Level 2 — RecallArithmetic & Number Systems

Arithmetic & Number Systems

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Level 2 — Recall (definitions, standard textbook problems, short derivations) Time: 30 minutes Total Marks: 40

Answer all questions. Show working where required.

Q1. Fill in the blanks. (4 marks) (a) The smallest whole number is ______. (b) In the number 45,08,317, the digit in the lakhs place is ______. (c) In a division, dividend = divisor × quotient + ______. (d) The absolute value of $-9$ is ______.

Q2. Evaluate using BODMAS: (4 marks) $36 \div 6 + 4 \times (7 - 2)^2$

Q3. (a) List all the factors of 48. (2 marks) (b) Write down all the prime numbers between 20 and 40. (2 marks)

Q4. Find the prime factorization of 180 using a factor tree, and write the answer in index (power) form. (4 marks)

Q5. Find the HCF and LCM of 24 and 36 by the prime factorization method. Verify that $\text{HCF} \times \text{LCM} = 24 \times 36$ . (5 marks)

Q6. Simplify and express as a single fraction in lowest terms: (4 marks) $\frac{2}{3} + \frac{5}{6} - \frac{1}{4}$

Q7. Convert: (4 marks) (a) The fraction $\dfrac{3}{5}$ into a decimal and a percentage. (b) $0.24$ into a fraction in lowest terms.

Q8. A shopkeeper buys a bag for ₹800 and sells it for ₹920. (4 marks) (a) Find the profit. (b) Find the profit percentage.

Q9. Divide ₹560 between two people in the ratio $3 : 4$ . (3 marks)

Q10. If 5 pens cost ₹75, find the cost of 8 pens using the unitary method. (3 marks)

End of paper

Answer keyMark scheme & solutions

Q1. (4 marks, 1 each) (a) 0 — whole numbers start at 0. (b) 5 — place values from right: 7(units),1(tens),3(hundreds),8(thousands),0(ten-thousands),5(lakhs),4(ten-lakhs). (c) remainder — the division algorithm: dividend = divisor × quotient + remainder. (d) 9 — modulus is distance from 0, always non-negative.

Q2. (4 marks) Brackets first: $7-2 = 5$ (1) Power: $5^2 = 25$ (1) Division & multiplication: $36\div6 = 6$ , $4\times25 = 100$ (1) Addition: $6 + 100 = \mathbf{106}$ (1)

Q3. (4 marks) (a) Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 (2 marks; −1 if one or two missing/extra). (b) Primes between 20 and 40: 23, 29, 31, 37 (2 marks; must all be present, no composites).

Q4. (4 marks) $180 = 2 \times 90 = 2 \times 2 \times 45 = 2 \times 2 \times 3 \times 15 = 2\times2\times3\times3\times5$ (2 marks for correct tree/breakdown). Index form: $180 = 2^2 \times 3^2 \times 5$ (2 marks).

Q5. (5 marks) $24 = 2^3 \times 3$ , $\;36 = 2^2 \times 3^2$ (1 mark). HCF = product of lowest powers common $= 2^2 \times 3 = \mathbf{12}$ (1). LCM = product of highest powers $= 2^3 \times 3^2 = \mathbf{72}$ (1). Verify: $\text{HCF}\times\text{LCM} = 12 \times 72 = 864$ (1). Product $= 24 \times 36 = 864$ . Equal ✓ (1).

Q6. (4 marks) LCM of 3, 6, 4 = 12 (1). $\frac{2}{3}=\frac{8}{12},\ \frac{5}{6}=\frac{10}{12},\ \frac{1}{4}=\frac{3}{12}$ (1). $\frac{8+10-3}{12} = \frac{15}{12}$ (1). Simplify: $\frac{15}{12} = \mathbf{\dfrac{5}{4}}$ (or $1\tfrac14$ ) (1).

Q7. (4 marks) (a) $\frac{3}{5} = 0.6$ (1); as percentage $= 0.6 \times 100 = \mathbf{60\%}$ (1). (b) $0.24 = \frac{24}{100} = \frac{6}{25}$ (2; dividing by HCF 4).

Q8. (4 marks) (a) Profit $= 920 - 800 = \mathbf{₹120}$ (2). (b) Profit % $= \frac{120}{800}\times100 = \mathbf{15\%}$ (2).

Q9. (3 marks) Total parts $= 3+4 = 7$ (1). One part $= 560\div7 = 80$ (1). Shares: $3\times80 = \mathbf{₹240}$ and $4\times80 = \mathbf{₹320}$ (1).

Q10. (3 marks) Cost of 1 pen $= 75\div5 = ₹15$ (1). Cost of 8 pens $= 15\times8 = \mathbf{₹120}$ (2).

[
  {"claim":"Q2 BODMAS equals 106","code":"result = (36//6 + 4*(7-2)**2) == 106"},
  {"claim":"Q5 HCF*LCM equals product 864","code":"h=gcd(24,36); l=lcm(24,36); result = (h==12 and l==72 and h*l==24*36==864)"},
  {"claim":"Q6 fraction sum equals 5/4","code":"result = (Rational(2,3)+Rational(5,6)-Rational(1,4)) == Rational(5,4)"},
  {"claim":"Q8 profit percent equals 15","code":"result = (Rational(920-800,800)*100) == 15"},
  {"claim":"Q7b decimal 0.24 equals 6/25","code":"result = Rational(24,100) == Rational(6,25)"}
]