Solving Word Problems |

**S olving 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__

- If the problem is about something you can draw in a diagram, do it. A visual image of the situation always helps in math.
- If the problem involves several statements about the variable, make a table to organize the information.
- Always
**define the variable with a "let" statement**. - 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.
- Underline or hilite the important details of the question.
- Write an algebraic sentence (equation) that says exactly what the English sentence says.
- Solve the equation(s).
- 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

(or any other variable) to be. Then, we make an algebra statement that says exactly what the words do. Like this:*x*Let

*x*= the number4(

*x*+ 13) = 72Now, perform the indicated operations and solve for x.

4

*x*+ 52 = 724

*x*= 20*x*= 5The 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

) to represent*x***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**4 *x*4 *x*+ 10The 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:4

*x*+ 10 = 2(*x*+ 10)(Harry's age in 10 years) = twice (Sally's age in 10 years)

4

*x*+ 10 = 2*x*+ 202

*x*= 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..

**Example 3:**coin problemFrank 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

= the number of pennies.*p*so

*p***+ 5**= the number of nickelsand

*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*+ 5*p*+ 25 + 10*p*+ 100 = 36516

*p*+ 125 = 36516

*p*= 240*p*= 15There 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.

.

mixture problem__Example 4:__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

to be the*x***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*500 *x*2.25(100) + 1.90(400) = 500

*x*225 + 760 = 500

*x*985 = 500

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

*x*+ 130*x*= 600 250*x*= 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

= the length of the rug*x*So

–*x***3**= the widthThe 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)] = 422[2

*x*– 3] = 424

*x*– 6 = 424

*x*= 48*x*= 12*x*– 3 = 9The rug is 9 meters by 12 meters. The area is 108 square meters.

.

**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)*.

.

**Solutions**1) Let

*x*= the number of peas in the first bag.*x*+*x*+ 5 +*x*+ 11 +*x*+ 18 +*x*+ 26 = 2955

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

**47 peas**in the first bag..

2) Let

*x*= the number of subscriptions she sellsso 4

*x*+ 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 2*x*= 312.**Jan**scored**156**points and**Dan**won with**177**points..

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**2 *x*5(2 *x*)**dimes***x*10 *x***quarters**3 *x*/225(3 *x*/2)230¢ since # of nickels = 2

*x*then if

**q**is the # of quarters, 4q = 3(2*x*)so q = 3(2

*x*)/4 = 3*x*/210

*x*+ 10*x*+ 37.5*x*= 23057.5

*x*= 230 so*x*= 4There are

**4 dimes**,**8 nickels**and**6 quarters**..

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 + 2.20*x*260 = 1.2

*x*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**5 *x*5 *x*– 5**Mike***x**x*– 55

*x*– 5 = 10(*x*– 5) so 5*x*– 5 = 10*x*– 5045 = 5

*x*so*x*= 9**Mike is 9 years old and his dad is 45**.(

*all content**© MathRoom Learning Service; 2004 -*).