Word Problems in Algebra

Word problems allow us to apply algebra to real life situations. Many students find

them difficult, but if approached in a logical manner, they can be easily solved.

Steps for Solving Word Problems• FIND: Determine what you are asked to FIND… This means read the problem several

times until it is clear what question is being asked. You may need to filter out irrelevant information. Define a variable to represent the value you are looking for.

• FACTS: Gather the FACTS – What information is given in the problem and how does that information relate to the original question? Creating a table or a picture is often helpful. Is there a known formula that applies to this problem? For example, If the problem involves geometry, fractions or percents, what do you already know that may help? Pay attention to units of measure. If the problem asks for hours and you have data in minutes, you may need to do some conversion.

• FORMULA: Set up the equation based upon the facts gathered. You may be able to use standard formulas for geometry, interest, distance, etc. In that case we will SUBSTITUTE the known facts or values into the formula.

• SOLVE: Solve the Equation. Use the methods you have already learned to solve the equation.

• ANSWER: Does the solution answer the question asked? Does it make sense? Did the problem ask for more than one answer? Do the units of measure match up? Check your answer in the original equation.

Some Basic Examples

Three less than twice a number is 13. What is the number?

• FIND: we need to find the number, use the variable: x• FACTS: twice the number is 2x, three less than that is 2x -3• FORMULA: 2X – 3 = 13• SOLVE: 2x - 3 = 13

2x – 3 + 3 = 13 + 3 2x = 16

X = 8

• ANSWER: Does the answer 8 make sense? Twice 8 is equal to 16. Three less than 16 is 13. YES it answers the original problem and makes sense.

•add 3 to both sides (addition property of equality)

•divide both sides by 2 (division property of equality)

Consecutive Integer problems

Twice the sum of three consecutive odd integers is 150. Find the three numbers.• FIND: the three numbers; we’ll use the variable x for one of them. Each consecutive

odd integer is two more than the previous, so we will represent the 3 integers as : x, x+2, x+4

• FACTS: twice the sum = 150, so we need to represent the sum of the integers, and twice that value. Sum of integers will be x + x+2 + x + 4 and twice that value will be 2(x + x+2 + x + 4)

• FORMULA: 2(x + x +2 + x + 4) = 150• SOLVE: 2(x + x +2 + x + 4) = 150

2(3x + 6) = 150 6x + 12 = 1506x + 12 – 12 = 150 - 12 6x = 138

x = 23 • ANSWER: We have solved for x, but we need to supply three answers: x = 23, x + 2 =

25, x + 4 = 27. Check: the sum of the integers is 75; twice that is 150

•Combine like terms•Distribute•Subtract 12 from both sides

•Divide both sides by 6

Fixed and Variable Cost

Cab fare in Metropolis is $2.50, plus $2.00 per mile, with a surcharge of $1.00 for each passenger in addition to the first one. Three friends wish to take a cab to the theater. Each is willing to spend $7.00 toward the cab ride. They figure that $3.00 out of the total will be allocated for the tip. How far can they go on the money they have budgeted for the ride?

• FIND: the number of miles they can travel, we’ll represent that as x.• FACTS: 3 people have $7 each, for a total of $21. However $3 is allocated for tip, so that

leaves $18 for actual cab fare. Cost of cab is $2.50 + $2 per mile + $2 for the extra passengers. So fixed cost will be $4.50 and variable cost will be 2x ($2 for each mile)

• FORMULA: 2x + 4.5 = 18• SOLVE: 2x + 4.5 = 18

2x + 4.5 – 4.5 = 18 – 4.5 2x = 13.5 x = 6.75

• ANSWER: We used x to represent number of miles, so that means the three friends can go 6.75 miles. Does this make sense? It seems reasonable. Check your answer by substituting 6.75 into the original equation: (2)(6.75) + 4.5 = 18

13.5 + 4.5 = 18 18 = 18

•Subtract 4.5 from each side•Divide both sides by 2

Percent of Increase or Decrease

John bought a sweater at 40% off. He paid $29.97. What was the original price?

• FIND: original price, x• FACTS: discount is 40% of the original price or .40x; sale price = original

price – discount • FORMULA: x - .40x = 29.97• SOLVE: x - .40x = 29.97 .6 x = 29.97

x = 49.95• ANSWER: Does this answer make sense? The item was on sale, so the

original price should be more. This answer looks reasonable. Substitute it into the original equation. Does it check?

•Combine like terms•Divide both sides by .6