Question 1

A number is divisible by 5 if its last digit is 0 or 5. How many two-digit numbers are divisible by 5?

Accepted Answer

Step 1: Two-digit numbers range from 10 to 99. Step 2: Numbers divisible by 5 must end in 0 or 5. Step 3: Numbers ending in 0: 10, 20, 30, 40, 50, 60, 70, 80, 90 = 9 numbers. Step 4: Numbers ending in 5: 15, 25, 35, 45, 55, 65, 75, 85, 95 = 9 numbers. Step 5: Total = 9 + 9 = 18 two-digit numbers divisible by 5.

Question 2

If 2/5 of a number is 48, what is the number?

Accepted Answer

Step 1: Let the number be x. Step 2: Set up equation: (2/5)x = 48. Step 3: Multiply both sides by 5/2: x = 48 × (5/2) = 240/2 = 120. Therefore, the number is 120.

Question 3

Which number is divisible by 11?

Accepted Answer

Step 1: A number is divisible by 11 if the difference between the sum of digits in odd positions and the sum of digits in even positions is either 0 or divisible by 11. Step 2: For 5742: Odd positions (right to left): 2 + 7 = 9. Even positions: 4 + 5 = 9. Difference = 9 - 9 = 0. Step 3: Since difference is 0, 5742 is divisible by 11. Step 4: Verify: 5742 ÷ 11 = 522. ✓

Question 4

Which of the following numbers is divisible by 3?

Accepted Answer

Step 1: A number is divisible by 3 if the sum of its digits is divisible by 3. Step 2: For 4527: digit sum = 4 + 5 + 2 + 7 = 18. Step 3: 18 ÷ 3 = 6. ✓ Therefore, 4527 is divisible by 3. Step 4: Verify: 4527 ÷ 3 = 1509. ✓

Question 5

Simplify: 3² + 4² − 5 × 2

Accepted Answer

Step 1: Calculate 3² = 9. Step 2: Calculate 4² = 16. Step 3: Calculate 5 × 2 = 10. Step 4: Substitute: 9 + 16 − 10 = 15.

Question 6

Express 0.625 as a fraction in its simplest form.

Accepted Answer

Step 1: Write 0.625 as 625/1000. Step 2: Find GCD(625, 1000) = 125. Step 3: Divide both by 125: 625÷125 = 5 and 1000÷125 = 8. Therefore, 0.625 = 5/8.

Question 7

Evaluate: 60 ÷ 5 + 4 × 3 − 2

Accepted Answer

Step 1: Apply BODMAS — Division and Multiplication first. 60 ÷ 5 = 12 and 4 × 3 = 12. Step 2: Rewrite as 12 + 12 − 2. Step 3: Addition and Subtraction (left to right): 12 + 12 = 24, then 24 − 2 = 22.

Question 8

What is 40% of 250?

Accepted Answer

Step 1: Convert percentage to decimal: 40% = 0.40. Step 2: Multiply: 0.40 × 250 = 100.

Question 9

Find the next number in the series: 2, 6, 12, 20, 30, ?

Accepted Answer

Step 1: Identify the pattern by finding differences. First differences: 6-2=4, 12-6=6, 20-12=8, 30-20=10. Step 2: The first differences form an arithmetic sequence (4, 6, 8, 10). Step 3: Next first difference = 12. Step 4: Next number = 30 + 12 = 42.

Question 10

A number is divisible by 4 if its last two digits form a number divisible by 4. Which of the following is NOT divisible by 4?

Accepted Answer

Step 1: For divisibility by 4, check only the last two digits. Step 2: For 3216: last two digits are 16; 16 ÷ 4 = 4. ✓ Divisible by 4. Step 3: For 5428: last two digits are 28; 28 ÷ 4 = 7. ✓ Divisible by 4. Step 4: For 7634: last two digits are 34; 34 ÷ 4 = 8.5. ✗ NOT divisible by 4. Step 5: For 8920: last two digits are 20; 20 ÷ 4 = 5. ✓ Divisible by 4. Therefore, 7634 is NOT divisible by 4.

Question 11

Which of the following numbers is divisible by both 6 and 8?

Accepted Answer

Step 1: A number is divisible by 6 if it is divisible by both 2 and 3. Step 2: A number is divisible by 8 if its last three digits form a number divisible by 8. Step 3: For divisibility by both 6 and 8, the number must be divisible by LCM(6,8) = 24. Step 4: Check 264: 264 ÷ 24 = 11. Therefore, 264 is divisible by both 6 and 8.

Question 12

Simplify: (75 × 4) ÷ (25 − 15) + 12

Accepted Answer

Step 1: Calculate the numerator: 75 × 4 = 300. Step 2: Calculate the denominator: 25 − 15 = 10. Step 3: Divide: 300 ÷ 10 = 30. Step 4: Add: 30 + 12 = 42.

Question 13

What is the LCM of 24, 36, and 48?

Accepted Answer

Step 1: Prime factorise each number. 24 = 2³ × 3, 36 = 2² × 3², 48 = 2⁴ × 3. Step 2: LCM = highest power of each prime = 2⁴ × 3² = 16 × 9 = 144.

Question 14

What is the missing number in the series: 5, 10, 20, 40, ?, 160?

Accepted Answer

Step 1: Identify the pattern by examining ratios. 10÷5=2, 20÷10=2, 40÷20=2. Step 2: Each term is double the previous term (geometric sequence with ratio 2). Step 3: Missing term = 40 × 2 = 80. Step 4: Verify: 80 × 2 = 160 ✓

Question 15

If 2^(x+1) × 2^(x-1) = 256, find the value of x.

Accepted Answer

Step 1: Use the law of indices: 2^(x+1) × 2^(x-1) = 2^((x+1)+(x-1)) = 2^(2x). Step 2: Rewrite 256 as a power of 2: 256 = 2^8. Step 3: Equate exponents: 2^(2x) = 2^8, so 2x = 8, therefore x = 4.

Question 16

Rationalize: 1/(√7 - √5). Express in the form (a√7 + b√5)/c where a, b, c are integers.

Accepted Answer

Step 1: Multiply numerator and denominator by the conjugate (√7 + √5). Step 2: Numerator = 1 × (√7 + √5) = √7 + √5. Step 3: Denominator = (√7 - √5)(√7 + √5) = 7 - 5 = 2. Step 4: Result = (√7 + √5)/2, so a = 1, b = 1, c = 2.

Question 17

Simplify: (3^(1/2) × 3^(1/3)) / 3^(1/6)

Accepted Answer

Step 1: Apply product rule in numerator: 3^(1/2) × 3^(1/3) = 3^(1/2 + 1/3) = 3^(3/6 + 2/6) = 3^(5/6). Step 2: Apply quotient rule: 3^(5/6) / 3^(1/6) = 3^(5/6 - 1/6) = 3^(4/6) = 3^(2/3). Step 3: Express as ∛(3²) = ∛9.

Question 18

If √(x + 5) = 7, find x.

Accepted Answer

Step 1: Square both sides: (√(x + 5))² = 7². Step 2: Simplify: x + 5 = 49. Step 3: Solve for x: x = 49 - 5 = 44.

Question 19

Simplify: (2^3)^(2/3) × (2^2)^(1/2)

Accepted Answer

Step 1: Apply power rule (a^m)^n = a^(mn): (2^3)^(2/3) = 2^(3 × 2/3) = 2^2 = 4. Step 2: Simplify second term: (2^2)^(1/2) = 2^(2 × 1/2) = 2^1 = 2. Step 3: Multiply: 4 × 2 = 8.

Question 20

If 3/7 of a tank is filled with water and the tank holds 280 litres, how many litres of water are in the tank?

Accepted Answer

Step 1: Water in tank = (3/7) × 280. Step 2: Calculate: (3 × 280)/7 = 840/7 = 120 litres. Therefore, there are 120 litres of water in the tank.

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