6.06 Solving contextual problems with algebra

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

Question 2

Question 3