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$$6.
If Adam sells $q$q chocolates, which expression can be used to find the amount of money he raises?
$\frac{6}{q}$6q
A$6q$6q
B$6-q$6−q
C$6+q$6+q
DWhich statement is correct?
It is possible to raise $\$13$$13 if Adam sells chocolates at $\$6$$6.
A$4$4 chocolates at $\$6$$6 each will cost $\$10$$10.
BIt is possible for Adam to raise only $\$1$$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.
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$5−b
C$5+b$5+b
DIf 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.
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$72−q
B$\frac{72}{q}$72q
C$q-72$q−72
DWhich statement is correct?
The more students that are in the class, the less pencils each student receives.
AThe more students that are in the class, the more pencils Judy gives out.
BIf there were $2$2 more students in the class, each student would receive $2$2 less pencils.
CThe less students that are in the class, the less pencils each student receives.
D