01-TARA Problem Solving Book · 15 Topics + Algorithm & Workflow · Hard Edition
TARA Problem Solving Book · 15 Topics + Algorithm & Workflow · Hard Edition
This version restores the full TARA Problem Solving topic range and adds Algorithm and Workflow reasoning. The earlier version had too few topics; this edition has all requested topics with detailed explanations, diagrams and hard decision-making MCQs.
Book map
| Coverage | Design |
|---|---|
| 15 official-style Problem Solving topics | Fractions through Practical Problem Solving |
| Additional Algorithm and Workflow topic | Conditional steps, loops, stop rules and process logic |
| Question level | Hard TARA-style decision-making traps, not easy drills |
Teacher guidance
For each topic, teach students to name the hidden structure before they calculate: original amount, remaining amount, group weight, unit, threshold, overlap, sample space, or workflow condition.
1 Fractions
Use this diagram to teach denominator choice: original amount, remaining amount, and change are different bases.
Detailed explanation
Fractions become hard when the reference whole changes. The key question is: does the fraction refer to the original amount, the amount left, the amount added, or the change between two states?
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A tank is \(\frac35\) full. After \(18\) litres are added, it is \(\frac45\) full. What is the tank capacity?
- A60 L
- B75 L
- C90 L
- D100 L
- E120 L
Show solution, teacher note and trap analysis
Correct answer: C
SolutionThe added amount is \(\frac45-\frac35=\frac15\) of the tank. If \(\frac15=18\), capacity is \(90\) L.
Trap AnalysisThe 18 litres is the change in fullness, not the final amount.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A student spends \(\frac14\) of her money, then \(\frac13\) of what remains. What fraction remains?
- A\(\frac13\)
- B\(\frac12\)
- C\(\frac{5}{12}\)
- D\(\frac{7}{12}\)
- E\(\frac23\)
Show solution, teacher note and trap analysis
Correct answer: B
SolutionAfter the first spend, \(\frac34\) remains. Then \(\frac23\) of that remains: \(\frac34\cdot\frac23=\frac12\).
Trap AnalysisThe second fraction is of the remaining money, not the original money.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Three-fifths of a class are girls. There are \(8\) more girls than boys. How many students are in the class?
- A24
- B32
- C40
- D48
- E60
Show solution, teacher note and trap analysis
Correct answer: C
SolutionGirls minus boys is \(\frac35-\frac25=\frac15\) of the class. If \(\frac15=8\), total is \(40\).
Trap AnalysisThe difference, not the number of girls, is the useful fraction.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A bottle is \(\frac78\) full. \(120\) ml is poured out and it becomes \(\frac58\) full. What is the capacity?
- A360 ml
- B420 ml
- C480 ml
- D540 ml
- E600 ml
Show solution, teacher note and trap analysis
Correct answer: C
SolutionThe change is \(\frac78-\frac58=\frac28=\frac14\). If \(\frac14=120\), capacity is \(480\) ml.
Trap AnalysisThe poured amount is a change, not the final volume.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
2 Percentages
Reverse percentage questions are solved by dividing by the multiplier, not by subtracting the percentage from the final.
Detailed explanation
Percentages are multipliers. Reverse percentages and repeated changes require multiplication or division by the correct multiplier, not adding and subtracting percentage points casually.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
After a \(25\%\) discount, a jacket costs \(£54\). What was the original price?
- A£67.50
- B£70
- C£72
- D£75
- E£81
Show solution, teacher note and trap analysis
Correct answer: C
SolutionAfter discount, \(75\%\) remains. Original price \(=54/0.75=72\).
Trap AnalysisDo not subtract 25% of the final price.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A value is increased by \(20\%\), then decreased by \(20\%\). What is the net effect?
- ANo change
- B4% decrease
- C4% increase
- D20% decrease
- E40% decrease
Show solution, teacher note and trap analysis
Correct answer: B
SolutionMultiplier \(=1.2\cdot0.8=0.96\), so the value is 4% lower.
Trap AnalysisEqual percentage increase and decrease do not cancel because the bases differ.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Two groups have pass rates \(80\%\) of \(30\) students and \(60\%\) of \(70\) students. What is the overall pass rate?
- A66%
- B68%
- C70%
- D72%
- E74%
Show solution, teacher note and trap analysis
Correct answer: A
SolutionPasses \(=24+42=66\) out of \(100\), so \(66\%\).
Trap AnalysisDo not average 80% and 60% when group sizes differ.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A shop gives \(15\%\) off, then \(10\%\) off the reduced price. What is the total discount?
- A21.5%
- B23.5%
- C24%
- D25%
- E26.5%
Show solution, teacher note and trap analysis
Correct answer: B
SolutionRemaining multiplier \(=0.85\cdot0.90=0.765\). Discount \(=23.5\%\).
Trap AnalysisSuccessive discounts are multiplicative, not additive.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
3 Place Value
Place value problems are about digit position: the same digit can represent 5, 50, 500, or 5000.
Detailed explanation
Place value questions test the value of a digit in its position. Swapping digits, inserting digits, or using divisibility rules changes values by powers of ten.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A two-digit number has tens digit \(a\) and units digit \(b\). The reversed number is \(27\) less. What is \(a-b\)?
- A1
- B2
- C3
- D4
- E5
Show solution, teacher note and trap analysis
Correct answer: C
SolutionOriginal \(=10a+b\), reversed \(=10b+a\). Difference \(=9(a-b)=27\), so \(a-b=3\).
Trap AnalysisCompare place values, not just digits.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A number has form \(4a7\) and is divisible by \(9\). Which digit is \(a\)?
- A3
- B4
- C5
- D6
- E7
Show solution, teacher note and trap analysis
Correct answer: E
SolutionDigit sum is \(4+a+7=11+a\). It must be \(18\), so \(a=7\).
Trap AnalysisDivisibility by 9 uses digit sum.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A 4-digit number has thousands digit twice its units digit, hundreds digit \(3\), tens digit \(0\), and digit sum \(12\). What is the number?
- A4304
- B6303
- C8304
- D9303
- E6306
Show solution, teacher note and trap analysis
Correct answer: B
SolutionLet units digit be \(u\). Then thousands is \(2u\). Sum: \(2u+3+u=12\), so \(u=3\). Number \(6303\).
Trap AnalysisThousands digit is constrained by units digit.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Swapping the tens and units digits of a two-digit number increases it by \(36\). What is the difference between the new tens digit and new units digit?
- A2
- B3
- C4
- D5
- E6
Show solution, teacher note and trap analysis
Correct answer: C
SolutionIncrease \(=9(b-a)=36\), so \(b-a=4\). In the new number, the digit difference is \(4\).
Trap AnalysisSwapping changes the sign of the digit difference.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
4 Four Operations
Four-operation problems become hard when fixed charges, grouping, and rounding rules are hidden.
Detailed explanation
Four operations questions become hard when operations are nested, repeated, or hidden inside a context. The main skill is writing the correct expression before computing.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A number is multiplied by \(6\), then \(14\) is subtracted. This equals multiplying the original by \(4\) and adding \(8\). Find the number.
- A7
- B8
- C9
- D10
- E11
Show solution, teacher note and trap analysis
Correct answer: E
SolutionSet \(6x-14=4x+8\). Then \(2x=22\), so \(x=11\).
Trap AnalysisModel both processes; do not compute one side only.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A shop packs \(18\) items per box. It has \(740\) items. How many boxes are needed?
- A39
- B40
- C41
- D42
- E43
Show solution, teacher note and trap analysis
Correct answer: D
Solution\(740/18=41.11...\), so \(42\) boxes are needed.
Trap AnalysisRounding down leaves items unpacked.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A number is divided by \(5\), then \(7\) is added, giving \(23\). What was the number?
- A60
- B70
- C75
- D80
- E85
Show solution, teacher note and trap analysis
Correct answer: D
Solution\(x/5+7=23\), so \(x/5=16\), \(x=80\).
Trap AnalysisUndo addition before undoing division.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Three machines make \(24,36,45\) parts per hour. How many whole hours to make at least \(1000\) parts together?
- A8
- B9
- C10
- D11
- E12
Show solution, teacher note and trap analysis
Correct answer: C
SolutionRate \(=24+36+45=105\) parts/hour. \(1000/105=9.52...\), so \(10\) hours.
Trap AnalysisAt least means round up.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Added Question · Four Operations Order Trap
This diagram reinforces the order convention: division is evaluated from left to right.
The expression in the image is:
\[ 5\div5\div5\div5 \] What is its value if the operations are performed using the standard left-to-right convention?
\[ 5\div5\div5\div5 \] What is its value if the operations are performed using the standard left-to-right convention?
- A\(1\)
- B\(\dfrac{1}{5}\)
- C\(\dfrac{1}{25}\)
- D\(5\)
- E\(25\)
Show solution, teacher note and trap analysis
Correct answer: C
Solution
Division has the same priority throughout this expression, so we calculate from left to right:
\[
5\div5=1,\qquad 1\div5=\frac15,\qquad \frac15\div5=\frac1{25}.
\]
Therefore,
\[
5\div5\div5\div5=\frac1{25}.
\]
Trap Analysis
The common wrong answer is \(1\), because students mentally group the expression as \((5\div5)\div(5\div5)\). That grouping is not allowed unless brackets are written.
Teacher's Note
This is a useful TARA-style operation-order trap. It tests whether the student follows the written workflow rather than inventing brackets.
EduCoach Note
Ask students to rewrite every division as multiplication by a reciprocal:
\[
5\div5\div5\div5=5\times\frac15\times\frac15\times\frac15=\frac1{25}.
\]
5 Mean and Average
A combined mean is pulled closer to the larger group; never average averages unless group sizes are equal.
Detailed explanation
Mean is total divided by count. Hard average problems hide the total, count, or group weights. Combined means must be weighted by group size.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A group of \(12\) has mean \(70\). Another group of \(8\) has mean \(85\). What is the combined mean?
- A75
- B76
- C77
- D78
- E79
Show solution, teacher note and trap analysis
Correct answer: B
SolutionTotal \(=12(70)+8(85)=1520\). Count \(=20\). Mean \(=76\).
Trap AnalysisDo not take the simple average of the two means.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A set with mean \(12\) is combined with one value \(30\). The new mean is \(15\). How many values were originally in the set?
- A3
- B4
- C5
- D6
- E7
Show solution, teacher note and trap analysis
Correct answer: C
Solution\((12n+30)/(n+1)=15\Rightarrow n=5\).
Trap AnalysisThe unknown count is hidden inside the mean equation.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Replacing one value \(9\) by \(27\) in a set of \(6\) numbers changes the mean from \(15\) to what?
- A16
- B17
- C18
- D19
- E20
Show solution, teacher note and trap analysis
Correct answer: C
SolutionThe total increases by \(18\). Mean increases by \(18/6=3\), so new mean is \(18\).
Trap AnalysisThe change is distributed across all values.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Group A has mean \(50\). Group B has mean \(80\). Combined mean is \(62\). What is ratio of sizes A:B?
- A2:3
- B3:2
- C5:2
- D4:3
- E1:2
Show solution, teacher note and trap analysis
Correct answer: B
Solution\((50a+80b)/(a+b)=62\Rightarrow 12a=18b\Rightarrow a:b=3:2\).
Trap AnalysisThe combined mean is closer to the larger group.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
6 Money Problems
Money questions often use fixed fee plus variable cost; remove the fixed fee before reversing the rule.
Detailed explanation
Money problems are usually functions: fixed fee plus variable cost, discount plus fee, or total divided by units. Compare equivalent quantities.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A subscription costs \(£18\) monthly or \(£190\) yearly. How much is saved by paying yearly?
- A£16
- B£18
- C£22
- D£24
- E£26
Show solution, teacher note and trap analysis
Correct answer: E
SolutionMonthly yearly total \(=12\cdot18=216\). Saving \(=216-190=26\).
Trap AnalysisCompare the same time period.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A taxi charges \(£3.50\) plus \(£1.80\) per km. What is the cost for \(12\) km?
- A£21.60
- B£23.10
- C£24.10
- D£25.10
- E£26.60
Show solution, teacher note and trap analysis
Correct answer: D
Solution\(3.50+12(1.80)=25.10\).
Trap AnalysisDo not ignore the fixed charge.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Pack A has \(750\) g for \(£3.60\). Pack B has \(1.2\) kg for \(£5.40\). Which is cheaper per kg?
- AA by £0.10/kg
- BA by £0.20/kg
- CB by £0.20/kg
- DB by £0.30/kg
- ESame
Show solution, teacher note and trap analysis
Correct answer: D
SolutionA: \(3.60/0.75=4.80\). B: \(5.40/1.2=4.50\). B cheaper by \(£0.30/kg\).
Trap AnalysisConvert grams to kilograms.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A repair bill is \(£45\) call-out plus \(£30\) per hour. The bill is \(£165\). How many hours?
- A2
- B3
- C4
- D5
- E6
Show solution, teacher note and trap analysis
Correct answer: C
Solution\(45+30h=165\Rightarrow h=4\).
Trap AnalysisRemove the fixed fee first.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
7 Time and Calendar
Time questions require elapsed time, not clock-face guessing; repeated events use LCM.
Detailed explanation
Time questions require careful units, cycles, calendars and elapsed time. Hard questions often use LCM, clock drift or crossing an hour boundary.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A train leaves every \(18\) minutes and a bus every \(24\) minutes. Both leave at 08:00. When next together?
- A08:36
- B08:48
- C09:00
- D09:12
- E09:24
Show solution, teacher note and trap analysis
Correct answer: D
SolutionLCM of \(18\) and \(24\) is \(72\). 08:00 + 72 min = 09:12.
Trap AnalysisUse LCM, not addition.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A clock gains \(4\) minutes every \(3\) hours. Correct at noon, what does it show when true time is 9 pm?
- A9:04
- B9:08
- C9:10
- D9:12
- E9:16
Show solution, teacher note and trap analysis
Correct answer: D
SolutionNine hours is three blocks of 3 hours. Gain \(=12\) minutes.
Trap AnalysisClock error is a rate.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A task starts 13:47 and lasts \(2\) h \(38\) min. Finish time?
- A15:15
- B15:25
- C16:15
- D16:25
- E16:35
Show solution, teacher note and trap analysis
Correct answer: D
Solution13:47 + 2 h = 15:47; +38 min = 16:25.
Trap AnalysisCross the hour carefully.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A cycle repeats every \(45\) seconds and another every \(72\) seconds. Starting together, when next together?
- A3 min
- B4 min
- C5 min
- D6 min
- E9 min
Show solution, teacher note and trap analysis
Correct answer: D
SolutionLCM \(=360\) seconds = 6 minutes.
Trap AnalysisConvert seconds to minutes after finding the LCM.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
8 Measures and Unit Conversion
Unit conversion must happen before division, rate comparison, or rounding.
Detailed explanation
Measurement questions become hard when two conversions are needed or when the answer must be rounded up. Convert first, then reason.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A rope is \(3.6\) m long. How many \(45\) cm pieces can be cut?
- A6
- B7
- C8
- D9
- E10
Show solution, teacher note and trap analysis
Correct answer: C
Solution\(3.6\) m \(=360\) cm. \(360/45=8\).
Trap AnalysisUse the same units first.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A \(2.5\) L jug is filled using a \(200\) ml cup. How many full cups are needed to exceed capacity?
- A11
- B12
- C13
- D14
- E15
Show solution, teacher note and trap analysis
Correct answer: C
Solution\(2.5\) L \(=2500\) ml. 12 cups = 2400 ml, 13 cups = 2600 ml.
Trap AnalysisExceed means first value above.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A car travels \(90\) km/h. What is this in m/s?
- A15
- B20
- C25
- D30
- E36
Show solution, teacher note and trap analysis
Correct answer: C
Solution\(90\text{ km/h}=90000/3600=25\text{ m/s}\).
Trap AnalysisConvert both distance and time.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A medicine dose is \(8\) mg per kg. Patient mass is \(62.5\) kg. Dose?
- A400 mg
- B450 mg
- C500 mg
- D520 mg
- E560 mg
Show solution, teacher note and trap analysis
Correct answer: C
Solution\(8\cdot62.5=500\) mg.
Trap AnalysisPer kg means multiply by kg.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
9 Area and Perimeter
Area is covering; perimeter is boundary. Composite problems require different reasoning for each.
Detailed explanation
Area measures covering space; perimeter measures boundary. Composite questions require adding or subtracting only the relevant regions or edges.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A \(12\) by \(8\) rectangle has a \(3\) by \(4\) rectangle removed. What area remains?
- A72
- B78
- C80
- D84
- E88
Show solution, teacher note and trap analysis
Correct answer: D
SolutionOriginal area \(=96\). Removed area \(=12\). Remaining \(=84\).
Trap AnalysisSubtract area, not perimeter.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A path \(1\) m wide surrounds a \(10\) m by \(6\) m garden. What is path area?
- A24
- B28
- C32
- D36
- E40
Show solution, teacher note and trap analysis
Correct answer: D
SolutionOuter dimensions \(12\) by \(8\), area \(96\). Garden area \(60\). Path area \(36\).
Trap AnalysisThe width is added on both sides.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A square has perimeter \(36\). What is its area?
- A64
- B72
- C81
- D90
- E100
Show solution, teacher note and trap analysis
Correct answer: C
SolutionSide \(=36/4=9\). Area \(=81\).
Trap AnalysisDo not square the perimeter.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A circle has radius doubled. What happens to its area?
- ADoubles
- BTriples
- CQuadruples
- DHalves
- EUnchanged
Show solution, teacher note and trap analysis
Correct answer: C
SolutionArea scale factor \(=2^2=4\).
Trap AnalysisArea scales with the square of length.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
10 Volume of Boxes
Packing questions are solved by counting how many fit along each dimension.
Detailed explanation
Box volume uses three dimensions. Packing questions require checking how many items fit along each direction, not just dividing volumes blindly.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A cuboid is \(3\) by \(4\) by \(5\). What is volume?
- A12
- B20
- C48
- D60
- E120
Show solution, teacher note and trap analysis
Correct answer: D
SolutionVolume \(=3\cdot4\cdot5=60\).
Trap AnalysisThree dimensions must be multiplied.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
How many \(2\) cm cubes fit in a box \(8\) by \(6\) by \(4\) cm?
- A12
- B18
- C24
- D36
- E48
Show solution, teacher note and trap analysis
Correct answer: C
SolutionCounts along dimensions are \(4,3,2\). Total \(=4\cdot3\cdot2=24\).
Trap AnalysisCount along dimensions.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A cube has volume \(125\). What is its surface area?
- A25
- B75
- C100
- D125
- E150
Show solution, teacher note and trap analysis
Correct answer: E
SolutionSide \(=5\). Surface area \(=6\cdot25=150\).
Trap AnalysisVolume and surface area are different.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A cuboid volume is \(240\). Base is \(8\) by \(6\). What is height?
- A3
- B4
- C5
- D6
- E8
Show solution, teacher note and trap analysis
Correct answer: C
SolutionBase area \(=48\). Height \(=240/48=5\).
Trap AnalysisDivide by base area.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
11 Tables
In table questions, identify whether the denominator is a row total, column total, or grand total.
Detailed explanation
Tables hide totals, overlaps and conditional denominators. Identify whether the problem asks for a row total, column total, grand total, or conditional proportion.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
Sales are Mon 12, Tue 18, Wed 15, Thu 25. What fraction were on Thu?
- A\(\frac14\)
- B\(\frac{25}{70}\)
- C\(\frac{25}{58}\)
- D\(\frac13\)
- E\(\frac{5}{12}\)
Show solution, teacher note and trap analysis
Correct answer: B
SolutionTotal \(=70\). Thu fraction \(=25/70\).
Trap AnalysisUse total sales across all listed days.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
In a two-way table of \(80\) people, \(35\) are in A, \(50\) in B, \(20\) in both. How many are in neither?
- A5
- B10
- C15
- D20
- E25
Show solution, teacher note and trap analysis
Correct answer: C
SolutionAt least one \(=35+50-20=65\). Neither \(=15\).
Trap AnalysisSubtract the overlap once.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A table has \(x:2,5,8\), \(y:11,20,29\). Predict \(y\) for \(x=14\) assuming a linear pattern.
- A35
- B38
- C41
- D44
- E47
Show solution, teacher note and trap analysis
Correct answer: E
SolutionRule \(y=3x+5\). At \(x=14\), \(y=47\).
Trap AnalysisOutput changes by 9 when input changes by 3.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A survey: \(40\%\) of \(150\) boys and \(60\%\) of \(100\) girls chose sport. How many chose sport?
- A100
- B110
- C120
- D130
- E140
Show solution, teacher note and trap analysis
Correct answer: C
SolutionBoys \(=60\), girls \(=60\), total \(120\).
Trap AnalysisDo not average percentages.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
12 Graphs and Charts
Charts require translating visual information into numbers before making conclusions.
Detailed explanation
Charts require scale reading and context interpretation. Translate bar height, sector angle, line trend or graph point into a quantity before calculating.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A bar chart uses scale 1 cm = 8 units. A bar is 6.5 cm high. Frequency?
- A48
- B50
- C52
- D54
- E56
Show solution, teacher note and trap analysis
Correct answer: C
Solution\(6.5\cdot8=52\).
Trap AnalysisRead the vertical scale.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A pie chart sector is \(54^\circ\) and represents \(18\) people. Total people?
- A90
- B100
- C110
- D120
- E130
Show solution, teacher note and trap analysis
Correct answer: D
Solution\(54/360=18/T\Rightarrow T=120\).
Trap AnalysisA sector angle is part of 360.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A line graph rises from \(120\) to \(150\). Percentage increase?
- A20%
- B25%
- C30%
- D35%
- E40%
Show solution, teacher note and trap analysis
Correct answer: B
SolutionIncrease \(=30\). \(30/120=25\%\).
Trap AnalysisUse the starting value.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A trend line is based on ages 10 to 16. Predicting for age 25 is called:
- AInterpolation
- BExtrapolation
- CMean
- DMedian
- ERange
Show solution, teacher note and trap analysis
Correct answer: B
SolutionAge 25 is outside the observed range, so it is extrapolation.
Trap AnalysisOutside-range predictions are less reliable.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
13 Multi-step Numerical Reasoning
Write intermediate totals; most mistakes come from skipping a stage.
Detailed explanation
Multi-step reasoning requires a plan. Identify stages, record intermediate totals and check whether the question asks for profit, revenue, remainder or final value.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A shop buys \(80\) items for \(£6\) each. It sells \(75\%\) for \(£10\) each and the rest for \(£4\). What is the profit?
- A£120
- B£160
- C£180
- D£200
- E£240
Show solution, teacher note and trap analysis
Correct answer: D
SolutionCost \(=80(6)=480\). Revenue \(=60(10)+20(4)=680\). Profit \(=200\).
Trap AnalysisProfit is revenue minus cost.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A school trip has \(126\) students. Coaches hold \(48\) students each. Each coach costs \(£390\). What is cost per student?
- A£8.75
- B£9.29
- C£9.52
- D£10.00
- E£12.38
Show solution, teacher note and trap analysis
Correct answer: B
SolutionNeed 3 coaches. Total cost \(=1170\). Cost per student \(=1170/126\approx£9.29\).
Trap AnalysisRound coaches up before dividing cost.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A value is doubled, then increased by \(30\), then divided by \(5\), giving \(26\). What was the original value?
- A40
- B45
- C50
- D55
- E60
Show solution, teacher note and trap analysis
Correct answer: C
Solution\((2x+30)/5=26\Rightarrow2x+30=130\Rightarrow x=50\).
Trap AnalysisReverse the whole process.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A store has \(360\) units. It sells \(\frac25\) on Monday, then \(25\%\) of the remainder on Tuesday. How many remain?
- A144
- B162
- C180
- D198
- E216
Show solution, teacher note and trap analysis
Correct answer: B
SolutionAfter Monday: \(216\). After Tuesday, \(75\%\) remains: \(0.75(216)=162\).
Trap AnalysisTuesday percentage is of the remainder.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
14 Spatial Reasoning
Spatial reasoning should be systematic: count lines, edges, faces, layers or positions.
Detailed explanation
Spatial reasoning is about systematic counting, transformations and hidden 3D structure. Do not rely on visual guessing.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A cube is painted on all outside faces and cut into \(3\times3\times3\) small cubes. How many have exactly two painted faces?
- A6
- B8
- C12
- D18
- E24
Show solution, teacher note and trap analysis
Correct answer: C
SolutionExactly two painted faces are edge-centre cubes. There are 12 edges, one such cube per edge.
Trap AnalysisCorners have three painted faces.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
How many rectangles are in a \(2\times3\) grid of small squares?
- A6
- B12
- C18
- D24
- E30
Show solution, teacher note and trap analysis
Correct answer: C
SolutionChoose 2 of 3 horizontal grid lines and 2 of 4 vertical grid lines: \(3\cdot6=18\).
Trap AnalysisCount all rectangle sizes.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A shape is rotated \(90^\circ\) clockwise three times. This is equivalent to which single rotation?
- A90° clockwise
- B180°
- C270° clockwise
- D90° anticlockwise
- EOriginal position
Show solution, teacher note and trap analysis
Correct answer: D
SolutionThree clockwise quarter-turns = 270° clockwise = 90° anticlockwise.
Trap AnalysisRotations can be expressed in equivalent directions.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Four people sit in a row. A and B must sit together. How many arrangements?
- A6
- B8
- C12
- D16
- E24
Show solution, teacher note and trap analysis
Correct answer: C
SolutionTreat AB as one block plus two others: \(3!\) arrangements, times 2 internal orders = 12.
Trap AnalysisUse a block for together conditions.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
15 Practical Problem Solving
Practical problems require feasible decisions, not just exact arithmetic.
Detailed explanation
Practical problem solving combines money, time, units, comparison and constraints. The correct answer must be feasible, not merely arithmetically close.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
A venue has tables seating \(8\). There are \(143\) guests. Each table costs \(£12\). What is table hire cost?
- A£204
- B£216
- C£228
- D£240
- E£252
Show solution, teacher note and trap analysis
Correct answer: B
SolutionTables needed \(=\lceil143/8\rceil=18\). Cost \(=18(12)=216\).
Trap AnalysisRound up number of tables.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A recipe for \(6\) people needs \(450\) g rice. How much for \(14\) people?
- A950 g
- B1000 g
- C1050 g
- D1100 g
- E1150 g
Show solution, teacher note and trap analysis
Correct answer: C
SolutionPer person \(=75\) g. For 14: \(1050\) g.
Trap AnalysisScale proportionally.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A printer has \(40\) sheets. Each booklet uses \(6\) sheets and there are \(3\) cover sheets total. How many complete booklets can be made?
- A5
- B6
- C7
- D8
- E9
Show solution, teacher note and trap analysis
Correct answer: B
SolutionSheets for booklets \(=40-3=37\). \(37/6=6\) complete booklets.
Trap AnalysisSubtract fixed requirement first.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A family ticket costs \(£42\) for up to 5 people. Individual tickets cost \(£9\). For 6 people, what is cheapest total?
- A£42
- B£45
- C£51
- D£54
- E£60
Show solution, teacher note and trap analysis
Correct answer: C
SolutionUse one family ticket for 5 plus one individual: \(42+9=51\).
Trap AnalysisA family ticket cannot cover all 6.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
16 Algorithm and Workflow Reasoning
Workflow questions test ordered steps, branches and loops; the condition must be checked at the correct time.
Detailed explanation
Algorithm and workflow questions use ordered steps, loops, conditions and stop rules. The main trap is doing the right operations in the wrong order or stopping too early.
How to teach this topic
Start with a short modelling question: What is the input? What is the total? What is being compared? Which value changes? Which value stays fixed? This prevents fast but wrong arithmetic.
An algorithm starts with \(n=5\). Step 1: double \(n\). Step 2: if \(n>12\), subtract 4; otherwise add 7. Step 3: double once more. Final \(n\)?
- A20
- B24
- C28
- D34
- E40
Show solution, teacher note and trap analysis
Correct answer: D
SolutionStart 5. Double to 10. Since 10 is not greater than 12, add 7 to get 17. Double to 34.
Trap AnalysisThe condition is checked after the first doubling.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A workflow accepts \(70\%\) of forms at first check. Of returned forms, \(40\%\) are accepted after correction. What percentage are eventually accepted?
- A70%
- B78%
- C82%
- D88%
- E92%
Show solution, teacher note and trap analysis
Correct answer: C
SolutionInitially accepted 70%. Returned 30%; corrected accepted 40% of 30% = 12%. Total 82%.
Trap AnalysisThe 40% applies only to returned forms.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A machine repeats: subtract 4, then halve. Starting from 36, what is the result after two complete repeats?
- A6
- B7
- C8
- D9
- E10
Show solution, teacher note and trap analysis
Correct answer: A
SolutionFirst repeat: (36−4)/2 = 16. Second: (16−4)/2 = 6.
Trap AnalysisA complete repeat includes both steps.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
A queue rule: serve 4 customers, then add 1 priority customer. Starting with 23 customers, after 3 cycles how many remain?
- A10
- B11
- C12
- D13
- E14
Show solution, teacher note and trap analysis
Correct answer: E
SolutionEach cycle reduces the queue by net 3. After 3 cycles: \(23-9=14\).
Trap AnalysisThe priority customer is added after service.
Teacher's NoteThis is a hard TARA-style problem: the arithmetic is manageable, but the modelling decision is the main challenge.
EduCoach NoteBefore calculating, students should identify the total, unit, rule, condition, or sample space. Then check whether the result answers the exact question.
Diagram Appendix · How to Use the Visuals
Each diagram is designed as a teaching anchor. Before students solve the MCQs, ask them to point to the relevant part of the diagram: denominator, multiplier, digit column, fixed fee, group weight, time cycle, conversion ladder, area boundary, graph scale, grid count or workflow branch.
| Diagram type | Student question |
|---|---|
| Part-whole diagram | What is the whole? What changed? |
| Multiplier diagram | Do I multiply forward or divide backward? |
| Weighted average balance | Which group has more weight? |
| Fixed fee + variable cost | What is paid once, and what is paid per unit? |
| Workflow diagram | When is the condition checked, and which branch is followed? |