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Middle Years

6.06 Solving contextual problems with algebra

Lesson

Algebraic equations can be used to interpret situations and find answers to questions in a wide range of contexts. The key to solving these problems is in constructing the algebraic expressions needed to represent the important details.

Worked examples

Example 1

Patrick sells bouquets made up of $15$15 flowers. If each flower in a bouquet has $n$n petals, how many petals are there per bouquet?

Think: If each flower has $n$n petals and there are $15$15 flowers, we can think of this as $15$15 groups of $n$n.

Do: $15$15 groups of $n$n given as an algebraic expression is $15\times n$15×n or $15n$15n. There are $15n$15n petals in total in Patrick's bouquets.

Reflect: In questions which state that for each $a$a there are $b$b, or there are $a$a per $b$b, multiplication or division is typically used. Telling the difference is usually all about understanding the context, as the next example shows.

Example 2

Alison has $72$72 balloons and she wants to share them equally between $m$m people. Construct an algebraic expression to find the number of balloons given to each person.

Think: When sharing $72$72 balloons among $m$m equal groups, we want to use division to express our answer.

Do: $72$72 divided into $m$m equal groups is written as $\frac{72}{m}$72m. Each person will have $\frac{72}{m}$72m balloons.

Reflect: When we split an amount of size $a$a, into smaller equal groups of size $b$b, we are performing a division and we therefore use the division notation, resulting in the expression $\frac{a}{b}$ab.

Example 3

Jack's parents count the number of videos he watches in his spare time. On the first day he watched $5$5 videos. On the second day he watched $9$9 videos. On the third day he watched $13$13 videos.

If this pattern continues, what day will he first watch more than $26$26 videos?

Think: We can use the variable $d$d to express the number of days since his parents started counting. Each time $d$d increases by $1$1, the number of videos he watches increases by $4$4. It can help to write it out using a table:

Day $1$1 $2$2 $3$3 ... $d$d
Videos $5$5 $9$9 $13$13 ... $\editable{}$

Do: We can find the number of videos he watches on a day by multiplying the day number by $4$4, then adding $1$1. In other words, he watches $4d+1$4d+1 videos each day.

If we replace the $d$d with $6$6 we find that he watches $4\times6+1=25$4×6+1=25 videos on the sixth day, which is just too small to be an answer to our question. If we replace $d$d with $7$7 we find that he watches $4\times7+1=29$4×7+1=29 videos on the seventh day, the first day he watches more than $26$26 videos.

Reflect: These sorts of models can quickly break down. In a few months (after, say, $100$100 days), we all hope Jack isn't watching more than $400$400 videos every day! As our algebraic skills develop we will be able to make better and better models for situations like these.

Practice questions

Question 1

Adam sells chocolates to raise money for charity. Each chocolate costs $\$6$$6.

  1. If Adam sells $q$q chocolates, which expression can be used to find the amount of money he raises?

    $\frac{6}{q}$6q

    A

    $6q$6q

    B

    $6-q$6q

    C

    $6+q$6+q

    D
  2. Which statement is correct?

    It is possible to raise $\$13$$13 if Adam sells chocolates at $\$6$$6.

    A

    $4$4 chocolates at $\$6$$6 each will cost $\$10$$10.

    B

    It is possible for Adam to raise only $\$1$$1 after sales.

    C

    $5$5 chocolates at $\$6$$6 each will cost $\$30$$30.

    D

Question 2

Robert visits a carnival that costs $\$5$$5 to enter, and each ride costs $\$1$$1 per person.

  1. If Robert decides to go on $b$b rides, which expression can we use to find the total amount he spends at the carnival?

    $5b$5b

    A

    $5\div b$5÷​b

    B

    $5-b$5b

    C

    $5+b$5+b

    D
  2. If Robert goes on $5$5 rides, how much does he spend in total?

Question 3

Judy has $72$72 pencils, which she shares evenly among the students in her class.

  1. If there are $q$q students in her class, which expression can be used to find the number of pencils given to each student?

    $\frac{q}{72}$q72

    A

    $72-q$72q

    B

    $\frac{72}{q}$72q

    C

    $q-72$q72

    D
  2. Which statement is correct?

    The more students that are in the class, the less pencils each student receives.

    A

    The more students that are in the class, the more pencils Judy gives out.

    B

    If there were $2$2 more students in the class, each student would receive $2$2 less pencils.

    C

    The less students that are in the class, the less pencils each student receives.

    D

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