Rational Numbers
Your phone says 75% battery. Your friend's phone has used up $\frac{5}{8}$ of its charge. Whose phone has more charge left, and by how much? To answer, you need to compare and subtract fractions. Fractions also show up in recipes ($\frac{3}{4}$ cup of flour), in sports stats, and on every road trip ("we're $\frac{2}{3}$ of the way to Chicago"). By the end of this page you will handle them with confidence.
Rational Numbers and Equivalent Fractions
A fraction $\frac{a}{b}$ has two parts. The denominator $b$ tells you how many equal parts the whole is cut into. The numerator $a$ tells you how many of those parts you take.
You can read the same fraction in three ways. Keep all three in your head:
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Part of a whole: $\frac{3}{4}$ of a pizza is 3 slices of a pizza cut into 4.
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Division: $\frac{3}{4} = 3 \div 4 = 0.75$. That is why 75% battery is $\frac{3}{4}$ of a full charge.
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Operator: $\frac{3}{4}$ of $\$200$ means $200 \times \frac{3}{4} = \$150$.
Integers are rational too: $5 = \frac{5}{1}$ and $-2 = \frac{-4}{2}$. A fraction can be negative: $-\frac{3}{4} = \frac{-3}{4} = \frac{3}{-4}$. The negative sign can sit anywhere, but we usually write it in front. A mixed number is a rational number too: $1\tfrac{3}{4} = \frac{4}{4} + \frac{3}{4} = \frac{7}{4}$.
Play with the numerator and denominator. Notice what happens when the numerator is bigger than the denominator: the fraction is worth more than 1 (an improper fraction).
Equivalent fractions
Half a pizza, two fourths, four eighths: same amount of pizza, cut differently. These are equivalent fractions.
If you multiply or divide the numerator and denominator by the same nonzero number, you get an equivalent fraction. Dividing is called simplifying. When you cannot simplify any more, the fraction is in simplest form (also called lowest terms). Some problems call this an irreducible fraction. It means the same thing.
Look at the following graphical representation of a fraction. Your task is to: 1. Write the irreducible fraction that represents the part colored in orange with respect…
Look at the following graphical representation of a fraction. Your task is to:
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Write the irreducible fraction that represents the part colored in orange with respect to the whole circle.
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State whether the fraction $\frac{9}{24}$ is equivalent to the one represented in the image. Justify your answer by simplifying both fractions.
Emma has bought a large rectangular pizza for dinner. When she opens the box, she sees that the pizza is cut into 24 equal slices. Emma eats 4 slices. Her brother Luke…
Emma has bought a large rectangular pizza for dinner. When she opens the box, she sees that the pizza is cut into 24 equal slices.
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Emma eats 4 slices.
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Her brother Luke eats 6 slices.
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Her parents eat 8 slices between them.
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Write the fraction of pizza that each person has eaten (Emma, Luke, and parents). Simplify them fully.
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What fraction of the pizza is left over? Express the result as an irreducible fraction.
Comparing, Ordering, and the Number Line
Who ate more chocolate, someone who ate $\frac{3}{4}$ of a bar or someone who ate $\frac{5}{7}$ of the same size bar? Look at the bars:
It is close, but $\frac{3}{4}$ is bigger. You can't always draw bars, so you need a method.
Placing fractions on the number line
Every rational number has an exact spot on the number line. To place $\frac{7}{4}$:
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If it is improper, pull out the whole part: $\frac{7}{4} = 1 + \frac{3}{4}$. It lies between 1 and 2.
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Cut the stretch from 1 to 2 into 4 equal parts (the denominator).
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Count 3 of those parts from 1.
How do you cut a segment into, say, 3 equal parts when you have no ruler marks? Use parallel lines. Draw a slanted ray from 0 and mark 3 equal steps along it with a compass. Join the last mark to the point 1. Then draw lines parallel to that one through the other marks. They cut the segment into 3 equal pieces. Some problems call this Thales's theorem. In US geometry it is the Triangle Proportionality Theorem: parallel lines cut slanted lines into proportional pieces.
For a negative fraction such as $-\frac{2}{3}$, do the same thing on the segment from 0 to $-1$, so the point lands to the left of 0. Drag the points below to see which fraction and decimal match each position.
Three friends are competing in a 100-foot obstacle race. At a given moment, their positions are as follows: Ashley has covered $\frac{2}{5}$ of the total. Ben has…
Three friends are competing in a 100-foot obstacle race. At a given moment, their positions are as follows:
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Ashley has covered $\frac{2}{5}$ of the total.
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Ben has covered $\frac{3}{4}$ of the total.
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Chloe has covered $\frac{7}{10}$ of the total.
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Calculate exactly how many feet each one has covered.
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Order the fractions from smallest to largest using the $<$ symbol.
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Approximately locate the position of Ashley (A), Ben (B), and Chloe (C) on the number line shown below.
Represent the rational numbers $\frac{7}{4}$ and $-\frac{2}{3}$ on the number line. To do this, use Thales's theorem or the division of line segments to locate the…
Represent the rational numbers $\frac{7}{4}$ and $-\frac{2}{3}$ on the number line.
To do this, use Thales's theorem or the division of line segments to locate the points exactly, not approximately.
Use the following diagram as a reference for your construction in your notebook:
Adding and Subtracting Fractions
Adding fractions is like adding coins from different countries: before you combine them, you convert them to the same "currency." Here the currency is the denominator.
For mixed numbers, convert to improper fractions first: $2\tfrac{1}{2} + 1\tfrac{3}{4} = \frac{5}{2} + \frac{7}{4} = \frac{10}{4} + \frac{7}{4} = \frac{17}{4} = 4\tfrac{1}{4}$.
Perform the following operations with fractions. Remember that to add or subtract you must first find the least common multiple (LCM) of the denominators. Express the…
Perform the following operations with fractions. Remember that to add or subtract you must first find the least common multiple (LCM) of the denominators. Express the result as an irreducible fraction.
- a) $\frac{2}{3} + \frac{1}{6}$
- b) $\frac{4}{5} - \frac{1}{2}$
- c) $\frac{3}{4} + \frac{5}{12} - \frac{1}{3}$
The storage of your cell phone is distributed as follows: Photos take up $\frac{2}{5}$ of the space. Apps take up $\frac{1}{3}$ of the space. 1. What fraction of the…
The storage of your cell phone is distributed as follows:
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Photos take up $\frac{2}{5}$ of the space.
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Apps take up $\frac{1}{3}$ of the space.
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What fraction of the total do the photos and apps take up together?
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If the rest of the space is free, what fraction represents the free space?
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If the total capacity of the phone is 60 GB, how many GB are free?
Multiplying and Dividing Fractions
Multiplying is easier than adding: numerator times numerator, denominator times denominator. The picture shows why. $\frac{2}{3}$ of $\frac{3}{4}$ is the region where the two strips overlap.
Signs work as with integers: a positive times a negative is negative, and two negatives make a positive. An integer is a fraction with denominator 1, so $-2 = \frac{-2}{1}$.
A fraction of a quantity
The word of means multiply. $\frac{2}{3}$ of 60 is $\frac{2}{3} \times 60 = 40$. When several fractions of a total are taken, add them and subtract from 1 to find what is left, because the whole is $1 = \frac{\text{all parts}}{\text{all parts}}$.
Calculate the following products and divisions of fractions. Remember to simplify the final result if possible. a) $\frac{3}{4} \cdot \frac{8}{9}$ b)…
Calculate the following products and divisions of fractions. Remember to simplify the final result if possible.
- a) $\frac{3}{4} \cdot \frac{8}{9}$
- b) $\frac{2}{5} : \frac{4}{15}$
- c) $\frac{5}{6} \cdot (-2)$
Note: Remember the rules of signs and that an integer has a denominator of 1.
In a strategy video game, the map is divided into different zones according to the terrain. The total map occupies 1,200 square pixels. The forest occupies $\frac{1}{3}$…
In a strategy video game, the map is divided into different zones according to the terrain.
The total map occupies 1,200 square pixels.
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The forest occupies $\frac{1}{3}$ of the map.
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The water occupies $\frac{1}{4}$ of the map.
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The rest is desert.
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How many square pixels does the forest occupy?
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How many square pixels does the water occupy?
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Calculate what irreducible fraction of the map the desert represents.
Three friends buy a birthday present that costs $\$60$. Ashley contributes $\frac{1}{3}$ of the total price. Ben contributes $\frac{2}{5}…
Three friends buy a birthday present that costs $\$60$.
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Ashley contributes $\frac{1}{3}$ of the total price.
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Ben contributes $\frac{2}{5}$ of the total price.
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Chloe pays the rest.
- a) What fraction of the total does Chloe pay?
- b) How much money in dollars does each one contribute?
Fractions and Decimals
Every rational number can be written as a decimal. Divide the numerator by the denominator and you get one of two kinds of decimal.
We write a repeating decimal with a bar over the repeating digits: $0.333\ldots = 0.\overline{3}$ and $0.1666\ldots = 0.1\overline{6}$. Some problems put a hat over the repeating digit instead. It means the same thing.
Terminating decimals to fractions
Write the digits without the decimal point on top. Put a 1 followed by one zero for each decimal place on the bottom. Then simplify.
$$0.875 = \frac{875}{1000} = \frac{7}{8} \qquad\qquad 2.4 = \frac{24}{10} = \frac{12}{5}$$Repeating decimals to fractions
Some problems call the fraction you get the generating fraction. It is just the fraction equal to the decimal. Use algebra:
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Let $x$ be the decimal.
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Multiply by a power of 10 so the repeating block starts right after the decimal point.
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Multiply again by a power of 10 that shifts one full repeating block to the left of the point.
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Subtract the two equations. The repeating tails cancel.
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Solve for $x$ and simplify.
Classify the following decimal numbers as terminating, pure repeating, or mixed repeating. Then, express the terminating ones as an irreducible fraction. a) $3,25$ b)…
Classify the following decimal numbers as terminating, pure repeating, or mixed repeating. Then, express the terminating ones as an irreducible fraction.
- a) $3,25$
- b) $7,1111\dots$
- c) $0,25$
- d) $5,0424242\dots$
Hint: Remember that to convert a terminating decimal to a fraction, you write the number without the decimal point in the numerator and a 1 followed by as many zeros as there are decimal places in the denominator.
Classify the following decimal numbers (terminating, pure repeating, or mixed repeating) and find their generating fraction. Once the fractions are obtained, perform the…
Classify the following decimal numbers (terminating, pure repeating, or mixed repeating) and find their generating fraction. Once the fractions are obtained, perform the following operation and simplify the result:
$$ 1,2\widehat{5} - 0,\widehat{6} $$Order of Operations with Fractions
When an expression mixes operations there is a required order. Without it, two people would get different answers to the same problem.
You may know this as PEMDAS. Remember that multiplication and division share a level, and so do addition and subtraction. Within a level, go left to right. A long fraction bar also acts like parentheses: simplify the numerator and the denominator separately, then divide.
Three facts about exponents and roots that you will need:
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$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$, for example $\left(\frac{3}{4}\right)^2 = \frac{9}{16}$.
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A negative exponent flips the base: $a^{-n} = \frac{1}{a^n}$ and $\left(\frac{2}{3}\right)^{-2} = \left(\frac{3}{2}\right)^2 = \frac{9}{4}$.
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A root of a fraction is the root of each part: $\sqrt{\frac{16}{81}} = \frac{4}{9}$.
Calculate the value of the following combined operation, respecting the order of operations (parentheses, powers, multiplications/divisions, additions/subtractions). You…
Calculate the value of the following combined operation, respecting the order of operations (parentheses, powers, multiplications/divisions, additions/subtractions). You must simplify the final result until you obtain the irreducible fraction.
$$ \left( \frac{3}{2} - \frac{1}{4} \right)^2 : \frac{5}{8} - \left( 1 - \frac{3}{5} \cdot \frac{10}{3} \right) $$Calculate the exact value of the following combined expression. To do this, you must first convert all decimal numbers to their irreducible generating fraction and…
Calculate the exact value of the following combined expression. To do this, you must first convert all decimal numbers to their irreducible generating fraction and simplify each step before operating. Be careful with the order of operations and negative signs.
$$ \frac{2^{-2} - \left( 1,1\widehat{6} - \frac{7}{6} \right)^{-1} \cdot \sqrt{\frac{16}{81}}}{0,\widehat{3} \cdot \left( 1 - \frac{3}{2} \right)^2 + 2,5} $$Note: Remember that $a^{-n} = \frac{1}{a^n}$.
Fractions of What's Left
Many problems take a fraction of a total, then a fraction of what remains. The key idea: if you use up $\frac{1}{4}$ of something, you keep $1 - \frac{1}{4} = \frac{3}{4}$ of it. "$\frac{2}{3}$ of the rest" means $\frac{2}{3}$ of that $\frac{3}{4}$, not $\frac{2}{3}$ of the total.
A tape diagram makes this visible. Priya has some savings. She spends $\frac{1}{4}$ on sneakers, then $\frac{2}{3}$ of what is left on a concert ticket.
We have an oil drum with a capacity of 120 gallons. We want to bottle all the oil in two types of containers: We fill 40 bottles of $\frac{3}{4}$ of a gallon each. The…
We have an oil drum with a capacity of 120 gallons.
We want to bottle all the oil in two types of containers:
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We fill 40 bottles of $\frac{3}{4}$ of a gallon each.
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The rest of the oil is bottled in $1,5$-gallon jugs.
How many 1.5-gallon jugs will we need to bottle the remaining oil?
Note: Remember that 1.5 gallons can be expressed as the fraction $\frac{3}{2}$.
Your cell phone has an unknown total storage capacity. The operating system takes up $\frac{1}{6}$ of the total memory. Of what is left free, your apps and games take up…
Your cell phone has an unknown total storage capacity. The operating system takes up $\frac{1}{6}$ of the total memory. Of what is left free, your apps and games take up $\frac{3}{5}$. If in the end you have 32 GB free for your photos and videos:
- a) What fraction of the total represents the free memory for photos?
- b) What is the total storage capacity of the phone in GB?
Use the following graphical diagram to help you visualize the problem:
A group of 8th grade students is organizing the funding for their end-of-year trip. They have designed a fundraising plan in stages: 1. In the first stage (lottery…
A group of 8th grade students is organizing the funding for their end-of-year trip. They have designed a fundraising plan in stages:
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In the first stage (lottery sales), they managed to raise $\frac{2}{5}$ of the total cost of the trip.
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In the second stage (e-sports tournament), they raised $\frac{3}{4}$ of what they still needed after the first stage.
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In the third stage (charity market), they managed to raise $\frac{2}{3}$ of the new remainder they had left.
If after these three stages they are still $\$150$ short of covering the total cost:
- a) What fraction of the total does the amount raised in the third stage represent?
- b) What is the total cost of the trip?
- c) How much money did they raise in the e-sports tournament?
Use the following bar chart to help you visualize the remainder fractions:
Working Together and Unknown Wholes
If a job takes you $t$ hours alone, you finish $\frac{1}{t}$ of it each hour. That is your rate. When people work at the same time, their rates add.
Emma mows a lawn alone in 3 hours, so her rate is $\frac{1}{3}$ of the lawn per hour. Jayden takes 6 hours, so his rate is $\frac{1}{6}$ per hour.
Unknown wholes
When the total is unknown, call it $x$ and write each piece as a fraction of $x$. Add the pieces, then solve. For example, "$\frac{2}{5}$ of a number plus $\frac{1}{3}$ of the same number is 22":
$$\frac{2}{5}x + \frac{1}{3}x = 22 \;\Rightarrow\; \frac{6}{15}x + \frac{5}{15}x = 22 \;\Rightarrow\; \frac{11}{15}x = 22 \;\Rightarrow\; x = 22 \times \frac{15}{11} = 30$$Check: $\frac{2}{5}(30) + \frac{1}{3}(30) = 12 + 10 = 22$. ✓
Two more translations you will need. "Tank A is $\frac{3}{5}$ full" means contents $= \frac{3}{5} \times$ capacity. "A's capacity is $\frac{2}{3}$ of B's" means $A = \frac{2}{3}B$.
Two 3D printers are working to print the parts for a technology project. Printer model A (older) would take 6 hours to print all the parts on its own. Printer model B…
Two 3D printers are working to print the parts for a technology project.
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Printer model A (older) would take 6 hours to print all the parts on its own.
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Printer model B (latest generation) would take only 4 hours to complete the same job on its own.
To speed up the process, both are set to run simultaneously. However, after 1.5 hours of working together, printer B breaks down and stops.
- a) What fraction of the total work have they completed together before the breakdown?
- b) What fraction of the work is left pending?
- c) How much extra time will printer A take to finish the remaining work on its own? Express the result in hours and minutes.
We have three juice tanks with different capacities: Tank A: Full to $\frac{3}{5}$ of its capacity. Tank B: Full to $\frac{5}{6}$ of its capacity. Tank C: Empty, with a…
We have three juice tanks with different capacities:
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Tank A: Full to $\frac{3}{5}$ of its capacity.
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Tank B: Full to $\frac{5}{6}$ of its capacity.
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Tank C: Empty, with a capacity of 100 gallons.
We know that the total capacity of tank A is $\frac{2}{3}$ of the capacity of tank B.
If we pour the current contents of A and B into tank C, the mixture occupies exactly $\frac{11}{20}$ of the capacity of C.
Calculate the maximum capacity (total volume) of tanks A and B.
Repeated Fractions: Bouncing Balls and Patterns
A ball dropped from 16 feet bounces back up to $\frac{3}{4}$ of the height it fell from. Each bounce multiplies the previous height by $\frac{3}{4}$ again.
The same idea describes cutting patterns. If each step keeps $\frac{4}{5}$ of a sheet, after 2 steps $\left(\frac{4}{5}\right)^2 = \frac{16}{25}$ is left. So $1 - \frac{16}{25} = \frac{9}{25}$ has been removed.
A rubber ball is dropped from a height of 27 feet. On each bounce, the ball rises to a height equivalent to $\frac{2}{3}$ of the previous height. a) Write the fraction…
A rubber ball is dropped from a height of 27 feet. On each bounce, the ball rises to a height equivalent to $\frac{2}{3}$ of the previous height.
- a) Write the fraction and the height in feet it reaches after the first bounce.
- b) Write the fraction and the height in feet it reaches after the second bounce.
- c) To what height will it rise after the third bounce?
We have a square piece of conductive material for a circuit. Successive cuts are made following a fractal geometric pattern: 1. The original square is divided into 4…
We have a square piece of conductive material for a circuit. Successive cuts are made following a fractal geometric pattern:
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The original square is divided into 4 equal squares and the top left square is removed (colored black).
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Of the 3 remaining green squares, each is again divided into 4 equal squares and the top left of each of them is removed.
Look at the figure generated after the second step:
- a) Write the combined fraction operation that calculates what part of the total area has been removed (shaded areas) after the second step.
- b) If the side of the original square measures $1 \text{ m}$, how many $\text{cm}^2$ of "white" material (not removed) are left after the second step?
A very elastic rubber ball is dropped from a height of 243 inches. Each time it bounces on the ground, the ball rises to a height that is equal to $\frac{2}{3}$ of the…
A very elastic rubber ball is dropped from a height of 243 inches.
Each time it bounces on the ground, the ball rises to a height that is equal to $\frac{2}{3}$ of the height from which it previously fell.
- a) Write the sequence of heights (in the form of powers and fractions) that the ball reaches on the first, second and third bounce.
- b) To what height will it rise after the fifth bounce? Express the result as an irreducible fraction and then convert it to a decimal.
- c) What total distance will the ball have traveled (adding descents and ascents) just as it touches the ground for the fourth time?
Absolute Value and Distance
The absolute value $|a - b|$ is the distance between $a$ and $b$ on the number line. So $\left|x - \frac{3}{2}\right| \le \frac{1}{4}$ says: "$x$ is at most $\frac{1}{4}$ away from $\frac{3}{2}$."
A biologist is preparing a bacteria culture and needs a very specific room temperature. The laboratory thermostat has a slightly strange scale marked in fractions of a…
A biologist is preparing a bacteria culture and needs a very specific room temperature.
The laboratory thermostat has a slightly strange scale marked in fractions of a degree.
The ideal temperature $T$ must satisfy the following distance condition (absolute value):
$$ \left| T - \frac{25}{4} \right| \leq \frac{3}{8} $$- a) Represent on the number line the set of all possible values of temperature $T$ (interval).
- b) Write the solution interval using inequalities and fractions.
- c) Would a temperature of $6,65^\circ \text{C}$ be valid? Justify your answer by calculating with fractions.
Measurement Error
No measurement is perfect. To describe how far off a measurement is, we use two numbers.
The absolute error tells you how big the mistake is. The relative error tells you how big it is compared to the thing you measured. That is why relative error is the fair way to judge precision.
Approximating a decimal is also a measurement error. If you use $0.33$ for $\frac{1}{3}$, you can find the error exactly by using fractions:
$$\text{absolute error} = \left|\frac{1}{3} - \frac{33}{100}\right| = \left|\frac{100}{300} - \frac{99}{300}\right| = \frac{1}{300} \qquad \text{relative error} = \frac{1/300}{1/3} = \frac{1}{100} = 1\%$$In the physics laboratory we have measured the length of a table and the thickness of a coin. The results were: Length of the table: true value $120 \text{ cm}$,…
In the physics laboratory we have measured the length of a table and the thickness of a coin. The results were:
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Length of the table: true value $120 \text{ cm}$, measured value $118 \text{ cm}$.
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Thickness of the coin: true value $2 \text{ mm}$, measured value $2,5 \text{ mm}$.
Calculate the absolute error and the relative error (as a percentage) of each measurement. Which of the two measurements was more precise? Justify your answer based on the relative error.
Find an irreducible fraction $\frac{a}{b}$ such that it is the result of calculating: $$ 2,3\widehat{1} - 0,\widehat{5} + \frac{1}{11} \cdot 0,5 $$ If you try to do this…
Find an irreducible fraction $\frac{a}{b}$ such that it is the result of calculating:
$$ 2,3\widehat{1} - 0,\widehat{5} + \frac{1}{11} \cdot 0,5 $$If you try to do this operation by converting to decimals and calculating, you will get an approximation error. Prove algebraically, using generating fractions, that the result is exact.
Then, calculate the absolute error and the relative error (in %) that would be made if we approximated the number $2,3\widehat{1}$ simply by $2,31$ before starting to calculate.
Back to the start: your phone has $75\% = \frac{3}{4} = \frac{6}{8}$ of its charge left. Your friend used $\frac{5}{8}$, so she has $1 - \frac{5}{8} = \frac{3}{8}$ left. Your phone has $\frac{6}{8} - \frac{3}{8} = \frac{3}{8}$ of a full charge more than hers. In other words, your phone has twice as much charge left as her phone does.