There are numerous types of word problems that involve linear equations. Some of these types are so popular, they are categorized under specific headings.We will take a look at a few examples of these more popular types of questions. Just as there are numerous types of questions, there are also numerous ways to set-up and solve each of these questions.
If your teacher asks for a particular type of solution, be sure to prepare the question using that method.

 All algebraic solutions to word problems have a few concepts in common: • translate the words into algebraic symbols with variables (or variable expressions). • tell everyone what your variables (or expressions) represent. • clearly state the linear equation you will be using. • show your work. • read carefully to be sure you are giving the desired answer. • clearly state your answer. • check your answer to be sure your results are correct.

 Type: NUMBER PROBLEM The larger of two numbers is four times the smaller. If the larger number is increased by 12, the result is 5 times the smaller number. Find the numbers. Notice that this question asks you to find BOTH numbers.   Check: 48 is four times 12. 48 + 12 = 5 (12) 60 = 60 True Let x = the smaller number 4x = the larger number 4x + 12 = the larger number increased by 12 5x = five times the smaller number Create the equation: 4x + 12 = 5x Solve the equation: The numbers are x = 12 and 4x = 48.

Type: PERCENT PROBLEM
A new design of sneakers is on sale for \$58.80 after being marked down 40%. What was the original price of the sneakers?
You worked with percent word problems in middle school, so this should look familiar.
 Remember that 40% is 0.40. Check: Marked down 40% means cost is 60%. 60% of \$98 = \$58.80.
Let p = the original price
p - 0.40p = the sale price (price minus a 40% mark down)

Create the equation: p - 0.40p = 58.80

Solve the equation:

The original price was \$98.00.

 Type: AGE PROBLEM Kyle is 6 years older than Melissa. Nine years ago he was twice Melissa's age. How old is Kyle now? Notice that this question asks specifically for Kyle's age. Let a = Melissa's age now (the smaller age) a + 6 = Kyle's age now a - 9 = Melissa's age nine years ago (a + 6) - 9 = Kyle's age nine years ago Create the equation: (a + 6) - 9 = 2(a - 9) Solve the equation: Kyle's age is a + 6 = 15 + 6 = 21. Check: Now, Kyle is 21 and Melissa is 15. Nine years ago, Kyle was12 and Melissa was 6. Kyle was twice her age.

 Type: DISTANCE PROBLEM A snowshoe hiker, Aaron, and a cross country skier, Kevin, are racing to see who can arrive at a point 200 feet away. Aaron travels at a constant speed of 30 feet per minute for the entire distance. Confident Kevin travels at a constant speed of 40 feet per minute for the first 30 feet, stops to throw snowballs for 3 minutes, and then continues at 35 feet per minute for the remainder of the distance. Who won the race? A picture is always a good idea with distance problems. Get a visual idea of what is happening. Distance = Rate x Time Think of "rate" as speed. Let's investigate each boy's race. Let t = Aaron's time in minutes. Aaron's equation: 200 = 30t Aaron: 6.667 minutes Kevin's time, m, occurs in sections. First 30 feet: 30 = 40m Last 170 feet: 170 = 35m Don't forget he stops for 3 minutes. m = 3           Add up Kevin's times: 0.75 + 4.857 + 3 = 8.607 minutes Aaron wins the race!

 Type: CONSECUTIVE INTEGER PROBLEM Find three consecutive even integers whose sum is 18. "Consecutive" means "one after the other". Consecutive integers (such as 1, 2, 3) are represented as x, x+1, x+2. Consecutive even integers (such as 2, 4, 6) are represented as x, x+2, x+4. Consecutive odd integers (such as 3, 5, 7) are represented as x, x+2, x+4. Check: 4 + (4 + 2) + (4 + 4) = 18 4 + 6 + 8 = 18 18 = 18 Let n = the first even number n + 2 = the second even number n + 4 = the third even number Create the equation: n + (n + 2) + (n + 4) = 18 Solve the equation: The numbers are 4, 6, and 8.

 Type: MONEY PROBLEM Mario has 20 coins in quarters and nickels. He has a total of \$2.00. How many quarters and nickels does he have? When working with money be careful to express the money all in cents (50¢) or all in dollars (\$0.50). This example is expressed all in dollars. Check: 0.25q + 0.05(20 - q) = 2.00 0.25(5) + 0.05(20 - 5) = 2.00 1.25 + 1 - 0.25 = 2.00 2.00 = 2.00 Let q = number of quarters 20 - q = number of nickels 0.25q = money from quarters 0.05(20 - q) = money from nickels Create the equation: 0.25q + 0.05(20 - q) = 2.00 Solve the equation: There are 5 quarters and 15 nickels.

Type: MIXTURE PROBLEM (dry mixture)
Sunflower seeds sell for \$0.50 a pound and cracked corn sells for \$0.30 a pound. How many pounds of each will be needed to create a 40 pound mixture of birdseed selling at \$16.40?
 Mixture problems are often solved in a chart (or table). It can, however, be difficult to remember how to set up the chart, and what to put where in the chart.
A simple diagram may help:

Let s = pounds of sunflower seeds
0.50s = cost of sunflower seeds in mix
40 - s = pounds of corn
0.30(40 - s) = cost of corn in mix

Create the equation:
0.50s + 0.30(40 - s) = 16.40

Solve the equation:

The 40 pound mix is 22 pounds of sunflower seeds and 18 pounds of corn.

 Type: MIXTURE PERCENT PROBLEM (liquid mixture) Your chemistry lab requires a 20% saline solution. The only solutions available for your use are 10% saline and 40% saline solutions. You decide to mix the two solutions to create the 20% solution you need. If you need 8 liters of 20% saline solution, how many liters of the 10% saline and 40% saline solutions should you mix? Notice how we used the physical amount of solutions we needed (in liters) and the amount of pure saline it would give to the mix (% times the liters). See diagram below. Let x = liters of 10% solution needed 0.10x = amount of saline in mix (10%) 8 - x = liters of 40% solution needed 0.40(8 - x) = amount of saline in mix (40%) 0.20(8) = amount of saline we need                   (8 liters of 20% solution) Create the equation: 0.10x + 0.40(8 - x) = 0.20(8) Solve the equation: ANSWER: