets they belong to (Natural numbers $\mathbb{N}$, Integers $\mathbb{Z}$ or Rational numbers $\mathbb{Q}$):
- a) $-5$
- b) $\frac{18}{3}$
- c) $0,75$
- d) $-\frac{1}{4}$
- e) $\frac{20}{5}$
Fractions as Parts and as Division
A fraction $\frac{a}{b}$ has two jobs. It tells you $a$ parts out of $b$ equal parts. It is also the division $a \div b$.
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A proper fraction is less than $1$ (numerator smaller than denominator), like $\frac{5}{8}$.
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An improper fraction is $1$ or more (numerator at least the denominator), like $\frac{7}{3}$ or $\frac{4}{4}$.
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A mixed number is a whole number plus a proper fraction: $\frac{7}{3} = 2\tfrac{1}{3}$. You get it by dividing: $7 \div 3 = 2$ remainder $1$.
The following image represents the loading bar of a YouTube video that is being processed. What fraction of the video has already loaded? What fraction is left to load?…
The following image represents the loading bar of a YouTube video that is being processed. What fraction of the video has already loaded? What fraction is left to load? Express both fractions in the simplest possible form (irreducible fraction).
Three friends, Ashley, Ben, and Carlos, have ordered $2$ family pizzas for dinner. If they want to share them so that everyone eats exactly the same amount, what…
Three friends, Ashley, Ben, and Carlos, have ordered $2$ family pizzas for dinner. If they want to share them so that everyone eats exactly the same amount, what fraction of a pizza does each person get? Is it a proper or an improper fraction? Reason your answer.
Finding a Fraction of a Quantity
To find $\frac{a}{b}$ of a quantity, divide by $b$ to get the size of one part, then multiply by $a$ to take $a$ parts.
Why does this work? Dividing by $b$ cuts the quantity into $b$ equal parts. Multiplying by $a$ collects $a$ of those parts.
To add fractions with different denominators, first rewrite them with a common denominator.
"Of" means multiply. When a problem says "a fraction of what is left," multiply the two fractions.
Of the $320$ followers Lucy has on her photography account, $\frac{3}{8}$ are from her school, $\frac{1}{4}$ are from her neighborhood and the rest are people she has…
Of the $320$ followers Lucy has on her photography account, $\frac{3}{8}$ are from her school, $\frac{1}{4}$ are from her neighborhood and the rest are people she has met on her travels. Calculate:
- a) The number of followers who go to the same school as her.
- b) The number of followers who live in her neighborhood.
A content creator has an external hard drive with a total capacity of $500$ GB. They have used $\frac{3}{20}$ of the space for video clips and $\frac{2}{5}$ for music…
A content creator has an external hard drive with a total capacity of $500$ GB. They have used $\frac{3}{20}$ of the space for video clips and $\frac{2}{5}$ for music and sound effects.
- a) Calculate what fraction of the hard drive the total occupied space represents.
- b) What fraction of the drive is still free?
- c) How many gigabytes (GB) of space are left free?
Design a square mosaic for a swimming pool. The mosaic is divided into three zones of different colors: Zone A takes up $\frac{2}{5}$ of the total. Zone B takes up…
Design a square mosaic for a swimming pool. The mosaic is divided into three zones of different colors:
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Zone A takes up $\frac{2}{5}$ of the total.
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Zone B takes up $\frac{3}{7}$ of what is left after marking out Zone A.
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Zone C takes up the rest of the surface area.
If the total surface area of the swimming pool is $140 \text{ ft}^2$, calculate what fraction of the total Zone C represents and how many square feet correspond to it. Express the result of the surface area of Zone C as an exact decimal number.
Equivalent Fractions
Different fractions can name the same amount. $\frac{3}{6}$, $\frac{4}{8}$, and $\frac{1}{2}$ are all equivalent. Back to Maya and Ethan: they ate the same amount.
Why? Multiplying top and bottom by $3$ cuts every piece into $3$ smaller pieces and counts $3$ times as many. The amount does not change. Some problems call this multiplying step amplification.
To find a missing term, ask what number the known top or bottom was multiplied by, then do the same to the other term.
To prepare a fruit smoothie, the recipe indicates that $\frac{2}{5}$ of a gallon of milk should be used. If we want to prepare a larger amount maintaining the…
To prepare a fruit smoothie, the recipe indicates that $\frac{2}{5}$ of a gallon of milk should be used. If we want to prepare a larger amount maintaining the proportion, write three equivalent fractions to $\frac{2}{5}$ by amplification.
Find the unknown term $x$ in each of the following equalities so that the fractions are equivalent: a) $\frac{x}{12} = \frac{20}{48}$ b) $\frac{5}{9} = \frac{x}{45}$ c)…
Find the unknown term $x$ in each of the following equalities so that the fractions are equivalent:
- a) $\frac{x}{12} = \frac{20}{48}$
- b) $\frac{5}{9} = \frac{x}{45}$
- c) $\frac{14}{x} = \frac{2}{7}$
- d) $\frac{27}{81} = \frac{1}{x}$
Simplest Form
A fraction is in simplest form (some problems say irreducible fraction) when the numerator and denominator share no common factor except $1$. To get there, divide both terms by their GCF, the greatest common factor.
You do not have to find the GCF in one jump. Dividing by $2$, then by $3$, and so on works too. Just keep going until nothing is left to divide out.
Simplify the following fractions until you obtain the irreducible fraction in each case: a) $\dfrac{12}{18}$ b) $\dfrac{20}{50}$ c) $\dfrac{15}{60}$
Simplify the following fractions until you obtain the irreducible fraction in each case:
- a) $\dfrac{12}{18}$
- b) $\dfrac{20}{50}$
- c) $\dfrac{15}{60}$
Simplify the following fractions until you obtain the irreducible fraction. For part c, use the prime factor decomposition of the numerator and the denominator. a)…
Simplify the following fractions until you obtain the irreducible fraction. For part c, use the prime factor decomposition of the numerator and the denominator.
- a) $\frac{48}{72}$
- b) $\frac{150}{375}$
- c) $\frac{420}{1260}$
Comparing and Ordering Rational Numbers
Between any two rational numbers there is always another one, and then another. A quick way to find one is to average the two. For $\frac{1}{3}$ and $\frac{1}{2}$, write $\frac{2}{6}$ and $\frac{3}{6}$. The average is $\frac{5}{12}$, which lies between them. To get a particular denominator, rewrite both numbers with a bigger multiple of it first.
Compare the following pairs of rational numbers using the signs $<$ (less than), $>$ (greater than) or $=$ (equal to): a) $\dfrac{4}{7} \square \dfrac{6}{7}$ b)…
Compare the following pairs of rational numbers using the signs $<$ (less than), $>$ (greater than) or $=$ (equal to):
- a) $\dfrac{4}{7} \square \dfrac{6}{7}$
- b) $\dfrac{3}{2} \square \dfrac{3}{5}$
- c) $\dfrac{1}{3} \square \dfrac{4}{12}$
Order the following rational numbers from smallest to largest. To do this, first reduce them to a common denominator using the least common multiple: $$\frac{2}{3},…
Order the following rational numbers from smallest to largest. To do this, first reduce them to a common denominator using the least common multiple:
$$\frac{2}{3}, \quad -\frac{5}{6}, \quad \frac{7}{12}, \quad -\frac{1}{2}, \quad \frac{3}{4}$$Three friends are completing a soccer sticker album. Alex has stuck in $\frac{1}{3}$ of the total stickers, Beatrice has stuck in $\frac{2}{5}$ of the total and Daniel…
Three friends are completing a soccer sticker album. Alex has stuck in $\frac{1}{3}$ of the total stickers, Beatrice has stuck in $\frac{2}{5}$ of the total and Daniel has the rest of the stickers.
- a) What fraction of the album has Daniel completed?
- b) If the album has a total of $300$ stickers, how many has each person stuck in?
- c) Which of the three friends has stuck in the most stickers?
Rational Numbers on the Number Line
To graph $\frac{a}{b}$, cut each unit of the number line into $b$ equal parts. Then count $a$ of those parts from $0$: to the right if the fraction is positive, to the left if it is negative.
Here each unit is cut into $4$ parts, so the tick marks are fourths. Point $B$ is $5$ ticks to the right of $0$, so $B = \frac{5}{4} = 1\tfrac{1}{4}$. Point $A$ is $3$ ticks to the left of $0$, so $A = -\frac{3}{4}$.
A negative sign can be written in front of the fraction or in the numerator. Those are the same number: $-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$.
Draw a number line and represent the following rational numbers on it approximately. Indicate whether they are greater than, less than or equal to one: a) $\dfrac{1}{2}$…
Draw a number line and represent the following rational numbers on it approximately. Indicate whether they are greater than, less than or equal to one:
- a) $\dfrac{1}{2}$
- b) $\dfrac{5}{4}$
- c) $\dfrac{3}{3}$
Identify the fractions that correspond to the points $A$, $B$ and $C$ marked on the following number line:
Identify the fractions that correspond to the points $A$, $B$ and $C$ marked on the following number line:
Terminating or Repeating?
Every rational number can be written as a decimal by dividing the numerator by the denominator. There are only two possibilities:
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Terminating decimal: the digits stop. $\frac{7}{20} = 0.35$.
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Repeating decimal: a block of digits repeats forever. $\frac{5}{12} = 0.41666\ldots = 0.41\overline{6}$.
The bar marks the digits that repeat.
Why? A terminating decimal is a fraction over $10$, $100$, $1{,}000$, and so on. Since $10^n = 2^n \cdot 5^n$, only $2$s and $5$s can appear in a denominator that can be scaled to a power of $10$.
To turn a terminating decimal into a fraction, read the place value of its last digit, then simplify.
This helps with measurements given as decimals. To find the biggest square that fits evenly into a rectangle, write both sides as fractions with the same denominator. The side of the square is the GCF of the numerators, over that denominator.
A rectangular plot of land $120,45$ feet long by $90,3$ feet wide is to be divided into equal squares of the largest possible surface area. a) Express the dimensions of…
A rectangular plot of land $120,45$ feet long by $90,3$ feet wide is to be divided into equal squares of the largest possible surface area.
- a) Express the dimensions of the land as irreducible fractions.
- b) Calculate the side length of each square, knowing that it must be an exact number of inches.
- c) How many squares will be obtained in total?
Repeating Decimals as Fractions
A repeating decimal is still a rational number, so it has a fraction form. Here is the idea: if you subtract a number from a shifted copy of itself, the infinite repeating tails cancel.
When a single digit $d$ repeats, the fraction is $\frac{d}{9}$. Its simplest form therefore has a denominator of $9$, $3$, or $1$.
Match each decimal number with its corresponding simplified generating fraction: 1. $0,125$ 2. $2,5$ 3. $1{,}\hat{3}$ 4. $0{,}\overline{45}$ a) $\frac{4}{3}$ b)…
Match each decimal number with its corresponding simplified generating fraction:
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$0,125$
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$2,5$
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$1{,}\hat{3}$
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$0{,}\overline{45}$
- a) $\frac{4}{3}$
- b) $\frac{5}{11}$
- c) $\frac{1}{8}$
- d) $\frac{5}{2}$
Find a proper fraction whose sum of its terms (numerator and denominator) is $24$ and which is equivalent to a pure repeating decimal whose repeating part has a single…
Find a proper fraction whose sum of its terms (numerator and denominator) is $24$ and which is equivalent to a pure repeating decimal whose repeating part has a single digit. How many possible solutions exist that meet this condition? Justify your answer by finding at least two of them.
Mixed Repeating Decimals
Some decimals have digits that do not repeat before the repeating block begins, like $0.4\overline{5} = 0.4555\ldots$ Use the same idea, with one extra step: make two copies of $x$ so that the repeating tails line up.
If the decimal has a whole-number part, like $2.0\overline{45}$, convert the decimal part and add the whole number back: $2 + 0.0\overline{45}$.
Calculate the irreducible generating fraction of the following repeating decimals. Then, find a rational number between them that has the number $11$ as its denominator.…
Calculate the irreducible generating fraction of the following repeating decimals. Then, find a rational number between them that has the number $11$ as its denominator.
- a) $2,0 \widehat{45}$
- b) $2, \widehat{045}$
A communications satellite sends an adjustment signal every $0,12\widehat{6}$ seconds. A receiving station is programmed to pick up signals only at intervals that are…
A communications satellite sends an adjustment signal every $0,12\widehat{6}$ seconds. A receiving station is programmed to pick up signals only at intervals that are exact multiples of $\frac{19}{150}$ of a second.
Reason mathematically whether the receiving station will be able to pick up the satellite's signal at some point. To do this, express the signal time as an irreducible fraction and compare it with the station's interval.
Consider the following rational numbers: $$x = \frac{25}{33} \quad , \quad y = 0,7\widehat{57}$$ a) Without performing the division, can you be sure what type of decimal…
Consider the following rational numbers:
$$x = \frac{25}{33} \quad , \quad y = 0,7\widehat{57}$$- a) Without performing the division, can you be sure what type of decimal the fraction $x$ will generate? Justify your answer by analyzing the prime factors of the denominator.
- b) Prove whether $x$ and $y$ are equivalent.
- c) Find the irreducible fraction of the sum $x + y$. Express the result as a mixed repeating decimal.
Ordering Numbers in Different Forms
Real problems mix fractions, decimals, and percents. Before you compare, convert everything to the same form. Decimals are usually easiest.
For negative numbers, convert to decimals and remember that the number closer to zero is greater.
In a high mountain weather station, the following temperature variations with respect to the seasonal mean in degrees Fahrenheit have been recorded: Monday:…
In a high mountain weather station, the following temperature variations with respect to the seasonal mean in degrees Fahrenheit have been recorded:
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Monday: $-\frac{9}{4} \text{ } ^\circ\text{C}$
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Tuesday: $-2,1 \widehat{6} \text{ } ^\circ\text{C}$
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Wednesday: $-| -2,2 \text{ } | ^\circ\text{C}$
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Thursday: $-\frac{22}{10} \text{ } ^\circ\text{C}$
Represent these four values on a number line and order them in increasing order (from smallest to largest) using the $<$ symbol. Justify your answer by converting all the numbers to their irreducible fraction form.
Three friends, Alex, Beatrice and Chris, are taking part in an obstacle course. Alex has completed $\frac{13}{20}$ of the course, Beatrice $64\%$ and Chris has covered…
Three friends, Alex, Beatrice and Chris, are taking part in an obstacle course. Alex has completed $\frac{13}{20}$ of the course, Beatrice $64\%$ and Chris has covered $0,6\widehat{3}$ of the total.
- a) Indicate who is in first position and who is last.
- b) If the total course is $4500$ feet, how many feet does Chris have left to reach the finish line?
- c) Express each person's position as an irreducible fraction with the same denominator.
Back to the start: Maya and Ethan ate the same amount, because $\frac{3}{6} = \frac{1}{2} = \frac{4}{8}$. And yes, $5 = \frac{5}{1}$ and $-2 = \frac{-2}{1}$ are fractions too. Every integer and every decimal that ends or repeats belongs to the rational numbers.