ALGEBRAIC OPERATIONS

QUESTION 1 [4 marks] — Criterion A Medium

Write an algebraic expression for each of the following:

a. 5 more than twice a number $x$

[1]

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$$2x + 5$$

b. The product of $a$ and $b$, decreased by 7

[1]

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$$ab - 7$$

c. Half of the sum of $p$ and $q$

[1]

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$$\frac{p+q}{2}$$

d. 3 less than the square of $n$

[1]

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$$n^2 - 3$$

QUESTION 2 [3 marks] — Criterion A Medium

A number $y$ is tripled, then 8 is subtracted, and the whole result is then halved.

a. Write an algebraic expression for the final result, using brackets where needed.

[2]

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$$\frac{3y-8}{2}$$

b. Evaluate your expression when $y=10$.

[1]

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$$\frac{3(10)-8}{2}=\frac{22}{2}=11$$

QUESTION 3 [6 marks] — Criterion B Medium

Investigate how to represent 3 consecutive integers algebraically, and the pattern in their sum.

a. If the first of 3 consecutive integers is $n$, write expressions for the second and third integers.

[2]

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Second: $n+1$. Third: $n+2$.

b. Write and simplify an expression for the sum of the 3 consecutive integers.

[2]

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$$n+(n+1)+(n+2) = 3n+3$$

c. Test your formula with $n=5,6,7$ (sum should be 18) and $n=10,11,12$ (sum should be 33). Does it work? What do you notice about $3n+3$?

[2]

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For $n=5$: $3(5)+3=18$ ?. For $n=10$: $3(10)+3=33$ ?. Notice $3n+3=3(n+1)$ — the sum of 3 consecutive integers is always 3 times the MIDDLE integer.

QUESTION 4 [4 marks] — Criterion C Medium

A student writes '2x' to mean 'the digit 2 followed by the digit x', like writing the number 25 for $x=5$.

a. Explain what '2x' actually means in algebra, and evaluate it correctly for $x=5$.

[2]

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In algebra, placing a number directly next to a variable means MULTIPLICATION — '2x' means '2 times x'. For $x=5$: $2x=2\times5=10$, not 25.

b. Explain why the student's interpretation would cause serious problems if $x$ could be a decimal or negative number (e.g. $x=0.5$ or $x=-3$).

[2]

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The 'digit-joining' interpretation only makes sense for single positive whole-number digits — it breaks down completely for $x=0.5$ (there's no digit '0.5') or $x=-3$ (you can't 'write a negative digit' next to the 2). The multiplication interpretation works consistently for ANY value of $x$.

QUESTION 5 [5 marks] — Criterion D Medium

A taxi company charges a \$4.50 flat fee plus \$2.20 per kilometre travelled.

a. Write an algebraic expression for the total cost $C$ of a trip of $k$ kilometres.

[2]

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$$C = 4.50 + 2.20k$$

b. Use your expression to find the cost of an 8 km trip.

[1]

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$$C=4.50+2.20(8)=4.50+17.60=\$22.10$$

c. A customer has exactly \$30. Find the maximum whole number of kilometres they can travel.

[2]

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$4.50+2.20k \le 30 \Rightarrow 2.20k\le25.5 \Rightarrow k\le11.59$, so the maximum whole number of km is 11.

QUESTION 6 [4 marks] — Criterion A Hard

Write an algebraic expression for each, using $x$ for the unknown number:

a. 7 less than three times the square of $x$

[1]

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$$3x^2-7$$

b. The sum of $x$ and 4, all divided by twice $x$

[1]

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$$\frac{x+4}{2x}$$

c. 5 more than $x$, then the whole result tripled, then 2 subtracted

[2]

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$$3(x+5)-2$$

QUESTION 7 [4 marks] — Criterion A Hard

A gym membership costs a \$60 joining fee plus \$35 per month. A student membership gets a 20\% discount on the MONTHLY fee only (not the joining fee).

a. Write an expression for the total cost of a standard membership for $m$ months.

[2]

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$$60+35m$$

b. Write a SEPARATE, simplified expression for the total cost of a STUDENT membership for $m$ months.

[2]

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Monthly student rate: $35\times0.8=28$. $$60+28m$$

QUESTION 8 [6 marks] — Criterion B Hard

Investigate how to represent 3 consecutive EVEN integers algebraically, and find a pattern in their product.

a. If the smallest of 3 consecutive even integers is $2n$, write expressions for the other two.

[2]

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Second: $2n+2$. Third: $2n+4$.

b. Write an expression for the SUM of the three consecutive even integers, and simplify it.

[2]

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$$2n+(2n+2)+(2n+4)=6n+6$$

c. Factor your simplified sum, and explain what this factored form tells you about consecutive-even-integer sums (i.e. what number must they always be a multiple of).

[2]

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$6n+6=6(n+1)$ — this shows the sum of any 3 consecutive even integers is always a multiple of 6.

QUESTION 9 [3 marks] — Criterion C Hard

A student translates 'half of $x$ plus 3' as $\frac{x+3}{2}$, but their classmate writes $\frac{x}{2}+3$.

a. Determine which student's expression correctly matches the ORIGINAL phrase, and explain the difference in meaning using bracket placement.

[3]

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The classmate ($\frac{x}{2}+3$) is correct. 'Half of $x$' means $\frac{x}{2}$, and THEN '3' is added afterward, giving $\frac{x}{2}+3$. The first student's version, $\frac{x+3}{2}$, would actually match a DIFFERENT phrase: 'half of ($x$ plus 3)' — the position of the addition relative to the division changes the meaning entirely.

QUESTION 10 [6 marks] — Criterion D Hard

A rideshare app charges a base fare of \$3.50 plus \$1.20 per km, plus a \$0.15 per-minute charge for time spent in traffic.

a. Write an expression for the total fare, using $k$ for kilometres travelled and $t$ for minutes spent in traffic.

[2]

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$$3.50+1.20k+0.15t$$

b. A trip covers 8km and includes 6 minutes in traffic. Find the total fare, then find how much of the SAME total fare could instead have been achieved with ZERO traffic time but MORE kilometres travelled (find the equivalent extra distance, to 1dp).

[4]

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Actual fare: $3.50+1.20(8)+0.15(6)=3.50+9.60+0.90=\$14$. Traffic charge portion: $\$0.90$. Extra distance equivalent: $0.90\div1.20=0.75$km — so travelling $0.75$km further (at 0 traffic minutes) would cost the same as the 6 minutes of traffic did.

QUESTION 11 [5 marks] — Criterion D Hard

A phone data plan costs \$25 base, plus \$4 per GB for usage beyond the included 5GB.

a. Write an expression for total cost, using $g$ for GB used BEYOND the included 5GB (assume $g\ge0$).

[2]

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$$25+4g$$

b. A customer's bill was \$53. Using your expression, determine how many GB beyond the included amount they used, and hence their TOTAL data usage for the month (including the included 5GB).

[3]

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$25+4g=53 \Rightarrow 4g=28 \Rightarrow g=7$GB beyond the included amount. Total usage: $5+7=12$GB.

Algebraic notation