NBT Mathematics: Number Sense and Calculations Flashcards

A mixture of alcohol and water is in the ratio 3:5 by volume. If 12 litres of pure alcohol are added, the new ratio of alcohol to water becomes 1:1. What is the volume of water in the final mixture?

Answer: 30 litres

Let the original quantities be 3k litres alcohol and 5k litres water. After adding 12 litres of alcohol, the ratio is 1:1, so 3k + 12 = 5k. Solving gives k = 6. The water volume never changes, so water = 5k = 5 × 6 = 30 litres. Note: the total final volume is 60 litres (30 alcohol + 30 water), confirming the 1:1 ratio.

A sum of money is invested at 8% per annum simple interest. After how many complete years will the accumulated amount first exceed twice the original principal?

Answer: 13 years

Using the simple interest formula A = P(1 + rt), we need A > 2P, i.e. P(1 + 0.08t) > 2P. Dividing by P: 1 + 0.08t > 2, so 0.08t > 1, giving t > 12.5. Since we need complete years and the amount only reaches exactly double at t = 12.5 years, the first complete year for which the amount exceeds twice the principal is t = 13 years.

Evaluate: (3.6 × 10⁻⁴) ÷ (1.2 × 10⁻⁷)

Answer: 3 × 10³

Divide the coefficients and apply the law of exponents separately: (3.6 ÷ 1.2) = 3, and 10⁻⁴ ÷ 10⁻⁷ = 10^(−4 − (−7)) = 10^(−4+7) = 10³. Therefore the result is 3 × 10³ = 3 000. A common error is adding the exponents instead of subtracting, or mishandling the double negative in the exponent.

What is the smallest positive integer by which 252 must be multiplied so that the product is a perfect square?

Answer: 7

Prime factorise 252: 252 = 2² × 3² × 7¹. For a number to be a perfect square, every prime factor must appear an even number of times. Here 2² and 3² already have even exponents, but 7 appears only once (odd exponent). Multiplying by 7 gives 252 × 7 = 1764 = 2² × 3² × 7² = 42², a perfect square. So the smallest multiplier is 7.

A retailer marks up goods by 35% above cost price, then offers two successive discounts: first 10%, then a further 8%. What is the net percentage profit or loss on the original cost price? (Round to one decimal place.)

Answer: 11.8% profit

The overall multiplier is 1.35 × 0.90 × 0.92. Calculate step by step: 1.35 × 0.90 = 1.215, then 1.215 × 0.92 = 1.1178. This means the final selling price is 111.78% of cost price, giving a net profit of approximately 11.8%. The common error is adding/subtracting percentages directly (35% − 10% − 8% = 17%), which ignores the compounding effect of successive percentage changes.

The sum of the first n terms of a sequence is given by Sₙ = 3n² − n. What is the value of the 10th term of the sequence?

Answer: 56

The nth term is found using Tₙ = Sₙ − Sₙ₋₁. Calculate S₁₀ = 3(100) − 10 = 290, and S₉ = 3(81) − 9 = 243 − 9 = 234. Therefore T₁₀ = 290 − 234 = 56. Alternatively, derive the general term: Tₙ = 3n² − n − [3(n−1)² − (n−1)] = 6n − 4, so T₁₀ = 60 − 4 = 56. The distractor 290 is S₁₀ (the sum, not the term).