Mr Roche's Christmas Test Practice Questions

Natural Numbers · Sets · Fractions · Decimals · Percentages · Algebra · Area & Perimeter

Question 1 — Natural Numbers

1. Write down all the factors of \(36\).

2. Express \(84\) as a product of its prime factors.

3. Find the LCM of \(8\) and \(12\).

4. Find the HCF of \(45\) and \(60\).

5. Evaluate: \(3^3 + 2^4 - 5\)
1. Factors of \(36\) are \(1,2,3,4,6,9,12,18,36\).

2. \(84 = 2 \times 42 = 2^2 \times 21 = 2^2 \times 3 \times 7\).

3. \(8 = 2^3,\; 12 = 2^2 \times 3\). So \(\text{LCM} = 2^3 \times 3 = 24\).

4. \(45 = 3^2 \times 5,\; 60 = 2^2 \times 3 \times 5\). Common prime factors: \(3 \times 5 = 15\). So \(\text{HCF} = 15\).

5. \(3^3 = 27,\; 2^4 = 16\). \(27 + 16 - 5 = 43 - 5 = 38\).

Question 2 — Integer Operations

1. Evaluate: \(18 - 4 \times 3 + 2\).

2. Evaluate: \((-7) + 12 - (-5)\).

3. Evaluate: \(30 \div (2 \times 3)\).
1. Do the multiplication first: \(4 \times 3 = 12\). \(18 - 12 + 2 = 6 + 2 = 8\).

2. \((-7) + 12 - (-5) = -7 + 12 + 5\). \(-7 + 12 = 5\), then \(5 + 5 = 10\).

3. Inside the brackets: \(2 \times 3 = 6\). \(30 \div 6 = 5\).

Question 3 — Sets & Venn Diagrams

Let \(U = \{1,2,3,4,5,6,7,8,9\}\), \(A = \{1,3,5,7\}\), \(B = \{2,3,6,7\}\).

1. Find \(A \cap B\).

2. Find \(A \cup B\).

3. Find \(A'\).
1. \(A \cap B\) = elements in both sets: \(A \cap B = \{3,7\}\).

2. \(A \cup B\) = elements in \(A\) or \(B\) (or both): \(\{1,2,3,5,6,7\}\).

3. \(A'\) = elements in \(U\) but not in \(A\): \(A' = \{2,4,6,8,9\}\).

Question 4 — Fractions

1. Simplify: \(\dfrac{18}{30}\).

2. Add: \(\dfrac{2}{3} + \dfrac{5}{9}\).

3. Subtract: \(\dfrac{7}{8} - \dfrac{1}{4}\).

4. Multiply: \(\dfrac{3}{5} \times \dfrac{10}{9}\).

5. Divide: \(\dfrac{4}{7} \div \dfrac{2}{3}\).
1. \(\dfrac{18}{30} = \dfrac{18 \div 6}{30 \div 6} = \dfrac{3}{5}\).

2. \(\dfrac{2}{3} = \dfrac{6}{9}\). \(\dfrac{6}{9} + \dfrac{5}{9} = \dfrac{11}{9} = 1\dfrac{2}{9}\).

3. \(\dfrac{7}{8} - \dfrac{1}{4} = \dfrac{7}{8} - \dfrac{2}{8} = \dfrac{5}{8}\).

4. \(\dfrac{3}{5} \times \dfrac{10}{9} = \dfrac{30}{45} = \dfrac{2}{3}\).

5. \(\dfrac{4}{7} \div \dfrac{2}{3} = \dfrac{4}{7} \times \dfrac{3}{2} = \dfrac{12}{14} = \dfrac{6}{7}\).

Question 5 — Decimals

1. Write \(0.375\) as a fraction in simplest form.

2. Convert \(\dfrac{4}{5}\) to a decimal.

3. Convert \(0.46\) to a percentage.

4. Convert \(27\%\) to a decimal.

5. Evaluate: \(4.6 + 3.27 - 1.9\).

6. Round \(7.486\) to \(2\) decimal places.

7. Round \(0.034879\) to \(2\) significant figures.
1. \(0.375 = \dfrac{375}{1000} = \dfrac{3}{8}\).

2. \(\dfrac{4}{5} = 0.8\).

3. \(0.46 \times 100 = 46\%\).

4. \(27\% = \dfrac{27}{100} = 0.27\).

5. \(4.6 + 3.27 = 7.87\). \(7.87 - 1.9 = 5.97\).

6. \(7.486\) to \(2\) decimal places: look at the third decimal place (6). \(7.486 \approx 7.49\) (to \(2\) d.p.).

7. \(0.034879\): the first two significant figures are 3 and 4, the next digit is 8 (so we round up). \(0.034879 \approx 0.035\) (to \(2\) significant figures).

Question 6 — Percentages

1. Find \(15\%\) of €240.

2. A laptop costs €800 before VAT. VAT is \(23\%\). Find the total cost including VAT.

3. A shop buys a jacket for €45 and sells it for €60. Find the percentage profit.

4. Reduce €56 by \(12\%\).
1. \(15\% = 0.15\). \(0.15 \times 240 = 36\). So the answer is €36.

2. VAT: \(23\% \text{ of } 800 = 0.23 \times 800 = 184\). Total cost \(= 800 + 184 = 984\). The laptop costs €984 including VAT.

3. Profit \(= 60 - 45 = 15\). Percentage profit \(= \dfrac{15}{45} \times 100 = 33\tfrac{1}{3}\%\).

4. \(12\% \text{ of } 56 = 0.12 \times 56 = 6.72\). New amount \(= 56 - 6.72 = 49.28\). So the reduced price is €49.28.

Question 7 — Algebra

1. If \(x = -3\), find \(4x - 2x + 7\).

2. Simplify: \(7y + 4y - 9y + 12\).

3. Expand: \(5(2x - 3)\).

4. Expand the binomial: \((x + 4)(x - 2)\).
1. \(4x - 2x + 7 = (4 - 2)x + 7 = 2x + 7\). For \(x = -3\): \(2(-3) + 7 = -6 + 7 = 1\).

2. \(7y + 4y - 9y + 12 = (7 + 4 - 9)y + 12 = 2y + 12\).

3. \(5(2x - 3) = 5 \cdot 2x - 5 \cdot 3 = 10x - 15\).

4. \((x + 4)(x - 2) = x^2 - 2x + 4x - 8 = x^2 + 2x - 8\).

Question 8 — Area & Perimeter

1. A rectangle measures \(12\text{ cm}\) by \(7\text{ cm}\). (a) Find its perimeter. (b) Find its area.

2. A triangle has base \(10\text{ cm}\) and height \(6\text{ cm}\). Find its area.
1(a). Perimeter of a rectangle \(= 2(\text{length} + \text{width})\). \(2(12 + 7) = 2 \times 19 = 38\text{ cm}\).

1(b). Area \(= \text{length} \times \text{width} = 12 \times 7 = 84\text{ cm}^2\).

2. Area of a triangle \(= \dfrac{1}{2} \times \text{base} \times \text{height}\). \(\dfrac{1}{2} \times 10 \times 6 = 5 \times 6 = 30\text{ cm}^2\).