Year 9 Mathematics Mock Assessment

Curriculum Levels 4/5: Single-Attempt Diagnostics

Designed by Mr.Shen

Number Knowledge

Teaching Session: Factor Counting & Integer Logic

1. Finding the Total Number of Factors

You don't need to list every factor of a massive number to know how many there are. Use prime factorization and a special shortcut!

  1. Write the number as a product of its prime factors using index form. (e.g., $24 = 2^3 \times 3^1$).
  2. Look ONLY at the powers (the exponents). Here, they are 3 and 1.
  3. Add 1 to each power. ($3+1=4$, and $1+1=2$).
  4. Multiply these new numbers together: $4 \times 2 = 8$. The number 24 has exactly 8 factors!

1. Using prime factorization, calculate exactly how many factors the number 2520 has.

2. Between which two consecutive integers does the value of $\sqrt{115}$ lie?

Detailed Solutions:

Q1 Step-by-Step (Factor Theorem):
  1. Create a prime factor tree for 2520. $2520 = 252 \times 10 = (4 \times 9 \times 7) \times (2 \times 5)$.
  2. Combine into index form: $2^3 \times 3^2 \times 5^1 \times 7^1$.
  3. Extract the powers: 3, 2, 1, and 1.
  4. Add 1 to each power: 4, 3, 2, and 2.
  5. Multiply the results: $4 \times 3 \times 2 \times 2 = 48$. (Answer: C)
Q2 Step-by-Step (Surd Estimation):
  1. Identify the perfect squares immediately below and above 115.
  2. $10^2 = 100$, which is smaller than 115.
  3. $11^2 = 121$, which is larger than 115.
  4. Therefore, $\sqrt{100} < \sqrt{115} < \sqrt{121}$, meaning it lies between 10 and 11. (Answer: B)

Proportions & Percentages

Teaching Session: Reverse Percentages

Working Backwards

If a population increases by 20% and becomes 120, you CANNOT find the original population by subtracting 20% of 120. Percentages don't work backwards like that! Instead, use algebra and multipliers.

The Multiplier Method:
  1. An increase of 20% means the new amount is 120% of the original.
  2. Write this as a decimal multiplier: 1.20.
  3. Set up the equation: $\text{Original} \times 1.20 = \text{New Amount}$.
  4. To find the Original, simply divide: $\text{Original} = \text{New Amount} \div 1.20$.

1. A monetary prize of \$5,200 is split between two people in the ratio of 5:3. What is the exact value of the smaller share?

2. A town's population grew by 18% to reach a total of 2,065,000 people. What was the original population of the town, expressed in standard scientific form?

Detailed Solutions:

Q1 Step-by-Step (Ratios):
  1. Find the total number of "parts" in the ratio: $5 + 3 = 8$ parts.
  2. Find the monetary value of a single part: $\$5200 \div 8 = \$650$.
  3. Calculate the smaller share by multiplying by 3: $\$650 \times 3 = \$1950$. (Answer: B)
Q2 Step-by-Step (Reverse Percentages & Standard Form):
  1. Establish the multiplier for an 18% increase: $100\% + 18\% = 118\%$, which is $1.18$.
  2. Set up the reverse equation: $\text{Original} \times 1.18 = 2,065,000$.
  3. Divide to find the original amount: $2,065,000 \div 1.18 = 1,750,000$.
  4. Convert $1,750,000$ to standard form by placing the decimal after the first digit and counting the jumps (6 jumps): $1.75 \times 10^6$. (Answer: B)

Finance & Fractions

Teaching Session: The Fractional Difference

Finding the "Whole" from a "Difference"

If a question tells you "Sarah ate $\frac{2}{7}$ and Mike ate $\frac{1}{4}$, and Sarah ate 150g more than Mike," how do you find the weight of the whole pizza? You must find what fraction of the pizza that 150g represents.

Step-by-Step Logic:
  1. Common Denominators: Change $\frac{2}{7}$ and $\frac{1}{4}$ to have the same bottom number (28). Sarah ate $\frac{8}{28}$ and Mike ate $\frac{7}{28}$.
  2. Find the Fractional Difference: Subtract Mike's fraction from Sarah's. $\frac{8}{28} - \frac{7}{28} = \frac{1}{28}$.
  3. Equate and Solve: We now know that $\frac{1}{28}$ of the pizza equals exactly 150g. To find the whole pizza ($\frac{28}{28}$), simply multiply 150g by 28.

1. Alice eats $\frac{2}{5}$ of a large competition pizza and Bob eats $\frac{1}{4}$ of the same pizza. If Alice ate exactly 120g more pizza than Bob, what is the total weight of the entire pizza?

2. James invests \$3,000 into a bank account that pays 6% compound interest per year. He leaves the money untouched for exactly 3 years. What is the final total amount in the account?

Algorithmic Resolution:

Q1 Step-by-Step (Fractional Difference):
  1. Convert the fractions $\frac{2}{5}$ and $\frac{1}{4}$ to a common denominator of 20: $\frac{8}{20}$ and $\frac{5}{20}$.
  2. Calculate the difference: $\frac{8}{20} - \frac{5}{20} = \frac{3}{20}$.
  3. Since this $\frac{3}{20}$ difference is equal to 120g, we first find $\frac{1}{20}$ by dividing 120g by 3 (which is 40g).
  4. To find the whole pizza ($\frac{20}{20}$), multiply 40g by 20: $40 \times 20 = 800$g. (Answer: C)
Q2 Step-by-Step (Compound Interest):
  1. Recall the compound interest formula: $A = P(1 + r)^n$.
  2. Identify variables: Principal $P = 3000$, Rate $r = 0.06$, Time $n = 3$.
  3. Calculate: $3000 \times (1.06)^3$.
  4. Execute: $3000 \times 1.191016 = 3573.048$. Round to two decimal places for currency: \$3,573.05. (Answer: B)

Pythagoras Theorem

Teaching Session: Pythagorean Triples Shortcut

Multiples of the Classics

Competitions rarely give you random numbers for geometry. They almost always use "Pythagorean Triples" (sets of 3 whole numbers that perfectly fit $a^2 + b^2 = c^2$). If you memorize the base triples, you can bypass long calculations!

The Base Triples:
  • 3-4-5 (Hypotenuse is 5)
  • 5-12-13 (Hypotenuse is 13)
  • 8-15-17 (Hypotenuse is 17)
  • 7-24-25 (Hypotenuse is 25)
The Multiplier Trick:

If you multiply a base triple by ANY whole number, it creates a new valid triple. For example, multiply the 3-4-5 triple by 2, and you get 6-8-10. Multiply it by 10, and you get 30-40-50!

1. Triangle ABC is a right-angled triangle with legs AB = 7 cm and BC = 24 cm. A second right-angled triangle, ADC, shares the hypotenuse AC. If leg AD = 15 cm, what is the length of the remaining leg DC?

2. Which of the following sets of numbers forms a valid Pythagorean Triple containing the number 12?

Algorithmic Resolution:

Q1 Step-by-Step (Compound Right Triangles):
  1. First, solve Triangle ABC to find the shared hypotenuse AC. The legs are 7 and 24. This matches the 7-24-25 Pythagorean triple, so AC = 25 cm.
  2. Focus on Triangle ADC. The hypotenuse $c = 25$, and one leg $a = 15$. We need the other leg $b$ (which is DC).
  3. Use rearranged Pythagoras: $b^2 = c^2 - a^2$.
  4. $b^2 = 25^2 - 15^2 = 625 - 225 = 400$.
  5. $\sqrt{400} = 20$. (Alternatively, recognize that 15 and 25 are multiples of the 3-4-5 triple, where $3 \times 5=15$ and $5 \times 5=25$, meaning the missing leg is $4 \times 5=20$). (Answer: B)
Q2 Step-by-Step (Pythagorean Triples):
  1. Test (A): Is $5^2 + 12^2 = 13^2$? $25 + 144 = 169$. Yes (This is a base primitive triple).
  2. Test (B): Is $9^2 + 12^2 = 15^2$? $81 + 144 = 225$. Yes (This is the 3-4-5 triple scaled by 3).
  3. Test (C): Is $12^2 + 16^2 = 20^2$? $144 + 256 = 400$. Yes (This is the 3-4-5 triple scaled by 4).
  4. Since all options work, the answer is D. (Answer: D)