Solving Word Problems

Solving Word Problems with Linear Equations

As with all things in math, the best approach to solving word problems is a disciplined, orderly one. The more vigilant we are about setting up the problem, the greater our chances of success.

Hints for solving word problems

  1. If the problem is about something you can draw in a diagram, do it. A visual image of the situation always helps in math.
  2. If the problem involves several statements about the variable, make a table to organize the information.
  3. Always define the variable with a "let" statement.
  4. If the problem involves a series of operations in order to reach the final answer, make a plan of attack -- a list of the things to do in order to complete the question.
  5. Underline or hilite the important details of the question.
  6. Write an algebraic sentence (equation) that says exactly what the English sentence says.
  7. Solve the equation(s).
  8. Write an answer to the question with words -- not just x = 3.5

    Example 1:

    If the sum of a number and 13 is multiplied by 4, the product is 72. Find the number.

    What are we looking for? -- the number -- so, that's what we define x (or any other variable) to be. Then, we make an algebra statement that says exactly what the words do. Like this:

    Let x = the number

    4(x + 13) = 72

    Now, perform the indicated operations and solve for x.

    4x + 52 = 72

    4x = 20

    x = 5

    The number is 5.

    Example 2: Now one best solved with a table:

    Harry is now four times as old as Sally. Ten years from now, his age will be only twice her age. How old is each now?

    If we define a variable (say x ) to represent Sally's age now, we can use it to express Harry's age now. Then it's simple to express both their ages 10 years from now.

    person age now age in 10 years
    Sally x x + 10
    Harry 4x 4x + 10

    The question now says: Ten years from now, Harry's age will be only twice her age, so this is what our equation must say, like this:

    4x + 10 = 2(x + 10)

    (Harry's age in 10 years) = twice (Sally's age in 10 years)

    4x + 10 = 2x + 20

    2x = 10

    x = 5

    Sally is 5 years old and Harry is 20 years old.

    Other types of problems where a table helps are coin, mixture, and distance-rate-time questions.

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    Example 3: coin problem

    Frank has $3.65 in his piggy bank made up of pennies, nickels and dimes. The number of nickels is 5 more than the number of pennies; the number of dimes is 10 more than the number of pennies. How many of each coin does Frank have?

    Coins in this question have 2 properties -- the number of them -- and their value -- so we must consider both. We'll use a column for the number of coins and one for the value. We will define the variable p to represent the number of pennies.

    Let p = the number of pennies.

    so p + 5 = the number of nickels

    and p + 10 = the number of dimes.

    The monetary value of each nickel is 5 ¢, so we must multiply the # of nickels by 5 to get their value. Similarly, each dime is worth 10 ¢, so the value of (p + 10) dimes is 10(p + 10) as shown in the value column of the table.

    coin number value
    pennies p p ¢
    nickels p + 5 5(p + 5) ¢
    dimes p + 10 10(p + 10) ¢

    Now, with an equation, we can say that the sum of the value column = the total $3.65 or 365 ¢.

    p + 5(p + 5) + 10(p + 10) = 365

    p + 5p + 25 + 10p + 100 = 365

    16p + 125 = 365

    16p = 240

    p = 15

    There are 15 pennies, 20 nickels and 25 dimes in Frank's piggy bank.

    To check: see if 15 + 20(5) + 25(10) = 365 and it does.

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    Example 4: mixture problem

    A grocer has 100 kg of almonds he sells at $2.25/kg and 400 kg of cashews at $1.90/kg that he wishes to sell as a mixture. For what price per kilogram should the mixture sell?

    The key here is that the grocer must make as much from the mixture as he would selling the almonds and cashews separately. We define x to be the price per kilogram of the mixture.

    The table below reflects all the information we have.

    item amount unit price($) value($)
    almonds 100kg 2.25 2.25(100)
    cashews 400kg 1.90 1.90(400)
    mixture 500kg x 500x
    Again, we write an equation which says that the sum of the 1st 2 values in column 4 must equal the 3rd value, like this:

    2.25(100) + 1.90(400) = 500x

    225 + 760 = 500x

    985 = 500x

    x = $1.97

    The mixture should sell for $1.97/kg.

    distance, rate, time problem:

    Montreal and Toronto are 600 km apart. Ryan starts from Toronto towards Montreal, travelling at 120 km/hr at exactly the same time that Norman,travelling at 130 km/hr., leaves Montreal for Toronto .

    a) How long will it take them to meet?

    b) How far from Toronto will they meet?

    d = rt The key here is that they will have covered the entire 600 km separating the cities when they meet. So, the sum of their distances must = 600 km.

    person dist. rate time
    Ryan 120x 120k/h x
    Norman 130x 130k/h x

    120x + 130x = 600 250x = 600

    x = 2.4 hours. d = 2.4 (120) = 288 km

    They meet after 2.4 hours at a distance of 288 km from Toronto.

    perimeter, area problem

    A rug's width is 3 m less than its length. The perimeter is 42 m

    a) Find the dimensions of the rug. b) Find the area of the rug.

    A diagram of the rug will help with this question.

    Let x = the length of the rug

    So x – 3 = the width

    The perimeter of a rectangular figure is found by doubling the sum of the side measures. This is what we write in our equation.

    2[x + (x – 3)] = 42

    2[2x – 3] = 42

    4x – 6 = 42

    4x = 48

    x = 12

    x – 3 = 9

    The rug is 9 meters by 12 meters. The area is 108 square meters.

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    Practice

    Solve these word problems with a table and/or a diagram.

    1) We have 5 bags of peas. The second bag has 5 more peas than the first. The third bag has 6 more peas than the second. The fourth bag has 7 more peas than the third. And the fifth bag contains 8 more peas than the fourth. There are 295 peas in all. How many are in the first bag?

    (solution)

    2) Jennifer sells magazine subscriptions. She is paid $20 per day and $4 per subscription sold. If she earned $232 yesterday, how many subscriptions did she sell? (solution)

    3) At the end of a game of SCRABBLE, Jan and Dan had a total of 333 points. If Dan beat Jan by 21 points, how many points did each of them score in the game? (solution)

    4) I reached into the pocket of my winter coat and found $2.30 made up of nickels, dimes and quarters. The number of nickels was twice the number of dimes and 4 times the number of quarters was equal to 3 times the number of nickels. How many of each coin did I find?

    (solution)

    5) A grocer has 200 kg of shelled pistachios which sell for $3.50/kg that he can't seem to sell. He decides to combine them with peanuts that sell for $1.00/kg to make a mixture that will sell for $2.20/kg. How many kilograms of peanuts should he add to the pistachios in order to break even?

    (solution)

    6) Mike's father's age is now 5 times Mike's age. Five years ago, Mike's father was 10 times as old as he was then. How old are they now? (solution)

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    Solutions

    1) Let x = the number of peas in the first bag.

    x + x + 5 + x + 11 + x + 18 + x + 26 = 295

    5x + 60 = 295 which means that 5x = 235 so x = 47

    There are 47 peas in the first bag.

    .

    2) Let x = the number of subscriptions she sells

    so 4x + 20 = 232. We transpose the 20 and divide by 4.

    She sold 53 magazine subscriptions yesterday.

    .

    3) Let x = Jan's score

    x + x + 21 = 333 which means 2x = 312.

    Jan scored 156 points and Dan won with 177 points.

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    4) I reached into the pocket of my winter coat and found $2.30 made up of nickels, dimes and quarters. The number of nickels was twice the number of dimes and 4 times the number of quarters was equal to 3 times the number of nickels. How many of each coin did I find?

    coin number value
    nickels 2x 5(2x)
    dimes x 10x
    quarters 3x/2 25(3x/2)
        230¢

    since # of nickels = 2x

    then if q is the # of quarters, 4q = 3(2x)

    so q = 3(2x)/4 = 3x/2

    10x + 10x + 37.5x = 230

    57.5x = 230 so x = 4

    There are 4 dimes, 8 nickels and 6 quarters.

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    5) A grocer has 200 kg of shelled pistachios which sell for $3.50/kg that he can't seem to sell. He decides to mix them with peanuts that sell for $1.00/kg to make a mixture that will sell for $2.20/kg. How many kilograms of peanuts should he add in order to break even?

    item amount unit price $ value $
    pistachios 200kg 3.50 3.50(200)
    peanuts x 1.00 x
    mixture 200 + x 2.20 2.20(200 + x)

    3.50(200) + x = 2.20(200 + x)

    700 + x = 440 + 2.20x

    260 = 1.2x so,

    The grocer will use 216.67 kg. of peanuts in the mixture.

    .

    6) Mike's father's age is now 5 times Mike's age. Five years ago, Mike's father was 10 times as old as he was then. How old are they now?

    person age now age 5 yrs. ago
    Father 5x 5x – 5
    Mike x x – 5

    5x – 5 = 10(x – 5) so 5x – 5 = 10x – 50

    45 = 5x so x = 9

    Mike is 9 years old and his dad is 45.

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