Algebra/Word Problems

From Wikibooks, open books for an open world
Jump to: navigation, search
Algebra
Algebra
 ← Solving Equations Word Problems Interval Notation → 

Word Problems are mathematical problems that are delivered in ordinary words, instead of mathematical symbols. Part of the problem with dealing with word problems that they first need to be translated into mathematical equations, and then the equations need to be solved.

Working with Word Equations[edit]

Math is a kind of language, with its own "grammar"; to become "fluent" in math, you must understand its grammar. Some of the "grammatic" terms used in Math include:

Addition
(think positive!)
  • "in addition to"
  • "plus"
  • "added to"
  • "more than"
  • "and"
Subtraction
(always negative)
  • "less than"
  • "take away"
  • "minus"
Multiplication
  • "times"
  • "by"
  • "each"
Division
  • "divided by"
  • "per"
  • "out of"
Equals
  • "is"
  • "is equal to"
  • "is equivalent"
  • "their values are equal to each other"
  • almost anything with the words "equal" or "same"

This is not a definitive listing. While problems found in text books usually include these phrasings, you may encounter one where such words are implicit.

Steps To Solving Word Problems[edit]

The majority of word problems in algebra can be rewritten as equations, which you can then solve. There are some steps to take as you analyze a word problem.

  1. Read Carefully
  2. Define A Variable (What You Are Looking For)
  3. Make A Plan
  4. Write An Equation
  5. Solve
  6. Is This Answer Reasonable?

These six steps may seem either obvious, but serve as the foundation to solving word problems.

Exercises[edit]

  1. Steven bought 4 packs of baseball cards, each with the same number of cards inside. If Steven had 32 cards overall, how many cards were in each pack?
  2. 4 barrels of equal volume are filled with water. 3 of the barrels are completely filled and the fourth barrel is filled with only 8 gallons. If 53 gallons of water have been pumped into the barrels overall, what is the volume of each barrel?
  3. Jason pays each of his employees equally. If Jason had to pay each employee $50 and he gives them each a 10% bonus, how many employees does Jason have if he pays $440 overall?
  4. A car travels 20 km/h. faster than a second car. The first car covers 180 km. in the same time, the second car covers 135 km. What is the average speed of each car?
  5. Quinton and Thomas work at a company. Quinton's salary is $20 per hour, while Thomas has a salary of $40 per hour. Quinton, however, being in sales, can expect a $3 raise (per hour) every month, while Thomas can only expect a raise of 50 cents. How long until they are paid the same amount per hour?

Answers[edit]

1. Let x = Number of cards per pack.

  • 4x = 32 [(Number of packs)(Cards per pack) = (Total number of cards)]
  • x = 8

2. Let x = Volume of each barrel

  • 3x + 8 = 53 [(Number of filled barrels)(Volume of each barrel) + (Volume of water in fourth barrel) = (Overall volume of water)]
  • 3x = 45
  • x = 15

3. Let x = Number of employees.

  • x(50)(1.1) = 440 [(Number of employees)(Wage per employee)(percentage of bonus + 1) = (Overall payment)]
  • 55x = 440
  • x = 8

4. Let s1 = speed of the first car, and s2 = speed of second car. Let t = a unit of time (indeterminte).

  • s2 = (135km/h)/t, and s1 = (180 km/h)/t
  • Since s1 = s2 + 20km/h:
    • s2+20km/h = (180 km/h)/t
  • Substitute s2 = (135km/h)/t
    • (135km/h)/t + 20km/h = (180 km/h)/t
    • 20km/h = (45 km/h)/t
    • t = 45km/h / 20 km/h
    • t = 9/4
  • Substitute t:
    • s1 = (4/9) * 180 km/h = 80 km/h
    • s2 = (4/9) * 135 km/h = 60 km/h

5. Let x = the number of months until the salaries are equal and let y = the amount of money they earn

  • y = 3x + 20
  • y = .5x + 40
  • 3x + 20 = .5x + 40
  • 2.5x = 20
  • x = \frac{20}{2.5}
  • x = 8 months