MYP 3 · Maths

EQUATIONS

Algebraic flowcharts

QUESTION 1 [3 marks] — Criterion A Medium
xx 4?+ 3= 23
The flowchart shows an unknown $x$ having two operations applied, resulting in 23.xx 4?+ 3= 23
a. Write the equation shown by the flowchart.
[1]
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$$4x+3=23$$
b. Solve the flowchart in REVERSE (starting from 23, undoing each operation in opposite order) to find $x$.
[2]
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Reverse: $23-3=20$, then $20\div4=5$. So $x=5$.
QUESTION 2 [4 marks] — Criterion A Medium
x- 6?x 2= 18
The flowchart shows two operations applied to $x$, resulting in 18.x- 6?x 2= 18
a. Write the equation shown, and solve it forward (using algebra, not the flowchart) to check your understanding.
[2]
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Equation: $2(x-6)=18$. Solving: $x-6=9 \Rightarrow x=15$.
b. Now solve using the REVERSE flowchart method (undo $\times2$ first, then undo $-6$), and confirm you get the same answer.
[2]
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Reverse: $18\div2=9$, then $9+6=15$. Matches, $x=15$.
QUESTION 3 [5 marks] — Criterion B Medium
Investigate why the REVERSE flowchart method requires undoing operations in the OPPOSITE order to how they were applied.
a. Consider $x\to(\times3)\to(+7)\to25$. If you tried to undo in the SAME order (subtract 7 first is correct, but what if you divided by 3 first instead)? Show what happens if you incorrectly divide by 3 before subtracting 7.
[3]
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Incorrect order: $25\div3\approx8.33$, then $8.33-7\approx1.33$ — this does NOT match the actual solution. Correct order: $25-7=18$, then $18\div3=6$, giving $x=6$ (verify: $3(6)+7=25$ ?).
b. Explain, like unpacking nested boxes, why operations must be undone in reverse (last-applied, first-undone) order.
[2]
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The LAST operation applied when building forward is the 'outermost' one, so it must be the FIRST one removed when working backward — like taking off your shoes before your socks, even though socks went on first. Undoing in the wrong order doesn't correctly peel back each layer.
QUESTION 4 [4 marks] — Criterion C Medium
A student builds a flowchart for $\dfrac{x+4}{3}=7$ as $x\to(+4)\to(\div3)\to7$, but then reverses it incorrectly as 'undo $+4$ first, then undo $\div3$' (same order, not reversed).
a. Explain the student's error, and perform the CORRECT reverse process.
[3]
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The student forgot to REVERSE the order — since $\div3$ was applied last (forward), it must be undone FIRST (backward). Correct: $7\times3=21$ (undo $\div3$), then $21-4=17$ (undo $+4$). So $x=17$.
b. Verify $x=17$ by substituting into the original equation.
[1]
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$\frac{17+4}{3}=\frac{21}{3}=7$ ?
QUESTION 5 [4 marks] — Criterion D Medium
tx 12?+ 45= 165
A taxi fare is modelled by a flowchart: take the time in hours $t$, multiply by 12, then add a base fee of 45, giving the total fare in dollars.tx 12?+ 45= 165
a. If a fare came to \$165, use the reverse flowchart method to find the trip's duration $t$.
[2]
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Reverse: $165-45=120$, then $120\div12=10$. So $t=10$ hours.
b. A 10-hour taxi trip seems unusually long. Suggest what the flowchart's numbers might actually represent instead of 'hours' and 'dollars per hour', to make the scenario more realistic (e.g. a different time unit or fare structure).
[2]
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A more realistic interpretation might be that $t$ represents distance in KILOMETRES (not hours), with \$12 per km plus a \$45 base/booking fee — a 10 km trip costing \$165 total is far more typical for a taxi fare than a 10-hour trip.
QUESTION 6 [3 marks] — Criterion A Medium
The flowchart shows two operations applied to $x$, giving 27.xx 5?- 8= 27
a. Write the equation shown, and solve it using the reverse flowchart method.
[3]
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Equation: $5x-8=27$. Reverse: $27+8=35$, then $35\div5=7$. So $x=7$.
QUESTION 7 [7 marks] — Criterion B Hard
Investigate how a THREE-operation flowchart is reversed, using $x \to (+6) \to (\times3) \to (-4) \to 41$.
a. Write the equation this flowchart represents.
[2]
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$$3(x+6)-4=41$$
b. Reverse the flowchart (undoing all THREE operations in the correct opposite order), showing each step.
[3]
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Reverse: undo $-4$ first: $41+4=45$. Undo $\times3$: $45\div3=15$. Undo $+6$: $15-6=9$. So $x=9$.
c. Verify $x=9$ by substituting into the original equation.
[2]
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$3(9+6)-4=3(15)-4=45-4=41$ ?.
QUESTION 8 [5 marks] — Criterion B Medium
Investigate why a flowchart involving DIVISION must be reversed using MULTIPLICATION, using the flowchart shown.x÷ 4?+ 9= 20
a. Write the equation, and identify the TWO operations shown (in order).
[2]
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Equation: $\frac{x}{4}+9=20$. Operations: divide by 4, then add 9.
b. Reverse the flowchart correctly, explaining specifically why the LAST operation (÷4, working backward from division) must be undone using MULTIPLICATION, not further division.
[3]
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Reverse: undo $+9$: $20-9=11$. Undo $\div4$: $11\times4=44$. So $x=44$. Division and multiplication are INVERSE operations — since the forward flowchart used $\div4$, reversing it requires the OPPOSITE (inverse) operation, $\times4$, to correctly 'undo' the division and recover the original value.
QUESTION 9 [4 marks] — Criterion C Medium
A student building a flowchart for $\dfrac{3x-5}{2}=8$ draws it as $x\to(\times3)\to(-5)\to(\div2)\to8$, then reverses it by undoing $\times3$ FIRST (in the same order as building it), instead of reversing the order.
a. Explain the student's error, and perform the CORRECT reverse process, step by step.
[4]
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The student forgot that reversing a flowchart requires undoing the operations in OPPOSITE order to how they were built (last-applied, first-undone) — undoing $\times3$ first (the FIRST operation applied) is incorrect. Correct reverse order: undo $\div2$ first: $8\times2=16$. Undo $-5$: $16+5=21$. Undo $\times3$: $21\div3=7$. So $x=7$.
QUESTION 10 [7 marks] — Criterion D Hard
tx 18?+ 65= 191
A cleaning service charges according to a flowchart: take the number of rooms cleaned $t$, multiply by 18, then add a fixed \$65 travel fee.tx 18?+ 65= 191
a. If a bill came to \$191, use the reverse flowchart method to find the number of rooms cleaned.
[3]
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Reverse: $191-65=126$, then $126\div18=7$. So $t=7$ rooms.
b. The cleaning service is considering changing their pricing to a flowchart with the operations REVERSED in order (add \$65 first, THEN multiply by 18), keeping the same numbers. Determine the NEW bill for 7 rooms under this alternative structure, and explain why changing the ORDER of operations in the flowchart (not just the numbers) produces a genuinely different pricing formula.
[4]
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New formula: $18(t+65)$. For $t=7$: $18(72)=1296$ — a MASSIVELY higher bill (\$1,296 vs \$191). Changing the order fundamentally changes the formula's structure: the original $(18t+65)$ applies the \$65 fee ONCE regardless of room count, while the reversed version $18(t+65)$ effectively multiplies the \$65 fee by 18 too, making it scale disproportionately — this shows that in a flowchart, the ORDER of operations is just as important as the individual operations themselves.