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4.04 Problem solving with equations


Translating a word problem into a mathematical equation is a skill we will use all through our mathematical studies. It is always helpful to identify keywords and know how they might translate to math. We have had some practice with this when writing equations, but now we want to put it all together and solve the problem.


Identifying keywords

For each part of the equation here are some keywords that might be helpful.

Variable: There is not necessarily keywords here, but this will be the unknown quantity we are looking for, we will often use $x$xor $n$n.

Equal sign: balanced, equivalent, gives, is, matches, same, and yields

Addition: add, both, combined, increase, join, more, plus, sum, together, total

Subtraction: decrease, difference, fewer, left over, less, minus, subtract, take away

Multiplication: by, double, groups of, multiply, of, per, product, times

Division: divide, evenly, half, quotient, shared, split

Once we have translated to a mathematical sentence, we can solve the problem.


When working to solve problems, always be sure to:

  • Read the question at least twice and find keywords
  • Be clear what your variable is representing, for example instead of saying that "$x$x is chickens" we could say "$x$x is the number of chickens" or "$x$x is the weight of each chicken"
  • Check your answer to see if it is reasonable and then check it by substituting into your equation


Worked examples

Question 1

The sum of a number and $24$24 is $35$35. Set-up an equation to find the number.

Think: We have an unknown, "the number", so let's call "the number" $x$x. We should note some keywords which can help us after we replace "a number" with $x$x.

Do: Using our keywords, we can translate to an equation and solve.

$x+24$x+24 $=$= $35$35

Translating to math

$x+24-24$x+2424 $=$= $35-24$3524

Subtract $24$24 from both sides

$x$x $=$= $11$11



Therefore, the unknown number is $11$11.

We should first check to see if our answer is reasonable and then check it. We can use lots of different strategies such as estimation to see if our answer makes sense. $25+10=35$25+10=35, so our answer does make sense. Using the exact values, $24+11=35$24+11=35, so our answer is correct.

Reflect: What other strategies could you use to solve this problem?

Question 2

The perimeter of a square is $56$56 cm. Use an equation to find the side length of the square. Then use this answer to find the area of the square.

Think: We are asked to set-up an equation, so we can start by declaring a variable. Let's let $x$x be the side length. Once solve for $x$x, we can find the area, $A=x^2$A=x2.


$P$P $=$= $4x$4x

The perimeter of a square is $4$4 times its side length

$56$56 $=$= $4x$4x

The perimeter is $56$56

$\frac{56}{4}$564 $=$= $\frac{4x}{4}$4x4

We need to divide both sides by $4$4

$14$14 $=$= $x$x



The square has side length of $14$14 cm.

The area of this square is then $A=14^2$A=142, or $A=196$A=196 cm2.

Reflect: If we were told the area was $56$56 cm2 and asked to find the perimeter, would that change anything?

Practice questions

Question 3

The sum of $8$8 and $12x$12x is equal to $92$92.

By constructing an equation, find the value of $x$x.

Question 4

Consider the following word statement.

The sum of a number and $7$7 is $17$17.

  1. Write the word statement as an equation.

    Use $x$x as the variable.

  2. Find the solution of the equation from the set $\left\{2,4,6,8,10\right\}${2,4,6,8,10}.



Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.


Solve word problems leading to equations of the form px q = r and p(x q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach.

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