ALGEBRAIC OPERATIONS

QUESTION 1 [3 marks] — Criterion A Medium

Translate each phrase into an algebraic expression:

a. The sum of $x$ and 3, multiplied by 4

[1]

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$$4(x+3)$$

b. The quotient of $y$ and 6, increased by 2

[1]

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$$\frac{y}{6}+2$$

c. 9 decreased by the product of 3 and $z$

[1]

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$$9-3z$$

QUESTION 2 [3 marks] — Criterion A Medium

Translate: 'The square of a number, minus 5 times the number.'

a. Write this as an algebraic expression, using $n$ for the number.

[1]

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$$n^2-5n$$

b. Evaluate the expression for $n=6$ and for $n=-2$.

[2]

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$n=6$: $36-30=6$. $n=-2$: $4-(-10)=14$.

QUESTION 3 [5 marks] — Criterion B Medium

Investigate different word phrases that can translate to the SAME algebraic expression, $3x+2$.

a. Write two different word phrases that would both translate to $3x+2$.

[2]

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Example phrases: 'triple a number, then add 2' and '2 more than 3 times a number' — both give $3x+2$.

b. Now write a word phrase for $3(x+2)$ (a DIFFERENT expression), and explain what makes it different from $3x+2$ despite looking similar in words.

[3]

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Phrase: 'a number increased by 2, then tripled'. The key difference is where the brackets fall: $3(x+2)$ means the ADDITION happens first, then multiplication — giving $3x+6$ when expanded, which is different from $3x+2$.

QUESTION 4 [5 marks] — Criterion C Medium

A classmate translates 'a number decreased by 5' and '5 decreased by a number' as the same expression, $x-5$.

a. Explain the classmate's error, writing the correct expression for each phrase.

[3]

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'A number decreased by 5' means starting with $x$ and subtracting 5: $x-5$. But '5 decreased by a number' means starting with 5 and subtracting $x$: $5-x$. These are different expressions (in fact, $5-x=-(x-5)$).

b. Verify the difference by evaluating both expressions at $x=8$.

[2]

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$x-5=8-5=3$. $5-x=5-8=-3$. Confirmed — they give opposite (different) results.

QUESTION 5 [4 marks] — Criterion D Medium

A phone plan is described as: 'a base cost of \$20, plus \$0.15 for every text message sent beyond the first 100 (which are free).'

a. Write a formula for the monthly cost $C$ if $t$ texts are sent beyond the free 100 (i.e. $t\ge0$).

[2]

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$$C = 20 + 0.15t$$

b. A customer sent 340 texts in total this month. Find their bill.

[2]

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Texts beyond free 100: $340-100=240$. $$C=20+0.15(240)=20+36=\$56$$

QUESTION 6 [5 marks] — Criterion A Hard

Translate each phrase into an algebraic expression, then simplify where possible:

a. The square of the sum of $x$ and 3, minus the square of $x$

[3]

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$$(x+3)^2 - x^2 = x^2+6x+9-x^2 = 6x+9$$

b. Twice the difference between $y$ and 5, increased by the product of $y$ and 3

[2]

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$$2(y-5)+3y = 2y-10+3y=5y-10$$

QUESTION 7 [7 marks] — Criterion B Hard

Investigate whether the phrase '3 less than double a number' ALWAYS means the same thing as 'double the amount that is 3 less than a number'.

a. Translate '3 less than double a number $x$' into an algebraic expression.

[2]

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$$2x-3$$

b. Translate '$double the amount that is 3 less than a number $x$' into a DIFFERENT algebraic expression.

[2]

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$$2(x-3)$$

c. Test both expressions at $x=10$, and explain why the two phrases (despite using similar words) give different results.

[3]

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First: $2(10)-3=17$. Second: $2(10-3)=14$. They differ because of WHEN the subtraction happens relative to the doubling — 'less than double' means subtract AFTER doubling, while 'double the amount that is less' means subtract BEFORE doubling. Word order and structure in English can be genuinely ambiguous, which is exactly why precise mathematical notation (using brackets) is needed to remove ambiguity.

QUESTION 8 [5 marks] — Criterion B Hard

Investigate how many DIFFERENT English phrases can describe the SAME expression, $5(x+2)$.

a. Write THREE different English phrases that would all correctly translate to $5(x+2)$.

[3]

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Examples: 'five times the sum of a number and 2'; '2 more than a number, all multiplied by 5'; 'a number increased by 2, then the result quintupled'. (Any three phrases that correctly capture 'add 2 first, then multiply by 5' are valid.)

b. Explain why, despite using very different WORDING, all three phrases must translate to the EXACT same algebraic expression.

[2]

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All three phrases describe the SAME sequence of mathematical operations in the SAME order (add 2 to the number, THEN multiply the result by 5) — different words can describe an identical process, and since the underlying mathematical operations and their order are what determines the expression, the wording variation doesn't change the algebra.

QUESTION 9 [3 marks] — Criterion C Hard

A student translates 'the difference between $x$ and $y$' as $x-y$, always, without exception.

a. Explain why this translation can be problematic if $y > x$ (giving a negative result), and discuss whether 'difference' in everyday mathematical language usually implies a POSITIVE result (a distance/magnitude) rather than a signed value.

[3]

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If $y>x$, then $x-y$ gives a NEGATIVE number, but the everyday/common meaning of 'difference' (e.g. 'the difference in our heights') usually implies a positive gap or distance, regardless of which quantity is larger. A more universally safe translation of 'difference between $x$ and $y$' would use absolute value, $|x-y|$, which is guaranteed non-negative regardless of which value is bigger — though in ALGEBRA specifically, 'difference' can also legitimately mean the signed subtraction $x-y$, so context matters.

QUESTION 10 [5 marks] — Criterion D Hard

An insurance company calculates a premium using the description: 'a base rate of \$400, reduced by \$25 for every year of a driver's claim-free history, down to a minimum of \$150.'

a. Write an algebraic expression for the premium based on $y$ claim-free years, IGNORING the minimum for now.

[2]

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$$400-25y$$

b. Find the number of claim-free years at which the premium would reach exactly the \$150 minimum, and explain why the algebraic expression $400-25y$ alone is NOT a complete/accurate model of the real premium for drivers with MORE claim-free years than this.

[3]

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$400-25y=150 \Rightarrow 25y=250 \Rightarrow y=10$ years. Beyond 10 claim-free years, the plain formula $400-25y$ would predict a premium BELOW \$150 (even negative for very large $y$), which doesn't match the real policy — the true model needs a MINIMUM condition applied (e.g. premium $=\max(400-25y, 150)$), since a real insurance premium can never keep decreasing indefinitely.

The language of mathematics