Fractions and Decimals
Your phone says it has $\frac{1}{4}$ of its battery left, and the calculator says that $\frac{1}{3} = 0.3333333$ and calmly stops there. But you know that 3 goes on forever. How do you write that without infinite ink? And how do you get the fraction back if all you have is the decimal? On this page you review fractions, learn to go from fraction to decimal and back, and sort out all the kinds of numbers you know.
Fractions: equivalent fractions, simplifying, and a fraction of a quantity
A fraction $\frac{a}{b}$ splits a whole into $b$ equal parts and takes $a$ of them. If a phone battery is divided into 8 bars and 2 are lit, it is $\frac{2}{8}$ charged.
Two fractions are equivalent if they represent the same amount. You get them by multiplying or dividing the numerator and denominator by the same number: $\frac{2}{8} = \frac{1}{4}$. When you can't divide any further, the fraction is irreducible (in lowest terms).
To check whether two fractions are equivalent without drawing, cross-multiply: $\frac{a}{b} = \frac{c}{d}$ if $a \cdot d = b \cdot c$.
To find a fraction of a quantity, divide the quantity by the denominator and multiply by the numerator. The $\frac{3}{5}$ of 200 passengers: $200 \div 5 = 40$ and $40 \times 3 = 120$.
Fractions on the number line
Every fraction has its place on the number line. Divide each unit into as many parts as the denominator says and count from zero. Negative fractions go to the left of 0.
State whether the following pairs of fractions are equivalent. To justify your answer, use cross-multiplication or simplify both fractions to their irreducible form. a)…
State whether the following pairs of fractions are equivalent. To justify your answer, use cross-multiplication or simplify both fractions to their irreducible form.
- a) $\frac{3}{5}$ and $\frac{9}{15}$
- b) $\frac{4}{7}$ and $\frac{10}{21}$
- c) $\frac{-2}{3}$ and $\frac{8}{-12}$
Represent approximately on the following number line the rational numbers indicated below. Numbers to represent: $A = \frac{3}{2}$ $B = -\frac{1}{2}$ $C = \frac{3}{4}$…
Represent approximately on the following number line the rational numbers indicated below.
Numbers to represent:
-
$A = \frac{3}{2}$
-
$B = -\frac{1}{2}$
-
$C = \frac{3}{4}$
-
$D = -1.5$
Look at the following illustration representing the battery level of a cell phone. 1. Write the fraction that represents the charged part of the battery relative to the…
Look at the following illustration representing the battery level of a cell phone.
-
Write the fraction that represents the charged part of the battery relative to the total.
-
Simplify this fraction to obtain its irreducible fraction.
-
If the battery has a total capacity of $4000 \text{ mAh}$, how many mAh of energy are available?
Operations with fractions
-
Adding and subtracting: first rewrite them with a common denominator; operate on the numerators and keep the denominator the same.
-
Multiplying: numerator times numerator and denominator times denominator.
-
Dividing: multiply by the reciprocal fraction: $\frac{a}{b} : \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}$.
-
Exponents: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ and $a^{-n} = \frac{1}{a^n}$, that is, a negative exponent flips the fraction: $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$.
In combined operations follow the order of operations: parentheses, exponents, multiplication and division (left to right), and finally addition and subtraction.
Perform the following combined operations with fractions. Remember to simplify the final result whenever possible and respect the order of operations (brackets first,…
Perform the following combined operations with fractions. Remember to simplify the final result whenever possible and respect the order of operations (brackets first, then multiplication/division, and finally addition/subtraction).
- a) $\frac{2}{3} + \frac{1}{4} \times \frac{2}{5}$
- b) $\left( \frac{3}{2} - \frac{1}{4} \right) : \frac{5}{2}$
- c) $\frac{5}{6} - \frac{1}{2} + \frac{2}{3}$
Perform the following combined operations with fractions, respecting the order of operations. Remember to simplify the final result to obtain the irreducible fraction.…
Perform the following combined operations with fractions, respecting the order of operations. Remember to simplify the final result to obtain the irreducible fraction.
$$ \left( \frac{3}{4} - \frac{1}{2} \right)^{-2} + \frac{5}{3} \times \left( \frac{2}{5} - 1 \right) - \frac{1}{2} : \frac{3}{4} $$Problems with fractions
Almost all fraction problems are "fraction of the remainder." Each step is calculated on what is left from the previous step.
A group of friends buys a giant pizza for dinner. Ana eats $\frac{1}{6}$ of the pizza. Carlos eats $\frac{2}{6}$ of the pizza. Beatriz eats $\frac{1}{3}$ of the pizza.…
A group of friends buys a giant pizza for dinner.
-
Ana eats $\frac{1}{6}$ of the pizza.
-
Carlos eats $\frac{2}{6}$ of the pizza.
-
Beatriz eats $\frac{1}{3}$ of the pizza.
- a) What fraction of the pizza have the three of them eaten altogether? (Remember to find a common denominator if necessary).
- b) What fraction of the pizza is left over?
Javier has been checking the storage on his cell phone, which has a total capacity of $128 \text{ GB}$. Look at the following distribution of his data: 1. The operating…
Javier has been checking the storage on his cell phone, which has a total capacity of $128 \text{ GB}$. Look at the following distribution of his data:
-
The operating system and pre-installed apps take up $\frac{1}{8}$ of the total.
-
His vacation photos and videos take up $\frac{2}{5}$ of the remainder (what is left free after accounting for the system).
-
His games take up $\frac{1}{4}$ of what remained after saving the photos.
What fraction of the total represents the space he still has free? How many Gigabytes (GB) is that? Use the following diagram to help visualize the problem.
From fraction to decimal
To convert a fraction to a decimal, just divide the numerator by the denominator. What's interesting is what kind of decimal you get:
| Type | Example | What it looks like |
|---|---|---|
| Terminating | $\frac{3}{8} = 0.375$ | It has a finite number of decimal places. |
| Purely repeating | $\frac{5}{11} = 0.454545\ldots = 0.\overline{45}$ | The group that repeats (the period) starts right after the decimal point. |
| Mixed repeating | $\frac{7}{6} = 1.1666\ldots = 1.1\overline{6}$ | Before the period there are digits that don't repeat (the non-repeating part). |
The period is marked with a bar over it: $0.\overline{45}$. In other books an arc is used: $0.\widehat{45}$. They mean the same thing.
Why do the digits repeat?
When dividing by 7, the remainder can only be 0, 1, 2, 3, 4, 5, or 6. If it never comes out 0, sooner or later a remainder repeats, and from there the division gives the same digits again. That's why $\frac{1}{7} = 0.\overline{142857}$ and the period has at most 6 digits.
From decimal to fraction: the generating fraction
Every terminating or repeating decimal comes from a fraction, called its generating fraction.
Terminating. Put the number without the decimal point in the numerator and a 1 followed by as many zeros as decimal places in the denominator. Then simplify.
$$3.75 = \frac{375}{100} = \frac{15}{4}$$Repeating. The trick is to multiply by a power of 10 so that, when you subtract, the infinite periods cancel out.
Classify the following decimal numbers into terminating decimals, pure recurring decimals, or mixed recurring decimals. Then, convert the first two (parts a and b) into…
Classify the following decimal numbers into terminating decimals, pure recurring decimals, or mixed recurring decimals. Then, convert the first two (parts a and b) into their corresponding generating fraction.
- a) $2.5$
- b) $1.333...$
- c) $4.1252525...$
- d) $0.08$
Calculate the exact value of the following expression by first converting all decimal numbers into their generating fractions. Do not use decimal approximations during…
Calculate the exact value of the following expression by first converting all decimal numbers into their generating fractions. Do not use decimal approximations during the process.
$$ \frac{1.2 - 0,\widehat{6}}{1 - 0.8\widehat{3}} : 3^{-1} $$Let $x$ be a mixed recurring decimal number whose integer part is 0. Knowing that: Its non-recurring part has 2 digits. Its recurring part has 1 digit. The irreducible…
Let $x$ be a mixed recurring decimal number whose integer part is 0.
Knowing that:
-
Its non-recurring part has 2 digits.
-
Its recurring part has 1 digit.
-
The irreducible generating fraction of this number has a denominator of 900.
-
The number satisfies: $\frac{1}{8} < x < \frac{1}{7}$
Find all possible values of $x$ and express them as a fraction and as a decimal number.
Percentages, fractions, and decimals
A percentage is a fraction with denominator 100: 25% means $\frac{25}{100} = \frac{1}{4} = 0.25$. The three forms are the same number.
To find a percentage of a quantity, multiply by the decimal: 15% of 80 pesos is $80 \times 0.15 = 12$ pesos. To find what percentage a part is, divide and multiply by 100: 12 out of 80 is $\frac{12}{80} = 0.15 = 15\%$.
Calculate the irreducible fraction equivalent to each of the following percentages and decimal numbers. a) $25\%$ b) $0.75$ c) $10\%$ d) $1.5$
Calculate the irreducible fraction equivalent to each of the following percentages and decimal numbers.
- a) $25\%$
- b) $0.75$
- c) $10\%$
- d) $1.5$
In a school of 500 students, a survey was conducted on their favorite sports. The results are as follows: $\frac{1}{2}$ of the students prefer soccer. $\frac{3}{10}$ of…
In a school of 500 students, a survey was conducted on their favorite sports. The results are as follows:
-
$\frac{1}{2}$ of the students prefer soccer.
-
$\frac{3}{10}$ of the students prefer basketball.
-
The rest of the students prefer other sports.
- a) How many students prefer soccer?
- b) How many students prefer basketball?
- c) What fraction of the total number of students prefers "other sports"?
Three friends, Ana, Beto and Carla, decide to buy a birthday present that costs $\text{RD\$}\ 7\,200$. Ana contributes $\frac{2}{5}…
Three friends, Ana, Beto and Carla, decide to buy a birthday present that costs $\text{RD\$}\ 7\,200$.
-
Ana contributes $\frac{2}{5}$ of the total.
-
Beto contributes $\frac{1}{3}$ of the total.
-
Carla pays the rest.
- a) What fraction of the total does Carla contribute?
- b) How many pesos does each person pay?
- c) If they had decided to pay equal shares, what fraction of the total would each person have paid, and how much money would that be?
All numbers: from the natural numbers to the real numbers
Every time an operation "couldn't be done," mathematicians expanded the numbers:
-
With the natural numbers $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$ you can't do $3 - 5$.
-
That's why the integers $\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$ appear. But with them you can't do $3 \div 5$.
-
So we get the rational numbers $\mathbb{Q}$: all those that can be written as a fraction (terminating and repeating decimals).
-
Even so, some numbers are missing: the irrational numbers $\mathbb{I}$, decimals with infinitely many digits with no period, such as $\sqrt{2} = 1.41421356\ldots$ or $\pi = 3.14159265\ldots$
-
Rational and irrational numbers together form the real numbers $\mathbb{R}$: all the points on the line.
The real number line and intervals
Every real number occupies a point on the line, and every point on the line is a real number. To talk about stretches of the line we use intervals: $(a, b)$ does not include the endpoints, $[a, b]$ does include them, and $[a, b)$ includes $a$ but not $b$. So "the rational numbers between $\frac{13}{9}$ and $1.4\overline{5}$" are the numbers in the interval $\left(\frac{13}{9}, 1.4\overline{5}\right)$.
Between any two different real numbers there are always infinitely many rational and infinitely many irrational numbers. To locate one, convert the endpoints to decimals and compare them digit by digit.
Where is $\sqrt{2}$ on the number line?
Even though its decimals never end, $\sqrt{2}$ has an exact place. Draw a square with side 1: by the Pythagorean theorem its diagonal measures $\sqrt{1^2 + 1^2} = \sqrt{2}$. With a compass you carry that diagonal onto the line:
Classify the following numbers into the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Q}$) and, for those that are decimal numbers…
Classify the following numbers into the smallest number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$ or $\mathbb{Q}$) and, for those that are decimal numbers (terminating or recurring), find their simplified generating fraction.
- a) $4.25$
- b) $-7$
- c) $3.1\widehat{6}$
- d) $5,\widehat{27}$
- e) $-\frac{18}{3}$
Once you have obtained the fractions, order them from smallest to largest.
Find two rational numbers and two irrational numbers that lie strictly within the interval: $$ \left( \frac{13}{9} \, , \, 1.4\widehat{5} \right) $$ Explain in detail…
Find two rational numbers and two irrational numbers that lie strictly within the interval:
$$ \left( \frac{13}{9} \, , \, 1.4\widehat{5} \right) $$Explain in detail the procedure you followed to find them and prove that they indeed belong to that interval.
Absolute error and relative error
When measuring, the value you get (the approximate value) is almost never the true value.
-
Absolute error $= |\text{approximate value} - \text{true value}|$. It is expressed in the same unit.
-
Relative error $= \dfrac{\text{absolute error}}{|\text{true value}|}$. It has no unit; it is usually given as a percentage.
The relative error tells you how much the error weighs compared to what was measured, and that is why it is useful for comparing the precision of different measurements.
If the data are given in scientific notation, operate with the powers of 10: $2.4 \cdot 10^{-4} - 2.3 \cdot 10^{-4} = 0.1 \cdot 10^{-4} = 1 \cdot 10^{-5}$.
We have two different scales. Scale A: Used in a laboratory and when weighing a test tube gives a value of $24.5 \text{ g}$. The exact actual value is…
We have two different scales.
-
Scale A: Used in a laboratory and when weighing a test tube gives a value of $24.5 \text{ g}$. The exact actual value is $\frac{49}{2} \text{ g}$.
-
Scale B: Used in a warehouse and when weighing a box gives a value of $5.4 \text{ kg}$. The exact actual value is $\frac{16}{3} \text{ kg}$.
Calculate the absolute error and the relative error made in each measurement. Which of the two measurements is more precise? Justify your answer based on the relative error.
In a chemistry laboratory, precision measurements are being taken. Two students, Alex and Bea, measure the length of a microscopic fiber. The actual value of the length…
In a chemistry laboratory, precision measurements are being taken. Two students, Alex and Bea, measure the length of a microscopic fiber.
-
The actual value of the length of the fiber is $2.345 \times 10^{-4}$ meters.
-
Alex obtains a measurement of $2.3 \times 10^{-4}$ meters.
-
Bea obtains a measurement of $2.4 \times 10^{-4}$ meters.
Answer, justifying your answers with calculations:
a) Calculate the absolute error made by each student. Express the result in scientific notation.
b) Calculate the relative error of each measurement and express it as a percentage.
c) Based on the relative error, which of the two measurements is more precise and why?
d) If we truncate the actual value to ten-thousandths, does it coincide with rounding to ten-thousandths? Explain your answer.
More complete fraction problems
Here the previous ideas come together. Three types that appear often:
Rates (taps and drains). If a tap fills a tank in $t$ hours, in one hour it fills $\frac{1}{t}$ of the tank. The rates add up, and drains are subtracted. If everything works together, the time is the reciprocal of the sum.
Mixtures with fractions. If the first material has a fraction $f_1$ of gold and the second $f_2$, and you mix $x$ g of the first with $T - x$ g of the second, the total gold is $f_1 x + f_2 (T - x)$. Set it equal to the desired fraction times the total and solve for $x$. A ratio of "5 parts gold to 2 parts copper" means that gold is $\frac{5}{7}$ of the total.
Fraction of an area. If a figure is split into equal parts, the shaded fraction is (shaded parts) $\div$ (total parts). If the parts are not equal, divide the figure with auxiliary lines until they are, or calculate the areas and divide.
A rectangular plot of land is to be fenced. We know that the length of its base is $\frac{15}{2}$ meters and its height is $\frac{2}{3}$ of the measure of the base. a)…
A rectangular plot of land is to be fenced. We know that the length of its base is $\frac{15}{2}$ meters and its height is $\frac{2}{3}$ of the measure of the base.
- a) Calculate the perimeter of the rectangle as a fraction and as a decimal.
- b) Calculate the area of the plot in square meters ($\text{m}^2$).
- c) If fencing costs $\text{RD\$}\ 510$ per meter, how much will it cost to fence the entire plot?
A bacterium reproduces by splitting in two every hour. Initially, in a culture there is a certain number of bacteria occupying $\frac{1}{64}$ of the total capacity of…
A bacterium reproduces by splitting in two every hour. Initially, in a culture there is a certain number of bacteria occupying $\frac{1}{64}$ of the total capacity of the Petri dish.
- a) If the population doubles every hour, what fraction of the dish will be occupied after 3 hours? Express the result as a power with base 2.
- b) How many hours in total will have to pass (from the beginning) for the dish to be completely full (that is, for the fraction to be 1)?
- c) If initially there were 500 bacteria (corresponding to that $\frac{1}{64}$), how many bacteria will there be when the dish is at half capacity?
Calculate the exact value of the following expression, converting all decimal numbers beforehand into their irreducible generating fractions and simplifying the result…
Calculate the exact value of the following expression, converting all decimal numbers beforehand into their irreducible generating fractions and simplifying the result step by step:
$$ \frac{1.1\widehat{6} - \left( \frac{1}{2} \right)^{-2} \times 0.1}{2 - \frac{1}{1 - \frac{1}{3}}} + \sqrt{0,\widehat{4}} $$Note: Remember to respect the order of operations and work with fractions throughout the process to avoid rounding errors.
A cloud storage server experienced a series of massive data uploads and accidental deletions over the weekend. 1. On Saturday morning, the server was full up to…
A cloud storage server experienced a series of massive data uploads and accidental deletions over the weekend.
-
On Saturday morning, the server was full up to $\frac{3}{4}$ of its total capacity.
-
At noon, an automatic script deleted $\frac{1}{5}$ of the data stored at that moment.
-
In the afternoon, new files were uploaded that occupied $\frac{2}{3}$ of the capacity that remained free after the deletion.
If at the end of the day there were still 120 GB of free space remaining on the server's hard drive:
a) What fraction of the total disk does the final free space represent?
b) What is the total storage capacity of the server in GB?
Observe the following geometric figure representing a square plot of land divided into different crop patches. Calculate what fraction of the total area of the square is…
Observe the following geometric figure representing a square plot of land divided into different crop patches.
Calculate what fraction of the total area of the square is represented by the shaded region (orange color). Express the result as an irreducible fraction.
Hint: You may assume that the dividing lines connect vertices to midpoints or midpoints to each other recursively.
Three taps ($A$, $B$ and $C$) fill a tank with water, and a drain ($D$) allows it to be emptied. Tap $A$ fills the tank in 4 hours working alone. Tap $B$ fills the tank…
Three taps ($A$, $B$ and $C$) fill a tank with water, and a drain ($D$) allows it to be emptied.
-
Tap $A$ fills the tank in 4 hours working alone.
-
Tap $B$ fills the tank in 6 hours working alone.
-
Tap $C$ fills the tank in 12 hours working alone.
-
Drain $D$ empties the full tank in 8 hours.
The tank is initially empty. All three taps ($A$, $B$, $C$) and the drain ($D$) are opened simultaneously. However, after 2 hours of operation, tap $A$ is turned off due to a fault, leaving the other two and the drain working.
Calculate the total time (in hours and minutes) it will take for the tank to fill completely from the start.
A jeweller has two ingots of a gold and copper alloy. The first ingot has a purity such that for every 5 parts of gold there are 2 of copper. The second ingot is of…
A jeweller has two ingots of a gold and copper alloy.
-
The first ingot has a purity such that for every 5 parts of gold there are 2 of copper.
-
The second ingot is of lower quality, having 3 parts of gold for every 4 parts of copper.
The jeweller melts a certain amount of the first ingot with a certain amount of the second to obtain a new ingot of 420 grams. If he wants the final fraction of gold in the new ingot to be exactly $\frac{2}{3}$ of the total weight:
How many grams must he cut and melt from each of the two original ingots?
Back to the beginning: the number on the screen is written without infinite ink as $0.\overline{3}$, and you recover its fraction by subtracting tails: $10x - x = 3.333\ldots - 0.333\ldots \Rightarrow 9x = 3 \Rightarrow x = \frac{3}{9} = \frac{1}{3}$.