Maths interleaved practice — Phase 1 (Maths)

Instructions: Solve all problems. Each problem may use a different method — read carefully and pick the right tool. Show full working. Total: 42 marks.

Evaluate using BODMAS: $48 \div (6 + 2) \times 3 - [5 \times (4 - 2)]$ . (4 marks)
Find the HCF of $84$ and $120$ using the Euclidean algorithm. (4 marks)
A shopkeeper bought a watch for $\$ 800 $and sold it at a$ 15%$ profit. Find the selling price. (4 marks)
Write the place value of the digit $7$ in the number $4\,73\,52\,168$ (Indian system). (3 marks)
Convert $\dfrac{3}{8}$ into a decimal and then into a percentage. (4 marks)
List all the factors of $36$ as factor pairs, and state whether $36$ is prime. (4 marks)
Simplify and write as a mixed number: $\dfrac{2}{3} + \dfrac{5}{6}$ . (4 marks)
Divide $\$ 960 $between two people in the ratio$ 5 : 7$. (4 marks)
Find the LCM of $12$ and $18$ using prime factorization, then verify that $\text{HCF} \times \text{LCM} = 12 \times 18$ . (5 marks)
Evaluate $|{-9}| + |4| - |{-3}|$ and mark the result's position relative to $0$ on the number line. State the type of angle measuring $135^\circ$ . (6 marks)

Q1. (Subtopic: Order of operations / BODMAS) Brackets first: $(6+2)=8$ , $(4-2)=2$ . $48 \div 8 \times 3 - [5 \times 2] = 6 \times 3 - 10 = 18 - 10 = \boxed{8}$ . Why: Looks like simple arithmetic but the nested brackets force strict BODMAS ordering — division/multiplication left-to-right.

Q2. (Subtopic: HCF — Euclidean algorithm) $120 = 84 \times 1 + 36$ $84 = 36 \times 2 + 12$ $36 = 12 \times 3 + 0$ HCF $= \boxed{12}$ . Why: Euclidean algorithm chosen over prime factorization for efficiency with two numbers; remainder vocabulary reused.

Q3. (Subtopic: Profit, loss, percentage application) Profit $= 15\%$ of $800 = \dfrac{15}{100}\times 800 = 120$ . SP $= 800 + 120 = \boxed{\$ 920}$. Why: Percentage-of-a-quantity applied to a profit context; must add to cost price, not just compute %.

Q4. (Subtopic: Place value — Indian system) $4\,73\,52\,168$ . The $7$ sits in the ten-lakhs place. Place value $= 7 \times 10\,00\,000 = \boxed{70\,00\,000}$ (seventy lakh). Why: Tests Indian place-value grouping (lakhs/crores), distinct from comma-grouping in decimals.

Q5. (Subtopic: Converting fractions ↔ decimals ↔ percentages) $\dfrac{3}{8} = 3 \div 8 = 0.375$ . As percent: $0.375 \times 100 = \boxed{37.5\%}$ . Why: Requires the conversion chain, not an operation — choose long division then ×100.

Q6. (Subtopic: Factors & primes) Factor pairs of $36$ : $1\times36,\ 2\times18,\ 3\times12,\ 4\times9,\ 6\times6$ . All factors: $1,2,3,4,6,9,12,18,36$ . $36$ has more than two factors, so it is not prime (it is composite). Why: Distinguishes "list factors" from "test primality"; $6\times6$ catches the perfect-square pair.

Q7. (Subtopic: Addition of fractions) LCD of $3,6$ is $6$ : $\dfrac{2}{3}=\dfrac{4}{6}$ . $\dfrac{4}{6}+\dfrac{5}{6}=\dfrac{9}{6}=\dfrac{3}{2}=\boxed{1\tfrac{1}{2}}$ . Why: Needs common denominator then conversion of improper to mixed number.

Q8. (Subtopic: Dividing in a ratio) Total parts $=5+7=12$ . One part $=960\div12=80$ . Shares: $5\times80=\boxed{\$ 400} $and$ 7\times80=\boxed{$560} $. **Why:** Ratio division (sum of parts) — not unitary direct proportion; check sum$ 400+560=960$.

Q9. (Subtopic: LCM via prime factorization + HCF×LCM relation) $12=2^2\times3$ , $18=2\times3^2$ . LCM $=2^2\times3^2=36$ . HCF $=2\times3=6$ . Check: $\text{HCF}\times\text{LCM}=6\times36=216$ and $12\times18=216$ . ✓ Why: Prime factorization gives both HCF and LCM; the identity verifies the result.

Q10. (Subtopic: Absolute value + number line + types of angles) $|{-9}|+|4|-|{-3}| = 9+4-3 = \boxed{10}$ . On the number line, $10$ lies $10$ units to the right of $0$ (positive direction). An angle of $135^\circ$ is between $90^\circ$ and $180^\circ$ → obtuse angle. Why: Interleaves modulus arithmetic (always non-negative output), number-line interpretation, and angle classification in one problem to force topic-switching.

[
  {"claim":"Q2 HCF(84,120)=12","code":"import math\nresult = (math.gcd(84,120)==12)"},
  {"claim":"Q9 LCM(12,18)=36 and HCF*LCM=12*18","code":"import math\nh=math.gcd(12,18)\nl=12*18//h\nresult = (l==36 and h*l==12*18)"},
  {"claim":"Q10 |-9|+|4|-|-3|=10","code":"result = (abs(-9)+abs(4)-abs(-3))==10"}
]