Applications of Algebra in Word Problems Study Guide
Introduction to Applications of Algebra in Word Problems
The human mind has never invented a labor-saving machine equal to algebra.
This lesson will cover some of the most common application questions involving algebra. Topics include consecutive integers, mixture problems, coin problems, age problems, and distance problems.
Consecutive Integer Problems
Consecutive integers are numbers in order one after the other. The integers 3, 4, and 5 are three consecutive integers. Because they are integers, there are no fractions or decimals.
- Consecutive odd integers are numbers such as 1, 3, 5, and –11, –9, –7.
- Consecutive even integers are numbers such as 4, 6, 8, and –20, –18, –16.
Since consecutive integers are always one apart from each other, use the expressions x, x + 1, x + 2, and so on, to define the variables in a consecutive integer problem.
Since consecutive odd integers are always two apart from each other, and consecutive even integers are always two apart as well, use the expressions x, x + 2, x + 4, and so on, to define the variables in a consecutive odd integer or consecutive even integer problem.
The sum of three consecutive integers is 60. What are the integers?
Read and understand the question. This question is looking for three consecutive integers that add to 60.
Make a plan. Use the expressions x, x + 1, and x + 2 to define the variables. Then, write an equation by adding these expressions and setting them equal to 60.
Carry out the plan. First, let x = the smallest integer, let x + 1 = the next integer, and let x + 2 = the greatest integer. Then, add these expressions together and set them equal to 60. The equation becomes x + x + 1 + x + 2 = 60. Combine like terms to get 3x + 3 = 60. Subtract 3 from each side of the equation.
- 3x + 3 – 3 = 60 – 3
- Simplify to get 3x = 57
- Divide each side by 3 to get the variable x alone.
- x = 19
- Because x = 19, then x + 1 = 20 and x + 2 = 21. The three integers are 19, 20, and 21.
- Check your answer. Check your answer by finding the sum of the three integers. The sum is 19 + 20 + 21 = 60, so this answer is checking.
Be sure to use your equation solving skills as outlined in Lesson 13 to help with solving word problems with equations. Refer back to the details in that lesson when necessary.
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