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Rational Numbers

Define and identify numbers that are rational.

Learning Objectives

After completing this section, you should be able to:

  1. Define and identify numbers that are rational.
  2. Simplify rational numbers and express in lowest terms.
  3. Add and subtract rational numbers.
  4. Convert between improper fractions and mixed numbers.
  5. Convert rational numbers between decimal and fraction form.
  6. Multiply and divide rational numbers.
  7. Apply the order of operations to rational numbers to simplify expressions.
  8. Apply density property of rational numbers.
  9. Solve problems involving rational numbers.
  10. Use fractions to convert between units.
  11. Define and apply percent.
  12. Solve problems using percent.

Defining and Identifying Numbers That Are Rational

A rational number (called rational since it is a ratio) is just a fraction where the numerator is an integer and the denominator is a non-zero integer. As simple as that is, they can be represented in many ways. It should be noted here that any integer is a rational number. An integer, \(n\), written as a fraction of two integers is \(\frac{n}{1}\).

In its most basic representation, a rational number is an integer divided by a non-zero integer, such as \(\frac{3}{12}\). Fractions may be used to represent parts of a whole. The denominator is the total number of parts to the object, and the numerator is how many of those parts are being used or selected. So, if a pizza is cut into 8 equal pieces, each piece is \(\frac{1}{8}\) of the pizza. If you take three slices, you have \(\frac{3}{8}\) of the pizza (). Similarly, if in a group of 20 people, 5 are wearing hats, then \(\frac{5}{20}\) of the people are wearing hats ().

Another representation of rational numbers is as a mixed number, such as \(2\frac{5}{8}\) (). This represents a whole number (2 in this case), plus a fraction (the \(\frac{5}{8}\)).

Rational numbers may also be expressed in decimal form; for instance, as 1.34. When 1.34 is written, the decimal part, 0.34, represents the fraction \(\frac{34}{100}\), and the number 1.34 is equal to \(1\frac{34}{100}\). However, not all decimal representations are rational numbers.

A number written in decimal form where there is a last decimal digit (after a given decimal digit, all following decimal digits are 0) is a terminating decimal, as in 1.34 above. Alternately, any decimal numeral that, after a finite number of decimal digits, has digits equal to 0 for all digits following the last non-zero digit.

All numbers that can be expressed as a terminating decimal are rational. This comes from what the decimal represents. The decimal part is the fraction of the decimal part divided by the appropriate power of 10. That power of 10 is the number of decimal digits present, as for 0.34, with two decimal digits, being equal to \(\frac{34}{100}\).

Another form that is a rational number is a decimal that repeats a pattern, such as 67.1313… When a rational number is expressed in decimal form and the decimal is a repeated pattern, we use special notation to designate the part that repeats. For example, if we have the repeating decimal 4.3636…, we write this as \(4.\overset{\bar}{36}\). The bar over the 36 indicates that the 36 repeats forever.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Simplifying Rational Numbers and Expressing in Lowest Terms

A rational number is one way to express the division of two integers. As such, there may be multiple ways to express the same value with different rational numbers. For instance, \(\frac{4}{5}\) and \(\frac{12}{15}\) are the same value. If we enter them into a calculator, they both equal 0.8. Another way to understand this is to consider what it looks like in a figure when two fractions are equal.

In , we see that \(\frac{3}{5}\) of the rectangle and \(\frac{9}{15}\) of the rectangle are equal areas.

They are the same proportion of the area of the rectangle. The left rectangle has 5 pieces, three of which are shaded. The right rectangle has 15 pieces, 9 of which are shaded. Each of the pieces of the left rectangle was divided equally into three pieces. This was a multiplication. The numerator describing the left rectangle was 3 but it becomes \(3\times 3\), or 9, as each piece was divided into three. Similarly, the denominator describing the left rectangle was 5, but became \(5\times 3\), or 15, as each piece was divided into 3. The fractions \(\frac{3}{5}\) and \(\frac{9}{15}\) are equivalent because they represent the same portion (often loosely referred to as equal).

This understanding of equivalent fractions is very useful for conceptualization, but it isn’t practical, in general, for determining when two fractions are equivalent. Generally, to determine if the two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent, we check to see that \(a\times d=b\times c\). If those two products are equal, then the fractions are equal also.

Determining If Two Fractions Are Equivalent

Try it.

Determine if \(\frac{12}{30}\) and \(\frac{14}{35}\) are equivalent fractions.

Solution

Applying the definition, \(a=12,b=30,c=14\) and \(d=35\). So \(a\times d=12\times 35=420\). Also, \(b\times c=30\times 14=420\). Since these values are equal, the fractions are equivalent.

That \(a\times d=b\times c\) indicates the fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent is due to some algebra. One property of natural numbers, integers, and rational numbers (also irrational numbers) is that for any three numbers \(a,b,\) and \(c\) with \(c\ne 0\), if \(a=b\), then \(a/c=b/c\). In other words, when two numbers are equal, then dividing both numbers by the same non-zero number, the two newly obtained numbers are also equal. We can apply that to \(a\times d\) and \(b\times c\), to show that \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if \(a\times d=b\times c\).

Condensed — the full section is in OpenStax Contemporary Mathematics.

Adding and Subtracting Rational Numbers

Adding or subtracting rational numbers can be done with a calculator, which often returns a decimal representation, or by finding a common denominator for the rational numbers being added or subtracted.

Adding Rational Numbers Using Desmos

Try it.

Calculate \(\frac{23}{42}+\frac{9}{56}\) using Desmos.

Solution

Enter \(\frac{23}{42}+\frac{9}{56}\) in Desmos. The result is displayed as \(0.70833333333\) (which is \(0.708\overset{\bar}{3}\)). Clicking the fraction button to the left on the calculation line yields \(\frac{17}{24}\).

Performing addition and subtraction without a calculator may be more involved. When the two rational numbers have a common denominator, then adding or subtracting the two numbers is straightforward. Add or subtract the numerators, and then place that value in the numerator and the common denominator in the denominator. Symbolically, we write this as \(\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm b}{c}\). This can be seen in the , which shows \(\frac{3}{20}+\frac{4}{20}=\frac{7}{20}\).

It is customary to then write the result in lowest terms.

Adding Rational Numbers with the Same Denominator

Try it.

Calculate \(\frac{13}{28}+\frac{7}{28}\).

Solution

Since the rational numbers have the same denominator, we perform the addition of the numerators, \(13+7\), and then place the result in the numerator and the common denominator, 28, in the denominator. \(\frac{13}{28}+\frac{7}{28}=\frac{13+7}{28}=\frac{20}{28}\)

Once we have that result, reduce to lowest terms, which gives \(\frac{20}{28}=\frac{4\times 5}{4\times 7}=\frac{4\times 5}{4\times 7}=\frac{5}{7}\).

When the rational numbers do not have common denominators, then we have to transform the rational numbers so that they do have common denominators. The common denominator that reduces work later in the problem is the LCM of the numerator and denominator. When adding or subtracting the rational numbers \(\frac{a}{b}\) and \(\frac{c}{d}\), we perform the following steps.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Converting Between Improper Fractions and Mixed Numbers

One way to visualize a fraction is as parts of a whole, as in \(\frac{5}{12}\) of a pizza. But when the numerator is larger than the denominator, as in \(\frac{23}{12}\), then the idea of parts of a whole seems not to make sense. Such a fraction is an improper fraction. That kind of fraction could be written as an integer plus a fraction, which is a mixed number. The fraction \(\frac{23}{12}\) rewritten as a mixed number would be \(1\frac{11}{12}\). Arithmetically, \(1\frac{11}{12}\) is equivalent to \(1+\frac{11}{12}\), which is read as “one and 11 twelfths.”

Improper fractions can be rewritten as mixed numbers using division and remainders. To find the mixed number representation of an improper fraction, divide the numerator by the denominator. The quotient is the integer part, and the remainder becomes the numerator of the remaining fraction.

Rewriting an Improper Fraction as a Mixed Number

Try it.

Rewrite \(\frac{48}{13}\) as a mixed number.

Solution

When 48 is divided by 13, the result is 3 with a remainder of 9. So, we can rewrite \(\frac{48}{13}\) as \(3\frac{9}{13}\).

Similarly, we can convert a mixed number into an improper fraction. To do so, first convert the whole number part to a fraction by writing the whole number as itself divided by 1, and then add the two fractions.

Alternately, we can multiply the whole number part and the denominator of the fractional part. Next, add that product to the numerator. Finally, express the number as that product divided by the denominator.

Rewriting a Mixed Number as an Improper Fraction

Try it.

Rewrite \(5\frac{4}{9}\) as an improper fraction.

Solution

Step 1: Multiply the integer part, 5, by the denominator, 9, which gives \(5\times 9=45\).

Step 2: Add that product to the numerator, which gives \(45+4=49\).

Step 3: Write the number as the sum, 49, divided by the denominator, 9, which gives \(\frac{49}{9}\).

Converting Rational Numbers Between Decimal and Fraction Forms

Understanding what decimals represent is needed before addressing conversions between the fractional form of a number and its decimal form, or writing a number in decimal notation. The decimal number 4.557 is equal to \(4\frac{557}{1,000}\). The decimal portion, .557, is 557 divided by 1,000. To write any decimal portion of a number expressed as a terminating decimal, divide the decimal number by 10 raised to the power equal to the number of decimal digits. Since there were three decimal digits in 4.557, we divided 557 by \({10}^{3}=1000\).

Decimal representations may be very long. It is convenient to round off the decimal form of the number to a certain number of decimal digits. To round off the decimal form of a number to \(n\) (decimal) digits, examine the (\(n+1\))st decimal digit. If that digit is 0, 1, 2, 3, or 4, the number is rounded off by writing the number to the \(n\)th decimal digit and no further. If the (\(n+1\))st decimal digit is 5, 6, 7, 8, or 9, the number is rounded off by writing the number to the \(n\)th digit, then replacing the \(n\)th digit by one more than the \(n\)th digit.

Rounding Off a Number in Decimal Form to Three Digits

Try it.

Round 5.67849 to three decimal digits.

Solution

The third decimal digit is 8. The digit following the 8 is 4. When the digit is 4, we write the number only to the third digit. So, 5.67849 rounded off to three decimal places is 5.678.

Rounding Off a Number in Decimal Form to Four Digits

Try it.

Round 45.11475 to four decimal digits.

Solution

The fourth decimal digit is 7. The digit following the 7 is 5. When the digit is 5, we write the number only to the fourth decimal digit, 45.1147. We then replace the fourth decimal digit by one more than the fourth digit, which yields 45.1148. So, 45.11475 rounded off to four decimal places is 45.1148.

To convert a rational number in fraction form to decimal form, use your calculator to perform the division.

Converting a Rational Number in Fraction Form into Decimal Form

Try it.

Convert \(\frac{47}{25}\) into decimal form.

Solution

Using a calculator to divide 47 by 25, the result is 1.88.

Converting a terminating decimal to the fractional form may be done in the following way:

Step 1: Count the number of digits in the decimal part of the number, labeled \(n\).

Step 2: Raise 10 to the \(n\)th power.

Step 3: Rewrite the number without the decimal.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Multiplying and Dividing Rational Numbers

Multiplying rational numbers is less complicated than adding or subtracting rational numbers, as there is no need to find common denominators. To multiply rational numbers, multiply the numerators, then multiply the denominators, and write the numerator product divided by the denominator product. Symbolically, \(\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}\). As always, rational numbers should be reduced to lowest terms.

Multiplying Rational Numbers

Try it.

Calculate \(\frac{12}{25}\times \frac{10}{21}\).

Solution

Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator.

\(\frac{12}{25}\times \frac{10}{21}=\frac{12\times 10}{25\times 21}=\frac{120}{525}\)

This is not in lowest terms, so this needs to be reduced. The GCD of 120 and 525 is 15.

\(\frac{120}{525}=\frac{15\times 8}{15\times 35}=\frac{8}{35}\)

As with multiplication, division of rational numbers can be done using a calculator.

Dividing Decimals with a Calculator

Try it.

Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary.

Solution

Using a calculator, we obtain 1.473729175565997. Rounding to three decimal places we have 1.474.

Before discussing division of fractions without a calculator, we should look at the reciprocal of a number. The reciprocal of a number is 1 divided by the number. For a fraction, the reciprocal is the fraction formed by switching the numerator and denominator. For the fraction \(\frac{a}{b}\), the reciprocal is \(\frac{b}{a}\). An important feature for a number and its reciprocal is that their product is 1.

When dividing two fractions by hand, find the reciprocal of the divisor (the number that is being divided into the other number). Next, replace the divisor by its reciprocal and change the division into multiplication. Then, perform the multiplication. Symbolically,\(\frac{b}{a}\div \frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{a\times d}{b\times c}\). As before, reduce to lowest terms.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Applying the Order of Operations to Simplify Expressions

The order of operations for rational numbers is the same as for integers, as discussed in Order of Operations. The order of operations makes it easier for anyone to correctly calculate and represent. The order follows the well-known acronym PEMDAS:

PParentheses
EExponents
M/DMultiplication and division
A/SAddition and subtraction

The first step in calculating using the order of operations is to perform operations inside the parentheses. Moving down the list, next perform all exponent operations moving from left to right. Next (left to right once more), perform all multiplications and divisions. Finally, perform the additions and subtractions.

Applying the Order of Operations with Rational Numbers

Try it.

Correctly apply the rules for the order of operations to accurately compute \((\frac{5}{7}-\frac{2}{7})\times {2}^{3}\).

Solution

Step 1: To calculate this, perform all calculations within the parentheses before other operations.

\((\frac{5}{7}-\frac{2}{7})\times {2}^{3}=(\frac{3}{7})\times {2}^{3}\)

Step 2: Since all parentheses have been cleared, we move left to right, and compute all the exponents next.

\((\frac{3}{7})\times {2}^{3}=(\frac{3}{7})\times 8\)

Step 3: Now, perform all multiplications and divisions, moving left to right.

\((\frac{3}{7})\times 8=\frac{24}{7}\)

Condensed — the full section is in OpenStax Contemporary Mathematics.

Applying the Density Property of Rational Numbers

Between any two rational numbers, there is another rational number. This is called the density property of the rational numbers.

Finding a rational number between any two rational numbers is very straightforward.

Step 1: Add the two rational numbers.

Step 2: Divide that result by 2.

The result is always a rational number. This follows what we know about rational numbers. If two fractions are added, then the result is a fraction. Also, when a fraction is divided by a fraction (and 2 is a fraction), then we get another fraction. This two-step process will give a rational number, provided the first two numbers were rational.

Applying the Density Property of Rational Numbers

Try it.

Demonstrate the density property of rational numbers by finding a rational number between \(\frac{4}{11}\) and \(\frac{7}{12}\).

Solution

To find a rational number between \(\frac{4}{11}\) and \(\frac{7}{12}\):

Step 1: Add the fractions.

\(\frac{4}{11}+\frac{7}{12}=\frac{4\times 12}{11\times 12}+\frac{7\times 11}{12\times 11}=\frac{48}{132}+\frac{77}{132}=\frac{125}{132}\)

Step 2: Divide the result by 2. Recall that to divide by 2, you multiply by the reciprocal of 2. The reciprocal of 2 is \(\frac{1}{2}\), as seen below.

\(\frac{125}{132}\div 2=\frac{125}{132}\times \frac{1}{2}=\frac{125}{264}\)

So, one rational number between \(\frac{4}{11}\) and \(\frac{7}{12}\) is \(\frac{125}{264}\).

We could check that the number we found is between the other two by finding the decimal representation of the numbers. Using a calculator, the decimal representations of the rational numbers are 0.363636…, 0.473484848…, and 0.5833333…. Here it is clear that \(\frac{125}{264}\) is between \(\frac{4}{11}\) and \(\frac{7}{12}\).

Solving Problems Involving Rational Numbers

Rational numbers are used in many situations, sometimes to express a portion of a whole, other times as an expression of a ratio between two quantities. For the sciences, converting between units is done using rational numbers, as when converting between gallons and cubic inches. In chemistry, mixing a solution with a given concentration of a chemical per unit volume can be solved with rational numbers. In demographics, rational numbers are used to describe the distribution of the population. In dietetics, rational numbers are used to express the appropriate amount of a given ingredient to include in a recipe. As discussed, the application of rational numbers crosses many disciplines.

Mixing Soil for Vegetables

Try it.

James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that \(\frac{2}{5}\) of the soil can be topsoil, but \(\frac{2}{5}\) needs to be peat moss and \(\frac{1}{5}\) has to be compost. To fill the raised garden bed with 60 cubic feet of soil, how much of each component does James need to use?

Solution

In this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For each component of the mixture, multiply 60 cubic feet, which is the total volume of the mixture we want, by the fraction required of the component.

Step 1: The required fraction of topsoil is \(\frac{2}{5}\), so James needs \(60\times \frac{2}{5}\) cubic feet of topsoil. Performing the multiplication, James needs \(60\times \frac{2}{5}=\frac{120}{5}=24\) (found by treating the fraction as division, and 120 divided by 5 is 24) cubic feet of topsoil.

Step 2: The required fraction of peat moss is also \(\frac{2}{5}\), so he also needs \(60\times \frac{2}{5}\) cubic feet, or \(60\times \frac{2}{5}=\frac{120}{5}=24\) cubic feet of peat moss.

Step 3: The required fraction of compost is \(\frac{1}{5}\). For the compost, he needs \(60\times \frac{1}{5}=\frac{60}{5}=12\) cubic feet.

Determining the Number of Specialty Pizzas

Try it.

At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas ordered, how many were specialty pizzas?

Solution

One-third of the whole are specialty pizzas, so we need one-third of 273, which gives \(\frac{1}{3}\times 273=\frac{273}{3}=91\), found by dividing 273 by 3. So, 91 of the pizzas that were ordered were specialty pizzas.

Using Fractions to Convert Between Units

A common application of fractions is called unit conversion, or converting units, which is the process of changing from the units used in making a measurement to different units of measurement.

For instance, 1 inch is (approximately) equal to 2.54 cm. To convert between units, the two equivalent values are made into a fraction. To convert from the first type of unit to the second type, the fraction has the second unit as the numerator, and the first unit as the denominator.

From the inches and centimeters example, to change from inches to centimeters, we use the fraction \(\frac{2.54\text{cm}}{1\text{in}}\). If, on the other hand, we wanted to convert from centimeters to inches, we’d use the fraction \(\frac{1\text{in}}{2.54\text{cm}}\). This fraction is multiplied by the number of units of the type you are converting from, which means the units of the denominator are the same as the units being multiplied.

Converting Liters to Gallons

Try it.

It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons.

Solution

We know that 1 liter = 0.264172 gal. Since we are converting from liters, when we create the fraction we use, make sure the liter part of the equivalence is in the denominator. So, to convert the 14 liters to gallons, we multiply 14 by \(\frac{1\text{gal}}{0.264172\text{gal}/1\text{liter}}\). Notice the gallon part is in the numerator since we’re converting to gallons, and the liter part is in the denominator since we are converting from liters. Performing this and rounding to three decimal places, we find that 14 liters is \(14\ \text{liter}\times \frac{0.264172\text{gal}}{1\text{liter}}=3.69841\text{gal}\).

Converting Centimeters to Inches

Try it.

It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches.

Solution

We know that 1 inch = 2.54 cm. Since we are converting from centimeters, when we create the fraction we use, make sure the centimeter part of the equivalence is in the denominator, \(\frac{1\ \text{in}}{2.54\ \text{cm}}\). To convert the 100 cm to inches, multiply 100 by \(\frac{1\ \text{in}}{2.54\ \text{cm}}\). Notice the inch part is in the numerator since we’re converting to inches, and the centimeter part is in the denominator since we are converting from centimeters. Performing this and rounding to three decimal places, we obtain \(100\ \text{cm}\times \frac{1\ \text{in}}{2.54\ \text{cm}}=39.370\ \text{in}\). This means 100 cm equals 39.370 in.

Defining and Applying Percent

A percent is a specific rational number and is literally per 100. \(n\) percent, denoted \(n\)%, is the fraction \(\frac{n}{100}\).

Rewriting a Percentage as a Fraction

Try it.

Rewrite the following as fractions:

  1. 31%
  2. 93%
Solution
  1. Using the definition and \(n=31\), 31% in fraction form is \(\frac{31}{100}\).
  2. Using the definition and \(n=93\), 93% in fraction form is \(\frac{93}{100}\).
Rewriting a Percentage as a Decimal

Rewrite the following percentages in decimal form:

Try it.

  1. 54%
  2. 83%
Solution

  1. Using the definition and \(n=54\), 54% in fraction form is \(\frac{54}{100}\). Dividing a number by 100 moves the decimal two places to the left; 54% in decimal form is then 0.54.
  2. Using the definition and \(n=83\), 83% in fraction form is \(\frac{83}{100}\). Dividing a number by 100 moves the decimal two places to the left; 83% in decimal form is then 0.83.

You should notice that you can simply move the decimal two places to the left without using the fractional definition of percent.

Percent is used to indicate a fraction of a total. If we want to find 30% of 90, we would perform a multiplication, with 30% written in either decimal form or fractional form. The 90 is the total, 30 is the percentage, and 27 (which is \(0.30\times 90\)) is the percentage of the total.

Finding a Percentage of a Total

Try it.

  1. Determine 40% of 300.
  2. Determine 64% of 190.
Solution
  1. The total is 300, and the percentage is 40. Using the decimal form of 40% and multiplying we obtain \(0.40\times 300=120\).
  2. The total is 190, and the percentage is 64. Using the decimal form of 64% and multiplying we obtain \(0.64\times 190=121.6\).

In the previous situation, we knew the total and we found the percentage of the total. It may be that we know the percentage of the total, and we know the percent, but we don't know the total. To find the total if we know the percentage of the total, use the following formula.

The percentage can be found if the total and the percentage of the total is known. If you know the total, and the percentage of the total, first divide the part by the total. Move the decimal two places to the right and append the symbol %. The percentage may be found using the following formula.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Solve Problems Using Percent

In the media, in research, and in casual conversation percentages are used frequently to express proportions. Understanding how to use percent is vital to consuming media and understanding numbers. Solving problems using percentages comes down to identifying which of the three components of a percentage you are given, the total, the percentage, or the percentage of the total. If you have two of those components, you can find the third using the methods outlined previously.

Percentage of Students Who Are Sleep Deprived

Try it.

A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night?

Solution

The percentage of interest is 70%. The total number of students is 400. With that, we can find how many were in the percentage of the total, or, how many were sleep deprived. Applying the formula from above, the number who were sleep deprived was \(0.70\times 400=280\); 280 students on the study were sleep deprived.

Amazon Prime Subscribers

Try it.

There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United States, what percentage of U.S. residents are Amazon Prime subscribers?

Solution

We are asked to find the percentage. To do so, we divide the percentage of the total, which is 126 million, by the total, which is 328.2 million. Performing this division and rounding to three decimal places yields \(\frac{126}{328.2}=0.384\). The decimal is moved to the right by two places, and a % sign is appended to the end. Doing this shows us that 38.4% of U.S. residents are Amazon Prime subscribers.

Condensed — the full section is in OpenStax Contemporary Mathematics.

Key Concepts

  • Rational numbers are fractions of integers, and can always be written as an integer divided by an integer.
  • The numerator and denominator of a fraction may have common factors. In such cases, the fraction can be reduced by canceling common factors. When the numerator and denominator of a fraction have no common factors, the fraction is said to be reduced.
  • An improper fraction is one with a numerator larger than the denominator. Such a fraction can be rewritten as an integer plus a proper fraction. This is called a mixed number.
  • Using division and remainder, an improper fraction may be written as a mixed number.
  • A mixed number can be converted to an improper fraction by reversing the process for changing an improper fraction to a mixed number.
  • The arithmetic operations or addition, subtraction, multiplication and division can all be performed on rational numbers.
  • Addition and subtraction of rational numbers can be performed after a common denominator has been identified, and the fractions have been converted to forms having the common denominator.
  • Multiplication and division of rational numbers can be performed without regard to common denominators.
  • Between any two rational numbers, there is always another rational number. This is the density property of the rational numbers.

Formulas

  • \(\frac{a}{c}\pm \frac{b}{c}=\frac{a\pm b}{c}\)
  • \(\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}\)
  • \(\frac{a}{b}\times \frac{c}{d}=\frac{a\times c}{b\times d}\frac{a}{b}\div \frac{c}{d}=\frac{a}{d}\times \frac{d}{c}=\frac{a\times d}{b\times c}\)

Videos

  • Introduction to Fractions
  • Equivalent Fractions
  • Reducing Fractions to Lowest Terms
  • Using Desmos to Reduce a Fraction
  • Adding and Subtracting Fractions with Different Denominators
  • Converting an Improper Fraction to a Mixed Number Using Desmos
  • Multiplying Fractions
  • Dividing Fractions
  • Order of Operations Using Fractions
  • Finding a Fraction of a Total
  • Converting Units

Practice (32)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

  1. Which of the following are perfect squares?

    1. 45
    2. 144
    Lafunua jibu

    1. We could attempt to find the perfect square by factoring. Writing all the factor pairs of 45 results in \(1\times 45,3\times 15\), and \(5\times 9\). None of the pairs is a square, so 45 is not a perfect square. Using a calculator to find the square root of 45, we obtain 6.708 (rounded to three decimal places). Since this was not an integer, the original number was not a perfect square.
    2. We could attempt to find the perfect square by factoring. Writing all the factor pairs of 144 results in \(1\times 144,2\times 72,3\times 48,6\times 24,8\times 18\), and \(12\times 12\). Since the last pair is an integer multiplied by itself, 144 is a perfect square. Alternately, using Desmos to find the square root of 144, we obtain 12. Since the square root of 144 is an integer, 144 is a perfect square.

  2. Determine which of the following are rational numbers:

    1. \(\sqrt{73}\)
    2. \(4.556\)
    3. \(3\frac{1}{5}\)
    4. \(\frac{41}{17}\)
    5. \(5.\overset{\bar}{64}\)
    Lafunua jibu

    1. Since 73 is not a perfect square, its square root is not a rational number. This can also be seen when a calculator is used. Entering \(\sqrt{73}\) into a calculator results in 8.544003745317 (and then more decimal values after that). There is no repeated pattern, so this is not a rational number.
    2. Since 4.556 is a decimal that terminates, this is a rational number.
    3. \(3\frac{1}{5}\) is a mixed number, so it is a rational number.
    4. \(\frac{41}{17}\) is an integer divided by an integer, so it is a rational number.
    5. \(5.646464...\) is a decimal that repeats a pattern, so it is a rational number.

  3. Determine if \(\frac{12}{30}\) and \(\frac{14}{35}\) are equivalent fractions.

    Lafunua jibu

    Applying the definition, \(a=12,b=30,c=14\) and \(d=35\). So \(a\times d=12\times 35=420\). Also, \(b\times c=30\times 14=420\). Since these values are equal, the fractions are equivalent.

  4. Express the following rational numbers in lowest terms:

    1. \(\frac{36}{48}\)
    2. \(\frac{100}{250}\)
    3. \(\frac{51}{136}\)
    Lafunua jibu

    1. One process to reduce \(\frac{36}{48}\) to lowest terms is to identify the GCD of 36 and 48 and divide out the GCD. The GCD of 36 and 48 is 12.

      Step 1: We can then rewrite the numerator and denominator by factoring 12 from both.

      \(\frac{36}{48}=\frac{12\times 3}{12\times 4}\)

      Step 2: We can now divide out the 12s from the numerator and denominator.

      \(\frac{36}{48}=\frac{12\times 3}{12\times 4}=\frac{3}{4}\)

      So, when \(\frac{36}{48}\) is reduced to lowest terms, the result is \(\frac{3}{4}\).

      Alternately, you could identify a common factor, divide out that common factor, and repeat the process until the remaining fraction is in lowest terms.

      Step 1: You may notice that 4 is a common factor of 36 and 48.

      Step 2: Divide out the 4, as in \(\frac{36}{48}=\frac{4\times 9}{4\times 12}=\frac{4\times 9}{4\times 12}=\frac{9}{12}\).

      Step 3: Examining the 9 and 12, you identify 3 as a common factor and divide out the 3, as in \(\frac{9}{12}=\frac{3\times 3}{3\times 4}=\frac{3}{4}\). The 3 and 4 have no common positive factors other than 1, so it is in lowest terms.

      So, when \(\frac{36}{48}\) is reduced to lowest terms, the result is \(\frac{3}{4}\).

    2. Step 1: To reduce \(\frac{100}{250}\) to lowest terms, identify the GCD of 100 and 250. This GCD is 50.

      Step 2: We can then rewrite the numerator and denominator by factoring 50 from both.

      \(\frac{100}{250}=\frac{50\times 2}{50\times 5}\).

      Step 3: We can now divide out the 50s from the numerator and denominator.

      \(\frac{100}{250}=\frac{50\times 2}{50\times 5}=\frac{2}{5}\)

      So, when \(\frac{100}{250}\) is reduced to lowest terms, the result is \(\frac{2}{5}\).

    3. Step 1: To reduce \(\frac{51}{136}\) to lowest terms, identify the GCD of 51 and 136. This GCD is 17.

      Step 2: We can then rewrite the numerator and denominator by factoring 17 from both.

      \(\frac{51}{136}=\frac{17\times 3}{17\times 8}\)

      Step 3: We can now divide out the 17s from the numerator and denominator.

      \(\frac{51}{136}=\frac{17\times 3}{17\times 8}=\frac{3}{8}\)

      So, when \(\frac{51}{136}\) is reduced to lowest terms, the result is \(\frac{3}{8}\).

  5. Calculate \(\frac{23}{42}+\frac{9}{56}\) using Desmos.

    Lafunua jibu

    Enter \(\frac{23}{42}+\frac{9}{56}\) in Desmos. The result is displayed as \(0.70833333333\) (which is \(0.708\overset{\bar}{3}\)). Clicking the fraction button to the left on the calculation line yields \(\frac{17}{24}\).

  6. Calculate \(\frac{13}{28}+\frac{7}{28}\).

    Lafunua jibu

    Since the rational numbers have the same denominator, we perform the addition of the numerators, \(13+7\), and then place the result in the numerator and the common denominator, 28, in the denominator. \(\frac{13}{28}+\frac{7}{28}=\frac{13+7}{28}=\frac{20}{28}\)

    Once we have that result, reduce to lowest terms, which gives \(\frac{20}{28}=\frac{4\times 5}{4\times 7}=\frac{4\times 5}{4\times 7}=\frac{5}{7}\).

  7. Calculate \(\frac{45}{136}-\frac{17}{136}\).

    Lafunua jibu

    Since the rational numbers have the same denominator, we perform the subtraction of the numerators, \(45-17\), and then place the result in the numerator and the common denominator, 136, in the denominator. \(\frac{45}{136}-\frac{17}{136}-\frac{45-17}{136}=\frac{28}{136}\)

    Once we have that result, reduce to lowest terms, this gives \(\frac{28}{136}=\frac{4\times 7}{4\times 34}=\frac{4\times 7}{4\times 34}=\frac{7}{34}\).

  8. Calculate \(\frac{11}{18}+\frac{2}{15}\).

    Lafunua jibu

    The denominators of the fractions are 18 and 15, so we label \(b=18\) and \(d=15\).

    Step 1: Find LCM(18,15). This is 90.

    Step 2: Calculate \(n\) and \(m\). \(n=\frac{90}{18}=5\) and \(m=\frac{90}{15}=6\).

    Step 3: Multiplying the numerator and denominator of \(\frac{11}{18}\) by \(n=5\) yields \(\frac{11\times 5}{18\times 5}=\frac{55}{90}\).

    Step 4: Multiply the numerator and denominator of \(\frac{2}{15}\) by \(m=6\) yields \(\frac{2\times 6}{15\times 6}=\frac{12}{90}\).

    Step 5: Now we add the values from Steps 3 and 4: \(\frac{55}{90}+\frac{12}{90}=\frac{67}{90}\).

    This is in lowest terms, so we have found that \(\frac{11}{18}+\frac{2}{15}=\frac{67}{90}\).

  9. Calculate \(\frac{14}{25}-\frac{9}{70}\).

    Lafunua jibu

    The denominators of the fractions are 25 and 70, so we label \(b=25\) and \(d=70\).

    Step 1: Find LCM(25,70). This is 350.

    Step 2: Calculate \(n\) and \(m\): \(n=\frac{350}{25}=14\) and \(m=\frac{350}{70}=5\).

    Step 3: Multiplying the numerator and denominator of \(\frac{14}{25}\) by \(n=14\) yields \(\frac{14\times 14}{25\times 14}=\frac{196}{350}\).

    Step 4: Multiplying the numerator and denominator of \(\frac{9}{70}\) by \(m=5\) yields \(\frac{9\times 5}{70\times 5}=\frac{45}{350}\).

    Step 5: Now we subtract the value from Step 4 from the value in Step 3: \(\frac{196}{350}-\frac{45}{350}=\frac{151}{350}\).

    This is in lowest terms, so we have found that \(\frac{14}{25}-\frac{9}{70}=\frac{151}{350}\).

  10. Rewrite \(\frac{48}{13}\) as a mixed number.

    Lafunua jibu

    When 48 is divided by 13, the result is 3 with a remainder of 9. So, we can rewrite \(\frac{48}{13}\) as \(3\frac{9}{13}\).

  11. Rewrite \(5\frac{4}{9}\) as an improper fraction.

    Lafunua jibu

    Step 1: Multiply the integer part, 5, by the denominator, 9, which gives \(5\times 9=45\).

    Step 2: Add that product to the numerator, which gives \(45+4=49\).

    Step 3: Write the number as the sum, 49, divided by the denominator, 9, which gives \(\frac{49}{9}\).

  12. Round 5.67849 to three decimal digits.

    Lafunua jibu

    The third decimal digit is 8. The digit following the 8 is 4. When the digit is 4, we write the number only to the third digit. So, 5.67849 rounded off to three decimal places is 5.678.

  13. Round 45.11475 to four decimal digits.

    Lafunua jibu

    The fourth decimal digit is 7. The digit following the 7 is 5. When the digit is 5, we write the number only to the fourth decimal digit, 45.1147. We then replace the fourth decimal digit by one more than the fourth digit, which yields 45.1148. So, 45.11475 rounded off to four decimal places is 45.1148.

  14. Convert \(\frac{47}{25}\) into decimal form.

    Lafunua jibu

    Using a calculator to divide 47 by 25, the result is 1.88.

  15. Convert 3.2117 to fraction form.

    Lafunua jibu

    Step 1: There are four digits after the decimal point, so \(n=4\).

    Step 2: Raise 10 to the fourth power, \({10}^{4}=10,000\).

    Step 3: When we remove the decimal point, we have 32,117.

    Step 4: The fraction has as its numerator the result from Step 3 and as its denominator the result of Step 2, which is the fraction \(\frac{32,117}{10,000}\).

  16. Calculate \(\frac{12}{25}\times \frac{10}{21}\).

    Lafunua jibu

    Multiply the numerators and place that in the numerator, and then multiply the denominators and place that in the denominator.

    \(\frac{12}{25}\times \frac{10}{21}=\frac{12\times 10}{25\times 21}=\frac{120}{525}\)

    This is not in lowest terms, so this needs to be reduced. The GCD of 120 and 525 is 15.

    \(\frac{120}{525}=\frac{15\times 8}{15\times 35}=\frac{8}{35}\)

  17. Calculate 3.45 ÷ 2.341 using a calculator. Round to three decimal places if necessary.

    Lafunua jibu

    Using a calculator, we obtain 1.473729175565997. Rounding to three decimal places we have 1.474.

    1. Calculate \(\frac{4}{21}\div \frac{6}{35}\).
    2. Calculate \(\frac{1}{8}\div \frac{5}{28}\).

    Lafunua jibu

    1. Step 1: Find the reciprocal of the number being divided by \(\frac{6}{35}\). The reciprocal of that is \(\frac{35}{6}\).

      Step 2: Multiply the first fraction by that reciprocal.

      \(\frac{4}{21}\div \frac{6}{35}=\frac{4}{21}\times \frac{35}{6}=\frac{140}{126}\)

      The answer, \(\frac{140}{126}\) is not in lowest terms. The GCD of 140 and 126 is 14. Factoring and canceling gives \(\frac{140}{126}=\frac{14\times 10}{14\times 9}=\frac{10}{9}\).

    2. Step 1: Find the reciprocal of the number being divided by, which is \(\frac{5}{28}\). The reciprocal of that is \(\frac{28}{5}\).

      Step 2: Multiply the first fraction by that reciprocal: \(\frac{1}{8}\div \frac{5}{28}=\frac{1}{8}\times \frac{28}{5}=\frac{28}{40}\)

      The answer, \(\frac{28}{40}\), is not in lowest reduced form. The GCD of 28 and 40 is 4. Factoring and canceling gives \(\frac{28}{40}=\frac{4\times 7}{4\times 10}=\frac{7}{10}\).

  19. Correctly apply the rules for the order of operations to accurately compute \((\frac{5}{7}-\frac{2}{7})\times {2}^{3}\).

    Lafunua jibu

    Step 1: To calculate this, perform all calculations within the parentheses before other operations.

    \((\frac{5}{7}-\frac{2}{7})\times {2}^{3}=(\frac{3}{7})\times {2}^{3}\)

    Step 2: Since all parentheses have been cleared, we move left to right, and compute all the exponents next.

    \((\frac{3}{7})\times {2}^{3}=(\frac{3}{7})\times 8\)

    Step 3: Now, perform all multiplications and divisions, moving left to right.

    \((\frac{3}{7})\times 8=\frac{24}{7}\)

  20. Correctly apply the rules for the order of operations to accurately compute \(4+\frac{2}{3}\div {({(\frac{5}{9})}^{2}-(\frac{2}{3}+5))}^{2}\).

    Lafunua jibu

    To calculate this, perform all calculations within the parentheses before other operations. Evaluate the innermost parentheses first. We can work separate parentheses expressions at the same time.

    Step 1: The innermost parentheses contain \(\frac{2}{3}+5\). Calculate that first, dividing after finding the common denominator.

    \(\begin{array}{l}4+\frac{2}{3}\div {({(\frac{5}{9})}^{2}-(\frac{2}{3}+5))}^{2} \\ =4+\frac{2}{3}\div {({(\frac{5}{9})}^{2}-(\frac{2}{3}+\frac{5}{1}))}^{2} \\ =4+\frac{2}{3}\div {({(\frac{5}{9})}^{2}-(\frac{2}{3}+\frac{15}{3}))}^{2} \\ =4+\frac{2}{3}\div {({(\frac{5}{9})}^{2}-(\frac{17}{3}))}^{2}\end{array}\)

    Step 2: Calculate the exponent in the parentheses, \({(\frac{5}{9})}^{2}\).

    \(\begin{array}{l}4+\frac{2}{3}\div {({(\frac{5}{9})}^{2}-(\frac{17}{3}))}^{2} \\ =4+\frac{2}{3}\div {((\frac{25}{81})-(\frac{17}{3}))}^{2}\end{array}\)

    Step 3: Subtract inside the parentheses is done, using a common denominator.

    \(\begin{array}{l}4+\frac{2}{3}\div {((\frac{25}{81})-(\frac{17}{3}))}^{2} \\ 4+\frac{2}{3}\div {((\frac{25}{81})-(\frac{17\times 27}{3\times 27}))}^{2} \\ 4+\frac{2}{3}\div {((\frac{25}{81})-(\frac{459}{81}))}^{2} \\ 4+\frac{2}{3}\div {((\frac{-434}{81}))}^{2}\end{array}\)

    Step 4: At this point, evaluate the exponent and divide.

    \(\begin{array}{l}4+\frac{2}{3}\div {((\frac{-434}{81}))}^{2} \\ 4+\frac{2}{3}\div (\frac{188,356}{6,561}) \\ =4+\frac{2}{3}\times (\frac{6,561}{188,356}) \\ =4+\frac{2,187}{94,178}\end{array}\)

    Step 5: Add.

    \(\begin{array}{l}4+\frac{2,187}{94,178} \\ =\frac{378,899}{94,178}\end{array}\)

    Had this been done on a calculator, the decimal form of the answer would be 4.0232 (rounded to four decimal places).

  21. Demonstrate the density property of rational numbers by finding a rational number between \(\frac{4}{11}\) and \(\frac{7}{12}\).

    Lafunua jibu

    To find a rational number between \(\frac{4}{11}\) and \(\frac{7}{12}\):

    Step 1: Add the fractions.

    \(\frac{4}{11}+\frac{7}{12}=\frac{4\times 12}{11\times 12}+\frac{7\times 11}{12\times 11}=\frac{48}{132}+\frac{77}{132}=\frac{125}{132}\)

    Step 2: Divide the result by 2. Recall that to divide by 2, you multiply by the reciprocal of 2. The reciprocal of 2 is \(\frac{1}{2}\), as seen below.

    \(\frac{125}{132}\div 2=\frac{125}{132}\times \frac{1}{2}=\frac{125}{264}\)

    So, one rational number between \(\frac{4}{11}\) and \(\frac{7}{12}\) is \(\frac{125}{264}\).

    We could check that the number we found is between the other two by finding the decimal representation of the numbers. Using a calculator, the decimal representations of the rational numbers are 0.363636…, 0.473484848…, and 0.5833333…. Here it is clear that \(\frac{125}{264}\) is between \(\frac{4}{11}\) and \(\frac{7}{12}\).

  22. James is mixing soil for a raised garden, in which he plans to grow a variety of vegetables. For the soil to be suitable, he determines that \(\frac{2}{5}\) of the soil can be topsoil, but \(\frac{2}{5}\) needs to be peat moss and \(\frac{1}{5}\) has to be compost. To fill the raised garden bed with 60 cubic feet of soil, how much of each component does James need to use?

    Lafunua jibu

    In this example, we know the proportion of each component to mix, and we know the total amount of the mix we need. In this kind of situation, we need to determine the appropriate amount of each component to include in the mixture. For each component of the mixture, multiply 60 cubic feet, which is the total volume of the mixture we want, by the fraction required of the component.

    Step 1: The required fraction of topsoil is \(\frac{2}{5}\), so James needs \(60\times \frac{2}{5}\) cubic feet of topsoil. Performing the multiplication, James needs \(60\times \frac{2}{5}=\frac{120}{5}=24\) (found by treating the fraction as division, and 120 divided by 5 is 24) cubic feet of topsoil.

    Step 2: The required fraction of peat moss is also \(\frac{2}{5}\), so he also needs \(60\times \frac{2}{5}\) cubic feet, or \(60\times \frac{2}{5}=\frac{120}{5}=24\) cubic feet of peat moss.

    Step 3: The required fraction of compost is \(\frac{1}{5}\). For the compost, he needs \(60\times \frac{1}{5}=\frac{60}{5}=12\) cubic feet.

  23. At Bella’s Pizza, one-third of the pizzas that are ordered are one of their specialty varieties. If there are 273 pizzas ordered, how many were specialty pizzas?

    Lafunua jibu

    One-third of the whole are specialty pizzas, so we need one-third of 273, which gives \(\frac{1}{3}\times 273=\frac{273}{3}=91\), found by dividing 273 by 3. So, 91 of the pizzas that were ordered were specialty pizzas.

  24. It is known that 1 liter (L) is 0.264172 gallons (gal). Use this to convert 14 liters into gallons.

    Lafunua jibu

    We know that 1 liter = 0.264172 gal. Since we are converting from liters, when we create the fraction we use, make sure the liter part of the equivalence is in the denominator. So, to convert the 14 liters to gallons, we multiply 14 by \(\frac{1\text{gal}}{0.264172\text{gal}/1\text{liter}}\). Notice the gallon part is in the numerator since we’re converting to gallons, and the liter part is in the denominator since we are converting from liters. Performing this and rounding to three decimal places, we find that 14 liters is \(14\ \text{liter}\times \frac{0.264172\text{gal}}{1\text{liter}}=3.69841\text{gal}\).

  25. It is known that 1 inch is 2.54 centimeters. Use this to convert 100 centimeters into inches.

    Lafunua jibu

    We know that 1 inch = 2.54 cm. Since we are converting from centimeters, when we create the fraction we use, make sure the centimeter part of the equivalence is in the denominator, \(\frac{1\ \text{in}}{2.54\ \text{cm}}\). To convert the 100 cm to inches, multiply 100 by \(\frac{1\ \text{in}}{2.54\ \text{cm}}\). Notice the inch part is in the numerator since we’re converting to inches, and the centimeter part is in the denominator since we are converting from centimeters. Performing this and rounding to three decimal places, we obtain \(100\ \text{cm}\times \frac{1\ \text{in}}{2.54\ \text{cm}}=39.370\ \text{in}\). This means 100 cm equals 39.370 in.

  26. Rewrite the following as fractions:

    1. 31%
    2. 93%
    Lafunua jibu
    1. Using the definition and \(n=31\), 31% in fraction form is \(\frac{31}{100}\).
    2. Using the definition and \(n=93\), 93% in fraction form is \(\frac{93}{100}\).
    1. Determine 40% of 300.
    2. Determine 64% of 190.
    Lafunua jibu
    1. The total is 300, and the percentage is 40. Using the decimal form of 40% and multiplying we obtain \(0.40\times 300=120\).
    2. The total is 190, and the percentage is 64. Using the decimal form of 64% and multiplying we obtain \(0.64\times 190=121.6\).

    1. What is the total if 28% of the total is 140?
    2. What is the total if 6% of the total is 91?

    Lafunua jibu

    1. 28 is the percentage, so \(n=28\). 28% of the total is 140, so \(x=140\). Using those we find that the total was \(\frac{100\times \text{140}}{\text{28}}=500\).
    2. 6 is the percentage, so \(n=6\). 6% of the total is 91, so \(x=91\). Using those we find that the total was \(\frac{100\times \text{91}}{\text{6}}=1,516.6\).

  29. Find the percentage in the following:

    1. Total is 300, percentage of the total is 60.
    2. Total is 440, percentage of the total is 176.
    Lafunua jibu

    1. The total is 300; the percentage of the total is 60. Calculating yields 0.2. Moving the decimal two places to the right gives 20. Appending the percentage to this number results in 20%. So, 60 is 20% of 300.
    2. The total is 440; the percentage of the total is 176. Calculating yields 0.4. Moving the decimal two places to the right gives 40. Appending the percentage to this number results in 40%. So, 176 is 40% of 440.

  30. A study revealed that 70% of students suffer from sleep deprivation, defined to be sleeping less than 8 hours per night. If the survey had 400 participants, how many of those participants had less than 8 hours of sleep per night?

    Lafunua jibu

    The percentage of interest is 70%. The total number of students is 400. With that, we can find how many were in the percentage of the total, or, how many were sleep deprived. Applying the formula from above, the number who were sleep deprived was \(0.70\times 400=280\); 280 students on the study were sleep deprived.

  31. There are 126 million users who are U.S. Amazon Prime subscribers. If there are 328.2 million residents in the United States, what percentage of U.S. residents are Amazon Prime subscribers?

    Lafunua jibu

    We are asked to find the percentage. To do so, we divide the percentage of the total, which is 126 million, by the total, which is 328.2 million. Performing this division and rounding to three decimal places yields \(\frac{126}{328.2}=0.384\). The decimal is moved to the right by two places, and a % sign is appended to the end. Doing this shows us that 38.4% of U.S. residents are Amazon Prime subscribers.

  32. Evander plays on the basketball team at their university and 73% of the athletes at their university receive some sort of scholarship for attending. If they know 219 of the student-athletes receive some sort of scholarship, how many student-athletes are at the university?

    Lafunua jibu

    We need to find the total number of student-athletes at Evander’s university.

    Step 1: Identify what we know. We know the percentage of students who receive some sort of scholarship, 73%. We also know the number of athletes that form the part of the whole, or 219 student-athletes.

    Step 2: To find the total number of student-athletes, use \(\frac{100\ \times \ x}{n}\), with \(x=219\) and \(n=73\). Calculating with those values yields \(\frac{100\ \times \ 219}{73}=300\).

    So, there are 300 total student-athletes at Evander’s university

Symbols used here

\sqrt{x},\ \sqrt[n]{x}
square root, n-th root
The non-negative number whose square (n-th power) is x.
\pm
plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.
\neq
not equal
The two sides are different.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\sum_{k=1}^{n} a_k
summation
Add a_k for k = 1 up to n.
\prod_{k=1}^{n} a_k
product
Multiply a_k for k = 1 up to n.
\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}
number sets
Naturals, integers, rationals, reals, complex numbers.
a \equiv b \pmod n
congruent modulo n
n divides a − b; a and b have the same remainder.
a \mid b,\ \gcd(a,b)
divides, greatest common divisor
b is a multiple of a; the largest number dividing both.
\varphi(n),\ \pi(x)
Euler's totient, prime-counting function
Count of 1..n coprime to n; number of primes up to x.
\mathbb{Z}/n\mathbb{Z},\ \mathbb{Z}_n
integers modulo n
The remainders 0…n−1 with clock arithmetic.
a \bmod n
remainder
What is left after dividing a by n.

How to: Rational Numbers

  1. Define and identify numbers that are rational.
  2. Simplify rational numbers and express in lowest terms.
  3. Add and subtract rational numbers.
  4. Convert between improper fractions and mixed numbers.
  5. Convert rational numbers between decimal and fraction form.
  6. Multiply and divide rational numbers.
  7. Apply the order of operations to rational numbers to simplify expressions.
  8. Apply density property of rational numbers.

Questions people ask

Why are primes so important?

Every integer factors into primes in exactly one way, so primes are the atoms of multiplication. Cryptography relies on that factoring being easy to state and hard to do.

How do I tell whether a big number is prime?

Trial division up to the square root works for small numbers. For large ones, probabilistic tests (Miller–Rabin) give an answer that is wrong with negligible probability, and deterministic tests (AKS) exist but are slower.

Jaribu kufanya mambo yako mwenyewe

Parts of this page are adapted from OpenStax Contemporary Mathematics (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mengi zaidi katika Number Theory

Prime factorisationPrime numbersGCD and LCMModular arithmeticDivisorsSequencesNumber basesDiophantine equationsFermat's little theorem and Euler's theoremRSA: cryptography from number theory