NBT Mathematics: Number Sense and Calculations Flashcards

An investment of R8 000 earns compound interest at 12% per annum, compounded monthly. What is the effective annual interest rate, correct to two decimal places?

Answer: 12.68%

The effective annual rate formula is (1 + i/n)^n − 1, where i = 0.12 and n = 12 (monthly compounding). So (1 + 0.01)^12 − 1 = (1.01)^12 − 1 ≈ 1.126825 − 1 = 0.126825 ≈ 12.68%. Option B (12.36%) is the result of semi-annual compounding, a common distractor.

If log₂(x) + log₂(x − 6) = 4, what is the value of x?

Answer: 8

Combine the logs: log₂[x(x − 6)] = 4, so x(x − 6) = 2⁴ = 16. This gives x² − 6x − 16 = 0, which factors as (x − 8)(x + 2) = 0. The solutions are x = 8 or x = −2. Since the argument of a logarithm must be positive, x = −2 is rejected. Therefore x = 8.

Simplify: (√5 + √3)² − (√5 − √3)²

Answer: 4√15

Expand each square: (√5 + √3)² = 5 + 2√15 + 3 = 8 + 2√15 and (√5 − √3)² = 5 − 2√15 + 3 = 8 − 2√15. Subtracting: (8 + 2√15) − (8 − 2√15) = 4√15. Alternatively, recognise the difference-of-squares identity: (a+b)² − (a−b)² = 4ab, where a = √5 and b = √3, giving 4·√5·√3 = 4√15.

Item A's price is increased by 25% and then discounted by 25%. Item B's price is increased by 40% and then discounted by 30%. Comparing the net percentage change for each item from the original price, which statement is correct?

Answer: Item A decreases by more, with a net change 4.25 percentage points lower than Item B

Item A: 1.25 × 0.75 = 0.9375, a net decrease of 6.25%. Item B: 1.40 × 0.70 = 0.98, a net decrease of 2%. Item A's loss (6.25%) exceeds Item B's loss (2%) by exactly 4.25 percentage points. The key insight is that the larger the symmetric swing, the greater the net loss.

Evaluate, giving the answer in scientific notation: (2.4 × 10⁵) × (3.1 × 10⁻³) ÷ (1.2 × 10⁴)

Answer: 6.2 × 10⁻²

Separate the coefficients and powers of 10. Coefficients: (2.4 × 3.1) ÷ 1.2 = 7.44 ÷ 1.2 = 6.2. Powers of 10: 10⁵ × 10⁻³ ÷ 10⁴ = 10^(5−3−4) = 10⁻². Final answer: 6.2 × 10⁻². A common error is mishandling the division of the power of 10, adding instead of subtracting the exponent.

Given that 4^x = 8, determine the value of 2^(3x − 1).

Answer: 8√2

First, solve for x: 4^x = 8 → (2²)^x = 2³ → 2^(2x) = 2³ → 2x = 3 → x = 3/2. Now substitute: 2^(3·(3/2) − 1) = 2^(9/2 − 1) = 2^(7/2) = 2³ × 2^(1/2) = 8√2. Option C (16 = 2⁴) is a common error from incorrectly computing 3x − 1 as 4 rather than 7/2.