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2.05 Practical problems with equations


Turning words into numbers

Writing word problems as algebraic expressions

In algebra, the variables that we write represent unknown values. It is these unknown values that we usually try to find.

As we have seen previously, each part of an algebraic expression (number sentence) can be expressed using words and vice versa. (This is why we call it a number sentence!)

Any operation can be used in algebraic expressions - in fact, any combination of operations can be used. To be able to turn written expressions into algebra, we need to be aware of all the words that can be used to describe different operations.

  • For the operation "$+$+" we might say "more than", "add", "plus" or "increased by", as well as other such words.
  • For the operation "$\times$×" we might say "groups of", "multiply", "double" (for $\times2$×2), "triple" (for $\times3$×3) or "times".
  • Write down all the words you can think of for subtraction "$-$" and division "$\div$÷​".


We usually use fractions to represent division, instead of using the "$\div$÷​" operator. This helps to avoid confusion about the order of operations in our number sentences.


Worked example

question 1

Write an algebraic expression for the following:  A number (call it $n$n) minus four equals ten.

Think:  What operation, number or symbol does each word in the question represent?

Do: In this case, "minus" is the operation "$-$". So the sentence can be written using algebra as $n-4=10$n4=10.


Finding unknown values

Once we can write these worded problems as algebraic equations, we can solve them by finding the unknown values. Let's look at an example of this:


Worked examples

question 2

In the equation $n-4=10$n4=10, what is the value of $n$n?

Think: One way of writing this equation in words is "take four away from a number and the result is ten". What number would we need to start with for this to be true?

Do: If we take $4$4 away from a number to get $10$10, then the number we started with must have been $4$4 larger than $10$10. Since $10+4=14$10+4=14, we can see that $n=14$n=14.


Question 3

Two less than a number (call the number $n$n) is equal to eighteen.  Write this sentence using mathematical symbols; and then find the number.

Think:  To write this sentence using mathematical symbols, we need to consider that to determine what $2$2 less than a number would be, that number must be first in the subtraction sentence.

Do:  Write $n-2=18$n2=18, then solve for n

$n-2$n2 $=$= $18$18

Represents the given equation written as a number sentence.

$n-2+2$n2+2 $=$= $18+2$18+2

To isolate $n$n add $2$2 to both sides of the equation.

$n$n $=$= $20$20

So, the number $n$n is equal to $20$20.


The unknown number is $20$20.

Reflect:  It is important to realize that the operations subtraction and division are not commutative.  So, it is important to consider which should be written first when you are translating a verbal expression into a mathematical sentence. 


More complex examples

We have already started to look at how to turn written sentences into algebraic equations. Let's continue now by looking at some more complex examples, involving more than one operation.


  • Addition "$+$+" can be expressed by words such as "more than", "sum", "plus", "add" and "increased by".
  • Subtraction "$-$" can be expressed by words such as "less than", "difference", "minus", "subtract" and "decreased by".
  • Multiplication "$\times$×" can be expressed by words such as "groups of", "times", "product" and "multiply".
  • Division "$\div$÷​" can be expressed by words such as "quotient" and "divided by". We usually represent division using fractions instead of using the "$\div$÷​" operator.
  • Equality "$=$=" can be expressed by words such as "is", "equal to" and "the same as". A number sentence needs one of these symbols to be an equation!


Worked example

question 4

Write down an equation in simplest form to represent "$v$v is $5$5 less than $3$3 groups of $v$v". 

Think: What symbol, number or variable can we use to represent each part of the sentence?

Do: "$v$v is" means that $v$v will be on one side of the "$=$=" sign and everything else will be on the other side. "$3$3 groups of $v$v" means $3\times v$3×v, and "$5$5 less than" means we are going to subtract $5$5 from this amount (using the "$-$" operator). So we have $v=3\times v-5$v=3×v5, which we can write more simply as $v=3v-5$v=3v5.


The order of the numbers in the sentence is not necessarily the same as the order in the equation!
In the example above, "$5$5 less than" meant that $5$5 was to be subtracted from the following term "$3$3 groups of $v$v". So the equation was written as $v=3v-5$v=3v5.


Writing the world as algebra

Sometimes we will come across questions that ask us to find an unknown, but the information they give us is not enough to immediately make the solution obvious. 

It is up to us to put the pieces together to construct an equation which will make it easier to solve the problem and find the unknown. 

The key is to find out what information is important. We can do this by looking for terms such as summinus, or equals and most importantly, by finding out what the question is asking us to solve.

Let's have a look at some problems that can be more easily solved by forming an equation.

Worked example

Question 5

Sally and Eileen do some fundraising for the sporting team. Together they raised $\$600$$600. Sally raised $\$272$$272 more than Eileen and Eileen raised $\$p$$p.  Write an equation in terms of $p$p that represents the relationship between the different amounts, and solve for $p$p.  Then calculate how much Sally raised.

Think:  If EIleen raised $\$p$$p and Sally raised $\$272$$272 more than Eileen, then we can express the money that Sally raised as $\$(p+272)$$(p+272).

Do:  Write an equation to represent the given situation and solve it.  

$p+(p+272)$p+(p+272)  $=$= $600$600

This equation shows the sum raising $$600$600 

$2p+272$2p+272 $=$= $600$600

Combine like terms

$2p+272-272$2p+272272 $=$= $600-272$600272

Subtract $272$272 from both sides to isolate $p$p

$2p$2p $=$= $328$328


$\frac{2p}{2}$2p2 $=$= $\frac{328}{2}$3282

Then divide both sides by $2$2 to solve for $p$p

$p$p $=$= $164$164



So if Eileen earned $\$164$$164, then Sally earned $\$164+\$272=\$436$$164+$272=$436.

Verifying that this makes sense, we know that the amounts that Eileen and Sally fundraised together should add up to $$600$600.  If EIleen earned $$164$164 and Sally earned $$436$436, then together they earned $\$164+\$436$$164+$436 = $$600$600.  So, our answer is reasonable.  

Reflect:  This problem described a situation in which Eileen and Sally together were able to raise $$600$600 in their fundraising efforts.  Because Sally's fundraising was defined in terms of how much money Eileen secured, we were able to use two different expressions with the same variable in them to represent the fundraising efforts of each of them.  It is very helpful to be able to express both unknowns as some expression of the same variable.  This way we will be able to apply what we know about solving linear equations to solve the resulting equation.


Practice questions

Question 6

When a number is added to both the numerator and denominator of $\frac{1}{5}$15, the result is $\frac{3}{7}$37.

  1. Let $n$n represent the number. Solve for $n$n.

Question 7

To manufacture sofas, the manufacturer has a fixed cost of $\$27600$$27600 plus a variable cost of $\$170$$170 per sofa. Find $n$n, the number of sofas that need to be produced so that the average cost per sofa is $\$290$$290.

Question 8

A commercial airplane has a total mass at take off of $51000$51000 kg. The luggage and fuel are $\frac{1}{3}$13 the mass of the unloaded plane, and the crew and passengers are $\frac{1}{4}$14 the mass of the fuel and luggage. Solve for $p$p, the mass of the unloaded plane.


The Overall Approach

So it seems that the following general steps can be taken to solve a problem through equation building:

1) Identify the unknown value you are trying to solve for and let it be represented by a variable (the question may already have given you the variable to use).

2) Identify any equations, concepts or formula that may be relevant to the problem. For example, if the question refers to averages, it may be useful to remember that $average=\frac{\text{sum of scores }}{\text{number of scores}}$average=sum of scores number of scores


The hardest step: Weaving it all together.

3) Try to relate the unknown to the other values given in the problem (either using words or mathematical symbols) to form an equation.

It may be useful to describe the relationship(s) you can see in words before writing them out as mathematical equations, or even to form smaller and more obvious mathematical expressions and see how these expressions relate to one another.

4) Solve the equation.

5) Check your answer to confirm that it makes sense in the original problem!



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