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  • 4.4 Solve Systems of Equations with Three Variables
  • Introduction
  • 1.1 Use the Language of Algebra
  • 1.2 Integers
  • 1.3 Fractions
  • 1.4 Decimals
  • 1.5 Properties of Real Numbers
  • Key Concepts
  • Review Exercises
  • Practice Test
  • 2.1 Use a General Strategy to Solve Linear Equations
  • 2.2 Use a Problem Solving Strategy
  • 2.3 Solve a Formula for a Specific Variable
  • 2.4 Solve Mixture and Uniform Motion Applications
  • 2.5 Solve Linear Inequalities
  • 2.6 Solve Compound Inequalities
  • 2.7 Solve Absolute Value Inequalities
  • 3.1 Graph Linear Equations in Two Variables
  • 3.2 Slope of a Line
  • 3.3 Find the Equation of a Line
  • 3.4 Graph Linear Inequalities in Two Variables
  • 3.5 Relations and Functions
  • 3.6 Graphs of Functions
  • 4.1 Solve Systems of Linear Equations with Two Variables
  • 4.2 Solve Applications with Systems of Equations
  • 4.3 Solve Mixture Applications with Systems of Equations
  • 4.5 Solve Systems of Equations Using Matrices
  • 4.6 Solve Systems of Equations Using Determinants
  • 4.7 Graphing Systems of Linear Inequalities
  • 5.1 Add and Subtract Polynomials
  • 5.2 Properties of Exponents and Scientific Notation
  • 5.3 Multiply Polynomials
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  • Introduction to Factoring
  • 6.1 Greatest Common Factor and Factor by Grouping
  • 6.2 Factor Trinomials
  • 6.3 Factor Special Products
  • 6.4 General Strategy for Factoring Polynomials
  • 6.5 Polynomial Equations
  • 7.1 Multiply and Divide Rational Expressions
  • 7.2 Add and Subtract Rational Expressions
  • 7.3 Simplify Complex Rational Expressions
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Learning Objectives

By the end of this section, you will be able to:

  • Determine whether an ordered triple is a solution of a system of three linear equations with three variables
  • Solve a system of linear equations with three variables
  • Solve applications using systems of linear equations with three variables

Be Prepared 4.10

Before you get started, take this readiness quiz.

Evaluate 5 x − 2 y + 3 z 5 x − 2 y + 3 z when x = −2 , x = −2 , y = −4 , y = −4 , and z = 3 . z = 3 . If you missed this problem, review Example 1.21 .

Be Prepared 4.11

Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. { − 2 x + y = −11 x + 3 y = 9 . { − 2 x + y = −11 x + 3 y = 9 . If you missed this problem, review Example 2.6 .

Be Prepared 4.12

Classify the equations as a conditional equation, an identity, or a contradiction and then state the solution. { 7 x + 8 y = 4 3 x − 5 y = 27 . { 7 x + 8 y = 4 3 x − 5 y = 27 . If you missed this problem, review Example 2.8 .

Determine Whether an Ordered Triple is a Solution of a System of Three Linear Equations with Three Variables

In this section, we will extend our work of solving a system of linear equations. So far we have worked with systems of equations with two equations and two variables. Now we will work with systems of three equations with three variables. But first let's review what we already know about solving equations and systems involving up to two variables.

We learned earlier that the graph of a linear equation , a x + b y = c , a x + b y = c , is a line. Each point on the line, an ordered pair ( x , y ) , ( x , y ) , is a solution to the equation. For a system of two equations with two variables, we graph two lines. Then we can see that all the points that are solutions to each equation form a line. And, by finding what the lines have in common, we’ll find the solution to the system.

Most linear equations in one variable have one solution, but we saw that some equations, called contradictions, have no solutions and for other equations, called identities, all numbers are solutions

We know when we solve a system of two linear equations represented by a graph of two lines in the same plane, there are three possible cases, as shown.

Similarly, for a linear equation with three variables a x + b y + c z = d , a x + b y + c z = d , every solution to the equation is an ordered triple, ( x , y , z ) ( x , y , z ) , that makes the equation true.

Linear Equation in Three Variables

A linear equation with three variables, where a, b, c, and d are real numbers and a, b , and c are not all 0, is of the form

Every solution to the equation is an ordered triple, ( x , y , z ) ( x , y , z ) that makes the equation true.

All the points that are solutions to one equation form a plane in three-dimensional space. And, by finding what the planes have in common, we’ll find the solution to the system.

When we solve a system of three linear equations represented by a graph of three planes in space, there are three possible cases.

To solve a system of three linear equations, we want to find the values of the variables that are solutions to all three equations. In other words, we are looking for the ordered triple ( x , y , z ) ( x , y , z ) that makes all three equations true. These are called the solutions of the system of three linear equations with three variables .

Solutions of a System of Linear Equations with Three Variables

Solutions of a system of equations are the values of the variables that make all the equations true. A solution is represented by an ordered triple ( x , y , z ) . ( x , y , z ) .

To determine if an ordered triple is a solution to a system of three equations, we substitute the values of the variables into each equation. If the ordered triple makes all three equations true, it is a solution to the system.

Example 4.31

Determine whether the ordered triple is a solution to the system: { x − y + z = 2 2 x − y − z = −6 2 x + 2 y + z = −3 . { x − y + z = 2 2 x − y − z = −6 2 x + 2 y + z = −3 .

ⓐ ( −2 , −1 , 3 ) ( −2 , −1 , 3 ) ⓑ ( −4 , −3 , 4 ) ( −4 , −3 , 4 )

Try It 4.61

Determine whether the ordered triple is a solution to the system: { 3 x + y + z = 2 x + 2 y + z = −3 3 x + y + 2 z = 4 . { 3 x + y + z = 2 x + 2 y + z = −3 3 x + y + 2 z = 4 .

ⓐ ( 1 , −3 , 2 ) ( 1 , −3 , 2 ) ⓑ ( 4 , −1 , −5 ) ( 4 , −1 , −5 )

Try It 4.62

Determine whether the ordered triple is a solution to the system: { x − 3 y + z = −5 − 3 x − y − z = 1 2 x − 2 y + 3 z = 1 . { x − 3 y + z = −5 − 3 x − y − z = 1 2 x − 2 y + 3 z = 1 .

ⓐ ( 2 , −2 , 3 ) ( 2 , −2 , 3 ) ⓑ ( −2 , 2 , 3 ) ( −2 , 2 , 3 )

Solve a System of Linear Equations with Three Variables

To solve a system of linear equations with three variables, we basically use the same techniques we used with systems that had two variables. We start with two pairs of equations and in each pair we eliminate the same variable. This will then give us a system of equations with only two variables and then we know how to solve that system!

Next, we use the values of the two variables we just found to go back to the original equation and find the third variable. We write our answer as an ordered triple and then check our results.

Example 4.32

How to solve a system of equations with three variables by elimination.

Solve the system by elimination: { x − 2 y + z = 3 2 x + y + z = 4 3 x + 4 y + 3 z = −1 . { x − 2 y + z = 3 2 x + y + z = 4 3 x + 4 y + 3 z = −1 .

Try It 4.63

Solve the system by elimination: { 3 x + y − z = 2 2 x − 3 y − 2 z = 1 4 x − y − 3 z = 0 . { 3 x + y − z = 2 2 x − 3 y − 2 z = 1 4 x − y − 3 z = 0 .

Try It 4.64

Solve the system by elimination: { 4 x + y + z = −1 − 2 x − 2 y + z = 2 2 x + 3 y − z = 1 . { 4 x + y + z = −1 − 2 x − 2 y + z = 2 2 x + 3 y − z = 1 .

The steps are summarized here.

Solve a system of linear equations with three variables.

  • If any coefficients are fractions, clear them.
  • Decide which variable you will eliminate.
  • Work with a pair of equations to eliminate the chosen variable.
  • Multiply one or both equations so that the coefficients of that variable are opposites.
  • Add the equations resulting from Step 2 to eliminate one variable
  • Step 3. Repeat Step 2 using two other equations and eliminate the same variable as in Step 2.
  • Step 4. The two new equations form a system of two equations with two variables. Solve this system.
  • Step 5. Use the values of the two variables found in Step 4 to find the third variable.
  • Step 6. Write the solution as an ordered triple.
  • Step 7. Check that the ordered triple is a solution to all three original equations.

Example 4.33

Solve: { 3 x − 4 z = 0 3 y + 2 z = −3 2 x + 3 y = −5 . { 3 x − 4 z = 0 3 y + 2 z = −3 2 x + 3 y = −5 .

We can eliminate z z from equations (1) and (2) by multiplying equation (2) by 2 and then adding the resulting equations.

Notice that equations (3) and (4) both have the variables x x and y y . We will solve this new system for x x and y y .

To solve for y , we substitute x = −4 x = −4 into equation (3).

We now have x = −4 x = −4 and y = 1 . y = 1 . We need to solve for z . We can substitute x = −4 x = −4 into equation (1) to find z .

We write the solution as an ordered triple. ( −4 , 1 , −3 ) ( −4 , 1 , −3 )

We check that the solution makes all three equations true.

3 x − 4 z = 0 ( 1 ) 3 ( −4 ) − 4 ( −3 ) = ? 0 0 = 0 ✓ 3 y + 2 z = −3 ( 2 ) 3 ( 1 ) + 2 ( −3 ) = ? − 3 −3 = −3 ✓ 2 x + 3 y = −5 ( 3 ) 2 ( −4 ) + 3 ( 1 ) = ? − 5 −5 = −5 ✓ The solution is ( −4 , 1 , −3 ) . 3 x − 4 z = 0 ( 1 ) 3 ( −4 ) − 4 ( −3 ) = ? 0 0 = 0 ✓ 3 y + 2 z = −3 ( 2 ) 3 ( 1 ) + 2 ( −3 ) = ? − 3 −3 = −3 ✓ 2 x + 3 y = −5 ( 3 ) 2 ( −4 ) + 3 ( 1 ) = ? − 5 −5 = −5 ✓ The solution is ( −4 , 1 , −3 ) .

Try It 4.65

Solve: { 3 x − 4 z = −1 2 y + 3 z = 2 2 x + 3 y = 6 . { 3 x − 4 z = −1 2 y + 3 z = 2 2 x + 3 y = 6 .

Try It 4.66

Solve: { 4 x − 3 z = −5 3 y + 2 z = 7 3 x + 4 y = 6 . { 4 x − 3 z = −5 3 y + 2 z = 7 3 x + 4 y = 6 .

When we solve a system and end up with no variables and a false statement, we know there are no solutions and that the system is inconsistent. The next example shows a system of equations that is inconsistent.

Example 4.34

Solve the system of equations: { x + 2 y − 3 z = −1 x − 3 y + z = 1 2 x − y − 2 z = 2 . { x + 2 y − 3 z = −1 x − 3 y + z = 1 2 x − y − 2 z = 2 .

Use equation (1) and (2) to eliminate z .

Use (2) and (3) to eliminate z z again.

Use (4) and (5) to eliminate a variable.

There is no solution.

We are left with a false statement and this tells us the system is inconsistent and has no solution.

Try It 4.67

Solve the system of equations: { x + 2 y + 6 z = 5 − x + y − 2 z = 3 x − 4 y − 2 z = 1 . { x + 2 y + 6 z = 5 − x + y − 2 z = 3 x − 4 y − 2 z = 1 .

Try It 4.68

Solve the system of equations: { 2 x − 2 y + 3 z = 6 4 x − 3 y + 2 z = 0 − 2 x + 3 y − 7 z = 1 . { 2 x − 2 y + 3 z = 6 4 x − 3 y + 2 z = 0 − 2 x + 3 y − 7 z = 1 .

When we solve a system and end up with no variables but a true statement, we know there are infinitely many solutions. The system is consistent with dependent equations. Our solution will show how two of the variables depend on the third.

Example 4.35

Solve the system of equations: { x + 2 y − z = 1 2 x + 7 y + 4 z = 11 x + 3 y + z = 4 . { x + 2 y − z = 1 2 x + 7 y + 4 z = 11 x + 3 y + z = 4 .

Use equation (1) and (3) to eliminate x .

Use equation (1) and (2) to eliminate x again.

Use equation (4) and (5) to eliminate y y .

The true statement 0 = 0 0 = 0 tells us that this is a dependent system that has infinitely many solutions. The solutions are of the form ( x , y , z ) ( x , y , z ) where x = 5 z − 5 ; y = −2 z + 3 x = 5 z − 5 ; y = −2 z + 3 and z is any real number.

Try It 4.69

Solve the system by equations: { x + y − z = 0 2 x + 4 y − 2 z = 6 3 x + 6 y − 3 z = 9 . { x + y − z = 0 2 x + 4 y − 2 z = 6 3 x + 6 y − 3 z = 9 .

Try It 4.70

Solve the system by equations: { x − y − z = 1 − x + 2 y − 3 z = −4 3 x − 2 y − 7 z = 0 . { x − y − z = 1 − x + 2 y − 3 z = −4 3 x − 2 y − 7 z = 0 .

Solve Applications using Systems of Linear Equations with Three Variables

Applications that are modeled by a systems of equations can be solved using the same techniques we used to solve the systems. Many of the application are just extensions to three variables of the types we have solved earlier.

Example 4.36

The community college theater department sold three kinds of tickets to its latest play production. The adult tickets sold for $15, the student tickets for $10 and the child tickets for $8. The theater department was thrilled to have sold 250 tickets and brought in $2,825 in one night. The number of student tickets sold is twice the number of adult tickets sold. How many of each type did the department sell?

Try It 4.71

The community college fine arts department sold three kinds of tickets to its latest dance presentation. The adult tickets sold for $20, the student tickets for $12 and the child tickets for $10.The fine arts department was thrilled to have sold 350 tickets and brought in $4,650 in one night. The number of child tickets sold is the same as the number of adult tickets sold. How many of each type did the department sell?

Try It 4.72

The community college soccer team sold three kinds of tickets to its latest game. The adult tickets sold for $10, the student tickets for $8 and the child tickets for $5. The soccer team was thrilled to have sold 600 tickets and brought in $4,900 for one game. The number of adult tickets is twice the number of child tickets. How many of each type did the soccer team sell?

Access this online resource for additional instruction and practice with solving a linear system in three variables with no or infinite solutions.

  • Solving a Linear System in Three Variables with No or Infinite Solutions
  • 3 Variable Application

Section 4.4 Exercises

Practice makes perfect.

In the following exercises, determine whether the ordered triple is a solution to the system.

{ 2 x − 6 y + z = 3 3 x − 4 y − 3 z = 2 2 x + 3 y − 2 z = 3 { 2 x − 6 y + z = 3 3 x − 4 y − 3 z = 2 2 x + 3 y − 2 z = 3

ⓐ ( 3 , 1 , 3 ) ( 3 , 1 , 3 ) ⓑ ( 4 , 3 , 7 ) ( 4 , 3 , 7 )

{ − 3 x + y + z = −4 − x + 2 y − 2 z = 1 2 x − y − z = −1 { − 3 x + y + z = −4 − x + 2 y − 2 z = 1 2 x − y − z = −1

ⓐ ( −5 , −7 , 4 ) ( −5 , −7 , 4 ) ⓑ ( 5 , 7 , 4 ) ( 5 , 7 , 4 )

{ y − 10 z = −8 2 x − y = 2 x − 5 z = 3 { y − 10 z = −8 2 x − y = 2 x − 5 z = 3

ⓐ ( 7 , 12 , 2 ) ( 7 , 12 , 2 ) ⓑ ( 2 , 2 , 1 ) ( 2 , 2 , 1 )

{ x + 3 y − z = 15 y = 2 3 x − 2 x − 3 y + z = −2 { x + 3 y − z = 15 y = 2 3 x − 2 x − 3 y + z = −2

ⓐ ( −6 , 5 , 1 2 ) ( −6 , 5 , 1 2 ) ⓑ ( 5 , 4 3 , −3 ) ( 5 , 4 3 , −3 )

In the following exercises, solve the system of equations.

{ 5 x + 2 y + z = 5 − 3 x − y + 2 z = 6 2 x + 3 y − 3 z = 5 { 5 x + 2 y + z = 5 − 3 x − y + 2 z = 6 2 x + 3 y − 3 z = 5

{ 6 x − 5 y + 2 z = 3 2 x + y − 4 z = 5 3 x − 3 y + z = −1 { 6 x − 5 y + 2 z = 3 2 x + y − 4 z = 5 3 x − 3 y + z = −1

{ 2 x − 5 y + 3 z = 8 3 x − y + 4 z = 7 x + 3 y + 2 z = −3 { 2 x − 5 y + 3 z = 8 3 x − y + 4 z = 7 x + 3 y + 2 z = −3

{ 5 x − 3 y + 2 z = −5 2 x − y − z = 4 3 x − 2 y + 2 z = −7 { 5 x − 3 y + 2 z = −5 2 x − y − z = 4 3 x − 2 y + 2 z = −7

{ 3 x − 5 y + 4 z = 5 5 x + 2 y + z = 0 2 x + 3 y − 2 z = 3 { 3 x − 5 y + 4 z = 5 5 x + 2 y + z = 0 2 x + 3 y − 2 z = 3

{ 4 x − 3 y + z = 7 2 x − 5 y − 4 z = 3 3 x − 2 y − 2 z = −7 { 4 x − 3 y + z = 7 2 x − 5 y − 4 z = 3 3 x − 2 y − 2 z = −7

{ 3 x + 8 y + 2 z = −5 2 x + 5 y − 3 z = 0 x + 2 y − 2 z = −1 { 3 x + 8 y + 2 z = −5 2 x + 5 y − 3 z = 0 x + 2 y − 2 z = −1

{ 11 x + 9 y + 2 z = −9 7 x + 5 y + 3 z = −7 4 x + 3 y + z = −3 { 11 x + 9 y + 2 z = −9 7 x + 5 y + 3 z = −7 4 x + 3 y + z = −3

{ 1 3 x − y − z = 1 x + 5 2 y + z = −2 2 x + 2 y + 1 2 z = −4 { 1 3 x − y − z = 1 x + 5 2 y + z = −2 2 x + 2 y + 1 2 z = −4

{ x + 1 2 y + 1 2 z = 0 1 5 x − 1 5 y + z = 0 1 3 x − 1 3 y + 2 z = −1 { x + 1 2 y + 1 2 z = 0 1 5 x − 1 5 y + z = 0 1 3 x − 1 3 y + 2 z = −1

{ x + 1 3 y − 2 z = −1 1 3 x + y + 1 2 z = 0 1 2 x + 1 3 y − 1 2 z = −1 { x + 1 3 y − 2 z = −1 1 3 x + y + 1 2 z = 0 1 2 x + 1 3 y − 1 2 z = −1

{ 1 3 x − y + 1 2 z = 4 2 3 x + 5 2 y − 4 z = 0 x − 1 2 y + 3 2 z = 2 { 1 3 x − y + 1 2 z = 4 2 3 x + 5 2 y − 4 z = 0 x − 1 2 y + 3 2 z = 2

{ x + 2 z = 0 4 y + 3 z = −2 2 x − 5 y = 3 { x + 2 z = 0 4 y + 3 z = −2 2 x − 5 y = 3

{ 2 x + 5 y = 4 3 y − z = 3 4 x + 3 z = −3 { 2 x + 5 y = 4 3 y − z = 3 4 x + 3 z = −3

{ 2 y + 3 z = −1 5 x + 3 y = −6 7 x + z = 1 { 2 y + 3 z = −1 5 x + 3 y = −6 7 x + z = 1

{ 3 x − z = −3 5 y + 2 z = −6 4 x + 3 y = −8 { 3 x − z = −3 5 y + 2 z = −6 4 x + 3 y = −8

{ 4 x − 3 y + 2 z = 0 − 2 x + 3 y − 7 z = 1 2 x − 2 y + 3 z = 6 { 4 x − 3 y + 2 z = 0 − 2 x + 3 y − 7 z = 1 2 x − 2 y + 3 z = 6

{ x − 2 y + 2 z = 1 − 2 x + y − z = 2 x − y + z = 5 { x − 2 y + 2 z = 1 − 2 x + y − z = 2 x − y + z = 5

{ 2 x + 3 y + z = 12 x + y + z = 9 3 x + 4 y + 2 z = 20 { 2 x + 3 y + z = 12 x + y + z = 9 3 x + 4 y + 2 z = 20

{ x + 4 y + z = −8 4 x − y + 3 z = 9 2 x + 7 y + z = 0 { x + 4 y + z = −8 4 x − y + 3 z = 9 2 x + 7 y + z = 0

{ x + 2 y + z = 4 x + y − 2 z = 3 − 2 x − 3 y + z = −7 { x + 2 y + z = 4 x + y − 2 z = 3 − 2 x − 3 y + z = −7

{ x + y − 2 z = 3 − 2 x − 3 y + z = −7 x + 2 y + z = 4 { x + y − 2 z = 3 − 2 x − 3 y + z = −7 x + 2 y + z = 4

{ x + y − 3 z = −1 y − z = 0 − x + 2 y = 1 { x + y − 3 z = −1 y − z = 0 − x + 2 y = 1

{ x − 2 y + 3 z = 1 x + y − 3 z = 7 3 x − 4 y + 5 z = 7 { x − 2 y + 3 z = 1 x + y − 3 z = 7 3 x − 4 y + 5 z = 7

In the following exercises, solve the given problem.

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is twice the measure of the first angle. The third angle is twelve more than the second. Find the measures of the three angles.

The sum of the measures of the angles of a triangle is 180. The sum of the measures of the second and third angles is three times the measure of the first angle. The third angle is fifteen more than the second. Find the measures of the three angles.

After watching a major musical production at the theater, the patrons can purchase souvenirs. If a family purchases 4 t-shirts, the video, and 1 stuffed animal, their total is $135.

A couple buys 2 t-shirts, the video, and 3 stuffed animals for their nieces and spends $115. Another couple buys 2 t-shirts, the video, and 1 stuffed animal and their total is $85. What is the cost of each item?

The church youth group is selling snacks to raise money to attend their convention. Amy sold 2 pounds of candy, 3 boxes of cookies and 1 can of popcorn for a total sales of $65. Brian sold 4 pounds of candy, 6 boxes of cookies and 3 cans of popcorn for a total sales of $140. Paulina sold 8 pounds of candy, 8 boxes of cookies and 5 cans of popcorn for a total sales of $250. What is the cost of each item?

Writing Exercises

In your own words explain the steps to solve a system of linear equations with three variables by elimination.

How can you tell when a system of three linear equations with three variables has no solution? Infinitely many solutions?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

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  • Publisher/website: OpenStax
  • Book title: Intermediate Algebra 2e
  • Publication date: May 6, 2020
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Fractions Calculator

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Use this calculator if the numerators or denominators are very big integers.

3 over 8

Unlike adding and subtracting integers such as 2 and 8, fractions require a common denominator to undergo these operations. One method for finding a common denominator involves multiplying the numerators and denominators of all of the fractions involved by the product of the denominators of each fraction. Multiplying all of the denominators ensures that the new denominator is certain to be a multiple of each individual denominator. The numerators also need to be multiplied by the appropriate factors to preserve the value of the fraction as a whole. This is arguably the simplest way to ensure that the fractions have a common denominator. However, in most cases, the solutions to these equations will not appear in simplified form (the provided calculator computes the simplification automatically). Below is an example using this method.

This process can be used for any number of fractions. Just multiply the numerators and denominators of each fraction in the problem by the product of the denominators of all the other fractions (not including its own respective denominator) in the problem.

An alternative method for finding a common denominator is to determine the least common multiple (LCM) for the denominators, then add or subtract the numerators as one would an integer. Using the least common multiple can be more efficient and is more likely to result in a fraction in simplified form. In the example above, the denominators were 4, 6, and 2. The least common multiple is the first shared multiple of these three numbers.

The first multiple they all share is 12, so this is the least common multiple. To complete an addition (or subtraction) problem, multiply the numerators and denominators of each fraction in the problem by whatever value will make the denominators 12, then add the numerators.

Subtraction:

Fraction subtraction is essentially the same as fraction addition. A common denominator is required for the operation to occur. Refer to the addition section as well as the equations below for clarification.

Multiplication:

Multiplying fractions is fairly straightforward. Unlike adding and subtracting, it is not necessary to compute a common denominator in order to multiply fractions. Simply, the numerators and denominators of each fraction are multiplied, and the result forms a new numerator and denominator. If possible, the solution should be simplified. Refer to the equations below for clarification.

Simplification:

Converting between fractions and decimals:, common engineering fraction to decimal conversions.

In engineering, fractions are widely used to describe the size of components such as pipes and bolts. The most common fractional and decimal equivalents are listed below.

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Four Fours Puzzle - Solution

The puzzle:.

A popular mathematical pastime:

Use exactly four 4's to form every integer from 0 to 50, using only the operators +, −, ×, /, () (brackets), . (decimal point), √ (square root) and ! (factorial).

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Calculator Use

Use this fraction calculator for adding, subtracting, multiplying and dividing fractions. Answers are fractions in lowest terms or mixed numbers in reduced form.

Input proper or improper fractions, select the math sign and click Calculate. This is a fraction calculator with steps shown in the solution.

If you have negative fractions insert a minus sign before the numerator. So if one of your fractions is -6/7, insert -6 in the numerator and 7 in the denominator.

To do math with mixed numbers (whole numbers and fractions) use the Mixed Numbers Calculator .

Math on Fractions with Unlike Denominators

There are 2 cases where you need to know if your fractions have different denominators:

  • if you are adding fractions
  • if you are subtracting fractions

How to Add or Subtract Fractions

  • Find the least common denominator
  • You can use the LCD Calculator to find the least common denominator for a set of fractions
  • For your first fraction, find what number you need to multiply the denominator by to result in the least common denominator
  • Multiply the numerator and denominator of your first fraction by that number
  • Repeat Steps 3 and 4 for each fraction
  • For addition equations, add the fraction numerators
  • For subtraction equations, subtract the fraction numerators
  • Convert improper fractions to mixed numbers
  • Reduce the fraction to lowest terms

How to Multiply Fractions

  • Multiply all numerators together
  • Multiply all denominators together
  • Reduce the result to lowest terms

How to Divide Fractions

  • Rewrite the equation as in "Keep, Change, Flip"
  • Keep the first fraction
  • Change the division sign to multiplication
  • Flip the second fraction by switching the top and bottom numbers

Fraction Formulas

There is a way to add or subtract fractions without finding the least common denominator (LCD) . This method involves cross multiplication of the fractions. See the formulas below.

You may find that it is easier to use these formulas than to do the math to find the least common denominator.

The formulas for multiplying and dividing fractions follow the same process as described above.

Adding Fractions

The formula for adding fractions is:

Example steps:

Subtracting Fractions

The formula for subtracting fractions is:

Multiplying Fractions

The formula for multiplying fractions is:

Dividing Fractions

The formula for dividing fractions is:

Related Calculators

To perform math operations on mixed number fractions use our Mixed Numbers Calculator . This calculator can also simplify improper fractions into mixed numbers and shows the work involved.

If you want to simplify an individual fraction into lowest terms use our Simplify Fractions Calculator .

For an explanation of how to factor numbers to find the greatest common factor (GCF) see the Greatest Common Factor Calculator .

If you are simplifying large fractions by hand you can use the Long Division with Remainders Calculator to find whole number and remainder values.

This calculator performs the reducing calculation faster than other calculators you might find. The primary reason is that it utilizes Euclid's Algorithm for reducing fractions which can be found on The Math Forum .

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The result:

4 * 3 /4 = 3 / 1 = 3, how do we solve fractions step by step.

  • Multiple: 4 * 3 / 4 = 4 · 3 / 1 · 4 = 12 / 4 = 3 · 4 / 1 · 4 = 3 Multiply both numerators and denominators. Result fraction keep to lowest possible denominator GCD(12, 4) = 4 . In the following intermediate step, cancel by a common factor of 4 gives 3 / 1 . In other words - four multiplied by three quarters is three.

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The Performance Review Problem

As the arcane annual assessment earns a failing grade, employers struggle to create a better system to measure and motivate their workers.

​After an annual review that lasted about 10 minutes, a New Jersey-based account coordinator knew it was time to leave the public relations agency where he had worked for almost a year. 

The 25-year-old, who requested anonymity, asked for the meeting because his boss had not mentioned any formal assessment process, nor had his manager ever critiqued his work. The coordinator says he sat with a trio of senior executives who did not ask him any questions beyond how he would rate himself. He says they ignored his requests for guidance on how to advance at the agency. 

Screen Shot 2023-03-15 at 85749 AM.png

This example also illustrates one of the common failures in performance management: limiting reviews to once or twice a year without having any other meaningful career discussions in between. Nearly half (49 percent) of companies give annual or semiannual reviews, according to a study of 1,000 full-time U.S. employees released late last year by software company Workhuman. 

The only situation that is worse than doing one review per year is doing none at all, experts say. The good news is that only 7 percent of companies are keeping employees in the dark about their performance, and 28 percent of organizations are conducting assessments quarterly, the Workhuman study found.  

A Pervasive Problem

Reviews generally do not work.

That doesn’t mean that more-frequent formal meetings or casual sit-downs between supervisors and their direct reports are solving the performance review quandary, either. Only about 1 in 4 companies in North America (26 percent) said their performance management systems were effective, according to a survey of 837 companies conducted last fall by consulting firm WTW. And only one-third of the organizations said employees felt their efforts were evaluated fairly. 

Meanwhile, a Gallup survey conducted last year found that 95 percent of managers are dissatisfied with their organization’s review system.

The problem is not new, though it is taking on greater importance, experts say. Millennials and members of Generation Z crave feedback and are focused on career development. Meanwhile, the tight labor market has companies searching for ways to keep high-performing employees in the fold. Fewer than 20 percent of employees feel inspired by their reviews, and disengaged employees cost U.S. companies a collective $1.6 trillion a year, according to Gallup.

Lesli Jennings, a senior director at WTW, says part of the issue is that reviews are now so much more than a discussion of past performance. They include conversations about career development, employee experience and compensation. 

“The performance management design itself is not evolving as quickly as the objectives and the purpose that we have set out for what we want it to do,” Jennings says. 

Screen Shot 2023-03-15 at 84340 AM.png

Poor Review Practices

Some argue that means it’s time to completely scrap annual reviews and stop using scales composed of numbers or adjectives to rate employees. 

“Every single human alive today is a horribly unreliable rater of other human beings,” says Marcus Buckingham, head of people and performance research at the Roseland, N.J.-based ADP Research Institute. He says people bring their own backgrounds and personalities to bear in the reviews in what is called the “idiosyncratic rating effect.” He says the ratings managers bestow on others are more a reflection of themselves than of those they’re reviewing.

Buckingham adds that very few positions have quantifiable outcomes that can be considered a measure of competence, talent or success. It’s possible to tally a salesperson’s results or test someone’s knowledge of a computer program, he says, but he’s baffled by attempts to measure attributes such as “leadership potential.”

“I’m going to rate you on a theoretical construct like ‘strategic thinking’? Everybody knows that’s rubbish,” Buckingham says. He adds that performance reviews that offer rankings give “data that’s just bad” and insists that companies rely on data analytics because they don’t trust their managers’ judgment. But instead of working on improving their managers’ skills, he says, they put data systems in place. 

“Because we don’t educate our managers on how to have some of these conversations, we’ve decided that the solution is to give them really bad ratings systems or really bad categorization systems,” Buckingham says. 

R eviewing the Data

A mong North American employers:

  • More than 9 in 10 (93 percent) cited driving organizational performance as a key objective for performance management, yet less than half (44 percent) said their performance management program is ­meeting that objective.
  • Nearly 3 in 4 (72 percent) said ­supporting the career development of their employees is a primary objective, but only 31 percent said their performance management program was meeting that objective.
  • Less than half (49 percent) agreed that managers at their organization are ­effective at assessing the performance of their direct reports. 
  • Only 1 in 3 indicated that employees feel their performance is evaluated fairly. 
  • Just 1 in 6 (16 percent) reported having altered their performance management approach to align with remote and hybrid work models, which are rapidly becoming more prevalent.

Source: WTW 2022 Performance Reset Survey of 837 organizations worldwide, including 150 North American employers.

Data Lovers

Ratings aren’t likely to disappear anytime soon, however. “Data-driven” has become a rallying cry for companies as they seek to operate more efficiently. Organizations are trying to measure everything from sales to productivity, though such efforts can cause turmoil and hurt some individuals’ careers.

A June 2022 study of nearly 30,000 workers at an unnamed North American retail chain found that women were more likely to receive higher overall ratings than men, though women were ranked lower on “potential.” 

In that study, women were 12 percent more likely to be given the lowest rating for potential, as well as 15 percent and 28 percent less likely to receive the middle and highest potential ratings, respectively, according to the professors who conducted the study, Alan Benson of the University of Minnesota, Danielle Li of MIT and Kelly Shue of Yale. The authors also said women were 14 percent less likely to get promoted than men. “Because potential is not directly observed,” they noted, “these assessments can be highly subjective, leaving room for bias.” 

Screen Shot 2023-03-15 at 85749 AM.png

Birmingham left abruptly one afternoon and did not go in to work the next day, which he says Blizzard interpreted as his resignation. Blizzard did not respond to requests for comment.

Stack ranking became popular in the 1980s after it was embraced by General Electric. Its adoption has waned, though several tech companies continue to use it. Google and Twitter relied on stack ranking to decide who to let go in their recent rounds of layoffs, according to published reports.

Birmingham says that the system can cause anxiety and competition, which can kill team cohesion, and that arbitrary lower ratings adversely affect compensation and promotion potential. These systems can also suggest that a manager is ineffective, he says. “It implies that as managers, we basically have not done our job to hire them and train them appropriately or terminate them if they really aren’t working out.”

Birmingham says he is not opposed to ranking systems but doesn’t think they’re necessary. “I feel like the conversation about how to improve your career, what the expectations are for your job and what it will take to get to the next level are all things you can do without a rating,” he says.

Measurements Matter

Grant Pruitt, president and co-founder of Whitebox Real Estate, does not give any type of rating in his performance reviews, though he believes in using data to track his employees’ performance. “What isn’t measured can’t be managed,” says Pruitt, whose company has about 20 employees in several offices across Texas. 

At the beginning of the year, Whitebox employees set goals with their managers. Discussions are held about what benchmarks are reasonable, and these targets can be changed if there is a meaningful shift in business conditions. Team leaders hold weekly department meetings with their direct reports to discuss what’s happening and track progress. Managers hold quarterly private reviews with individuals to dig deeper into whether they’re meeting their goals and if not, why.

“Was it an achievable goal? Realistic? If it was, then what do we need to do to make sure we don’t miss it the next time?” Pruitt says. Whitebox switched to quarterly reviews about four years ago to address problems earlier and avoid having issues fester, Pruitt adds.

It’s easier to set goals for people in sales than for those in other departments, Pruitt concedes. However, he adds that executives need to brainstorm about targets they can use for other roles. For example, administrative employees can be rated on how quickly and efficiently they handle requests.

Pruitt maintains that the goal system makes it easier to respond when an employee disagrees with their manager about their performance review because there are quantitative measures to examine. The data also helps eliminate any unconscious bias a manager may have and helps ensure that a leader isn’t just giving an employee a good rating because they work out at the same gym or their children go to school together.

“I think that’s really where the numbers and the data are important,” Pruitt says. “The data doesn’t know whose kids play on the same sports team.”

Whitebox employees are also judged on how well they embrace the company’s core values, such as integrity, tenacity and coachability. Some of those values may require more-subjective judgments that can be more important than hitting quantifiable goals. 

Pruitt admits that there were occasions when he looked the other way with a few individuals who were “hitting it out of the park,” even though he believed they lacked integrity. But eventually, he had to let them go and the company lost money.

“They really came back to bite me,” Pruitt says.

Screen Shot 2023-03-15 at 84352 AM.png

Grades Are Good

Diane Dooley, CHRO of Iselin, N.J.-based World Insurance Associates LLC, also believes establishing quantitative methods to gauge employees’ performance is essential. “We are living in a world of data analytics,” she says. The broker’s roughly 2,000 employees are rated on a scale of 1 to 5.

World Insurance has taken numerous steps to remove bias from reviews. For example, last year the company conducted unconscious-bias training to help managers separate personal feelings from performance reviews. And all people managers convene to go over the reviews they’ve conducted. Dooley says that process gives everyone a chance to discuss why an employee was given a certain rank and to question some decisions. “We want to make sure we’re using the same standards,” she explains.

Currently, World Insurance conducts reviews only once a year because it has been on an acquisition binge and there hasn’t been time to institute a more frequent schedule. That will change eventually, says Dooley, who adds that she wants to introduce department grids that show how an employee’s rank compares to others’ on the team. 

“It’s just a tool that helps the department or the division understand where their people are and how we can help them collectively,” says Dooley, who has used the system at other companies. 

Dooley says she isn’t worried about World Insurance holding reviews only annually, because good managers regularly check in with their employees regardless of how frequently reviews are mandated.

Such conversations can easily fall through the cracks, however. “Managers want to manage the employees, but they get so caught up in the company’s KPIs [key performance indicators] and making sure that they’re doing everything that they need to do,” says Jennifer Currence, SHRM-SCP, CEO of WithIn Leadership, a leadership development and coaching firm in Tampa, Fla. “It’s hard to set aside the time.” 

WTW’s Jennings adds that managers sometimes avoid initiating conversations with employees who are not performing well. Such discussions are often difficult, and managers may not feel equipped to conduct them. 

“Having to address underperformers is hard work,” Jennings says. 

Additionally, experts say, coaching managers to engage in such sensitive discourse can be expensive and time-consuming.

Improve Your Performance Reviews

H ere’s how to make the review process more ­palatable for both managers and their direct reports:

  • Don’t limit conversations to once or twice per year. Every team is different, so leaders should decide what schedule is most appropriate for their departments. However, it’s important to deal with any problems as they arise; don’t let them fester.
  • Set performance goals and expectations at the beginning of the year so employees understand their responsibilities. This helps lend objectivity to the process by introducing measurable targets. However, the goals should be adjusted if there are major changes to the business or an employee’s circumstances. 
  • Explain how each employee’s position, as well as each department, fits into the company’s overall ­strategy. This will help employees understand why their job matters and why it’s important.
  • Simplify the process. There’s no need for a ­double-digit number of steps or numerous
  • questions that require long-winded answers. 
  • Consider a 360-degree approach. Input from employees’ colleagues or from other managers can help give a fuller picture of employees’ capabilities and contributions.
  • Eliminate proximity bias. You may not see some employees as often as others, especially if they work remotely, but that doesn’t mean they’re not working hard. 
  • End recency bias, which is basing a review on an employee’s most recent performance while ignoring earlier efforts. Don’t let recent mistakes overshadow the employee’s other impressive accomplishments.
  • Solicit feedback from employees. Reviews should be a two-way conversation, not a lecture.
  • Train managers to give advice calmly and helpfully. This is especially important when leaders must call out an employee’s subpar performance. 
  • Don’t discuss compensation during reviews. Employees are likely to be so focused on learning about a raise or bonus that they won’t pay much attention to anything else.

Increase Conversations

Finding the right formula for performance reviews is tricky. The company’s size, values, industry and age all play a role. Currence says businesses need to think about the frequency and purpose of these meetings. Some managers may have weekly discussions with their direct reports, but the conversations might center on status updates as opposed to performance. 

“We need to have more regular conversations,” Currence says. “There has to be a happy balance.”

San Jose, Calif.-based software maker Adobe Inc. was a pioneer when it eliminated annual reviews in 2012 after employees said assessments that look backward weren’t useful and managers lamented how time-consuming they were. Instead, Adobe introduced quarterly check-ins and did away with its numerical ratings system, even though the company is “data-driven,” according to Arden Madsen, senior director of talent management.

Screen Shot 2023-03-15 at 85749 AM.png

Adobe’s system has changed over the years as the company grew from about 11,000 employees in 2012 to around 28,000 today. In the beginning, employees were not asked a universal set of questions and the information gathered was not stored in a central place accessible to all. In 2020, Adobe instituted three or four questions that must be asked at each quarterly meeting, one of which is whether the employee has feedback for the manager. Other topics covered depend on the employee, their role and their goals.

Madsen says asking consistent questions and making reviews easily accessible are important, as internal mobility within the company has grown. 

Adobe, like many businesses, separates conversations about performance from discussions about raises and bonuses, even though they’re intertwined. 

“Money is so emotionally charged,” says WithIn Leadership’s Currence. “When we tie performance review conversations with money, we as human beings do not hear anything about performance. We only focus on the money.”    

Theresa Agovino is the workplace editor for SHRM.

Illustrations by Neil Jamieson.

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7 Power Skills That Are in Demand in 2024 and How You Can Help Your Employees Develop Them

As the world of work changes, so do your needs. Many organizations now realize that power skills are just as important as technical skills in the workplace. Learn what power skills are in demand and how you can help your employees hone them.

[Featured image] Two women in a meeting discussing a marketing plan

Your employees need to know how to do their jobs and have the knowledge, education, and experience to back them up. But what many companies have traditionally overlooked are power skills. These differ from software literacy, copywriting, or speaking a foreign language. They are the human skills that help employees better interact with others and the work environment.

Power skills may be more critical now than ever, and helping your employees develop and sharpen them can benefit your organization. According to a 2022 Pearson study, power skills are more desirable than technical skills and will be through at least 2026 [ 1 ]. Look at some of the in-demand power skills in 2024 and how you can help your employees develop them.

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What are power skills?

Power skills is simply another name for workplace skills or human skills. It's considered a modern rebranding of these skills to prove their importance in the workplace. In the past, organizations focused more on whether a candidate could, for example, operate a computer rather than whether they could communicate well. Still, business leaders discovered in recent years that both soft and technical skills are important for efficiently running their businesses.

Human skills vs. technical skills

Technical skills are the skills you can learn through training and education that apply directly to your job. Human skills have more to do with how you interact with your environment and the people in it, and they typically apply to almost any job. These human skills can take more time and practice than technical skills. While you can take a course and learn how to use a particular type of software, you likely need some real-world experience to learn how to solve problems or make good decisions. These are skills you develop over time through various experiences.

Examples of human (power) skills include:

Critical thinking

Time management

Communication

Decision-making

Examples of technical skills include:

Video production

Copywriting 

Project management 

Foreign languages 

Graphic design

Data entry 

Search engine optimization (SEO)

Read more: Hard Skills vs. Soft Skills: What’s the Difference?

How employees with strong power skills benefit an organization

Strong power skills can help an employee's career, but your organization also benefits when employees possess these skills. These benefits might include:

Discovering untapped talent: You may find that an entry-level employee has a knack for customer service and sales. Helping them hone these skills can lead to your organization hiring from within when the need for a new salesperson arises.

Creating an agile workforce:  When your employees have strong power skills, like problem-solving, they adapt quicker to the unknown—whether it's new technology or a worldwide pandemic that disrupts everything.

Improving company culture: Improving power skills can lead to an inclusive workplace that often leads to a more positive and inviting company culture.

Increasing productivity: Employees with strong power skills often work faster and are more productive overall.

Improving employee retention: When you offer training and other opportunities to help employees improve their skills, it shows them that you value them. This can lead to greater retention numbers.

Attracting high-quality talent: When your company culture is positive and your employees are happy, your reputation as a desirable workplace grows. This can help you attract new high-quality talent when necessary.

Creating greater customer satisfaction: Employees with strong power skills can quickly adapt to customers' changing needs. This often leads to higher customer satisfaction rates.

Creating successful managers and leaders:  Good managers and leaders often have strong power skills. Honing these skills can help you develop future leaders that you can hire from within your workforce.

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7 power skills that are in demand in 2024

Although any combination of power skills can help improve your employees' jobs, some are considered critical, particularly in 2024. Here are seven that you may want to focus on with your employees in 2024 and for the foreseeable future:

1. Communication

Every job requires communication on some level, including speaking, active listening, body language, observing, video calling, emailing, creating documents, and much more. With globalization and remote work on the rise, knowing how to communicate well, both in person and virtually, with people worldwide is more important than ever. An employee who can clearly express their thoughts and ideas is vital to your organization, as is an employee who listens well and understands what other people have to say.

2. Customer service

The Pearson Study identifies customer service as one of the world's top five desired power skills [ 1 ]. A report from Salesforce revealed that 90 percent of customers say that the service they receive is just as important as the product they buy [ 2 ].  For this reason, many organizations report that a need for employees with strong customer service skills is on the rise in 2024. When a customer has a pleasant experience with any member of your staff, it can help build loyalty and may generate new customers through word of mouth and online reviews.

3. Leadership

While not every employee aspires to a management or leadership position, taking on a leadership role is vital across all departments and levels of your workforce, making it a critical power skill for modern times and a desirable skill for younger workers. Having leaders scattered throughout your team can inspire other members, spark creativity and innovation, and help you stay competitive.

4. Decision-making

Every organization needs employees who make good decisions, and in some cases, you need employees who make quick decisions on the spot. Helping employees build confidence in all areas of their job can lead them to make better decisions. The more employees who can make decisions on their own in a timely manner, the more efficient and competitive your company can be.

5. Problem-solving and critical thinking

Being a good problem solver usually means knowing how to identify a problem and going through a series of steps to develop a solution. From entry-level employees up to your executives, those who can solve problems independently often become more critical thinkers, leading to better overall job performance. Being a critical thinker lets your employees better understand a problem and apply logic and reason.

Read more: 7 Problem-Solving Skills That Can Help You Be a More Successful Manager

6. Collaboration and teamwork

With many employees working remotely or keeping hybrid schedules, applying collaboration and teamwork skills can be tricky, but they're more important than ever. Creating teams of individuals with unique backgrounds may help your organization solve problems faster, increase innovation, and become more efficient.

7. Emotional intelligence

As a positive work-life balance becomes increasingly important for employees worldwide, so does the need for emotional intelligence. This has to do with how a person controls their emotions. According to Mental Health America, emotional intelligence involves five key areas [ 3 ]:

Self-awareness

Self-regulation

Motivation 

Social skills

Mastering emotional intelligence means recognizing that you cannot control how other people act but can control how you react. This can help your employees with various issues, ranging from stress management to working with angry customers.

Ways to help employees improve their power skills

Many organizations have taken steps to help foster and grow their employees' power skills through upskilling programs or creating job-embedded opportunities to develop and refine these skills. For example, if you want to help your staff build teamwork and communication skills, try team-building exercises or group goal-setting activities. You also can assign employees to new projects they may be unfamiliar with so they can practice skills like problem-solving and decision-making.

Make an investment in your employees and the future of your company with one or more of these other upskilling ideas:

Encourage coaching or mentoring.  

Create eLearning opportunities that employees can participate in on their own time. 

Offer lectures and talks from experts.

Provide classroom-style training.

Reward employees who demonstrate strong power skills in certain situations. 

Create a culture of lifelong learning.

Spotlight employees with strong power skills.

Provide opportunities in which employees can practice specific power skills.

Create professional development plans that center around certain skills.

Keep the lines of communication and feedback open from management and employees.

An excellent way for employees to develop their power skills is to take self-guided online courses that allow them to practice and learn on their schedules. Respected businesses and educational institutions worldwide offer several options on Coursera. Focus on in-demand power skills with courses like Improving Communication Skills offered by the University of Pennsylvania, Introduction to Customer Service offered by CVS Health, and Principles of Management by Johns Hopkins University.  

Let's talk about making talent your advantage

Connect with our team to learn how you can prepare your business for rapid change.

Article sources

Pearson. " New Pearson Study Identifies Human Skills as the ‘Power Skills’ Most in Demand in World’s Major Job Markets " Accessed October 23, 2023.

Salesforce. " Salesforce Report: Nearly 90% of Buyers Say Experience a Company Provides Matters as Much as Products or Services " Accessed October 23, 2023.

Mental Health America. " What is Emotional Intelligence and How Does it Apply to the Workplace? " Accessed October 23, 2023.

This content has been made available for informational purposes only. Learners are advised to conduct additional research to ensure that courses and other credentials pursued meet their personal, professional, and financial goals.

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