IB MYP Mathematics: Number sense, estimation, exponents and proportion across MYP (MYP Mathematics) – Practice Questions

These are original questions written for Marlbridge, for revision and practice on this content. They are not reproduced past-paper questions, and they do not replicate the exam’s exact structure, question count or mark tariffs – the IB holds copyright in its own papers. Use these alongside the official past papers available through your school or the IB store.

These questions cover the Number rows of the numerical and abstract reasoning branch in the IB MYP Mathematics guide (from 2020, first assessment 2022). The marks shown are Marlbridge’s own, for self-marking. Questions 4(b), 11 and 12 use extended or enrichment content and are labelled. Explanations are in the study guide and revision notes. See also the MYP Mathematics hub and checklist.

Questions

1. Consider the numbers 14, -6, 5/7, √45, 1.2 and √121. (a) List the numbers that are integers. [2] (b) Write down the number that is irrational. [1]

2. Write 0.4777… (the 7 recurs) as a fraction in its simplest form. [3]

3. Two bus routes leave a depot every 84 minutes and every 120 minutes. Both leave together at 06:00. (a) Write 84 and 120 as products of prime factors. [2] (b) Find the time of their next joint departure. [2]

4. (a) Simplify √98 - √32. [2] (b) (Enrichment) Rationalize the denominator of 10/√5. [1] (c) Expand and simplify (3 + √5)(3 - √5). [2]

5. (a) Write 5⁻² × 5⁵ ÷ 5² as a single number. [2] (b) Calculate (2.4 × 10⁶) ÷ (6 × 10⁻³), giving the answer in standard form. [2] (c) Write 0.00305 in standard form. [1]

6. By rounding each number to 1 significant figure, estimate (305.8 × 0.0489) ÷ 1.97. Then evaluate it exactly to 3 significant figures and comment on your estimate. [4]

7. (a) List the integers x such that -2 < x ≤ 4. [2] (b) A number line has a closed circle at -3 with a ray going left, and an open circle at 2 with a ray going right. Write the two inequalities this shows. [2]

8. (a) Convert 72 km/h to m/s. [1] (b) Convert a density of 0.9 g/cm³ to kg/m³. [1] (c) A flight leaves a city on UTC+8 at 23:30 and lasts 6 h 40 min. It lands in a city on UTC+1. Find the local arrival time and day. [3]

9. (a) y is directly proportional to x, and y = 28 when x = 8. Find y when x = 18. [2] (b) Five pumps empty a tank in 9 hours. All pumps work at the same rate. Find the time taken by 4 pumps, in hours and minutes. [3]

10. (a) 5000 is invested at 3.5% compound interest per year. Find its value after 4 years, to the nearest cent. [2] (b) Find how much more interest this earns than simple interest at 3.5% over the same 4 years. [3] (c) A jacket costs 156 including 20% tax. Find the price before tax. [1]

11. (Extended) A rectangle has sides 14.2 cm and 6.5 cm, each measured to 1 decimal place. (a) Write the bounds for each side. [2] (b) Find the lower and upper bounds of the area. [2] (c) Find the lower and upper bounds of the perimeter. [2] (d) Give the area to a degree of accuracy that is justified by the bounds. [1]

12. (Extended) (a) Evaluate 32^(3/5). [2] (b) Evaluate log₅ 125. [1] (c) Write log 8 + log 125 as a single number (base 10). [2] (d) Solve 3ˣ = 40, giving x to 3 significant figures. [3]

Answers

1. (a) √121 = 11 [1]. The integers are 14, -6 and √121 [1]. (b) √45 = 3√5 does not terminate or recur, so √45 [1]. [3] Examiner insight: Evaluate roots before classifying. Leaving √121 unevaluated makes it easy to miss that it is an integer.

2. Let x = 0.4777… Then 10x = 4.777… and 100x = 47.777… [1] Subtract: 90x = 43 [1] x = 43/90 [1] [3] Examiner insight: Subtract so that the recurring tails cancel, and cancel the fraction at the end. Check whether the fraction cancels; here 43 is prime, so it does not.

3. (a) 84 = 2² × 3 × 7 [1]; 120 = 2³ × 3 × 5 [1]. (b) LCM = 2³ × 3 × 5 × 7 = 840 minutes [1]. 840 minutes = 14 hours, so the next joint departure is at 20:00 [1]. [4] Examiner insight: Finish in the form the question uses, here a clock time. The number 840 alone does not say when.

4. (a) √98 = 7√2 and √32 = 4√2 [1]. Difference = 3√2 [1]. (b) 10/√5 = 10√5/5 = 2√5 [1]. (c) 9 - 3√5 + 3√5 - 5 [1] = 4 [1]. [5] Examiner insight: Show the simplification of each surd before combining. A decimal approximation for part (a) loses the exact form the question asks for.

5. (a) 5⁻²⁺⁵⁻² = 5¹ [1] = 5 [1]. (b) 2.4 ÷ 6 = 0.4 and 10⁶ ÷ 10⁻³ = 10⁹ [1]. 0.4 × 10⁹ = 4 × 10⁸ [1]. (c) 3.05 × 10⁻³ [1]. [5] Examiner insight: After dividing the numbers you may get 0.4. It is not in standard form until you adjust it so that the first factor is between 1 and 10.

6. Estimate: 300 × 0.05 ÷ 2 [1] = 7.5 [1]. Exact: 305.8 × 0.0489 ÷ 1.97 = 7.59 [1]. The estimate is close, so the exact value is reasonable [1]. [4] Examiner insight: Show the rounded numbers. A bare estimate with no rounded values is hard to credit, and a comment must refer to the two values.

7. (a) Excluding -2 and including 4: -1, 0, 1, 2, 3, 4 [2]. (b) Closed circle with a left ray: x ≤ -3 [1]. Open circle with a right ray: x > 2 [1]. [4] Examiner insight: The circle type decides the inequality symbol. Join the two parts with “or”, not “and”, since no number satisfies both.

8. (a) 72 000 m ÷ 3600 s = 20 m/s [1]. (b) 0.9 g/cm³ = 900 kg/m³ [1], as 1 g/cm³ = 1000 kg/m³. (c) Arrival in departure-city time: 23:30 + 6:40 = 06:10 next day [1]. UTC+1 is 7 hours behind UTC+8 [1]. Local arrival 23:10, the same day as departure [1]. [5] Examiner insight: For time zones, state the day as well as the time. The arrival here is earlier on the clock than the departure, which is a good check.

9. (a) k = 28 ÷ 8 = 3.5 [1]; y = 3.5 × 18 = 63 [1]. (b) Pumps × time is constant: 5 × 9 = 45 [1]. Time = 45 ÷ 4 = 11.25 hours [1] = 11 hours 15 minutes [1]. [5] Examiner insight: Name the type of proportion in words. Converting 0.25 hours to minutes is a separate step that is easy to skip.

10. (a) 5000 × 1.035⁴ [1] = 5737.62 [1]. (b) Compound interest = 737.62 [1]. Simple interest = 5000 × 0.035 × 4 = 700 [1]. Difference = 37.62 [1]. (c) 156 ÷ 1.2 = 130 [1]. [6] Examiner insight: Carry full unrounded values forward and round once. Rounding the multiplier 1.035⁴ early can change the final cent. In (c), dividing by 1.2 is correct because 156 is already the increased price.

11. (a) 14.15 ≤ length < 14.25 [1] and 6.45 ≤ width < 6.55 [1]. (b) Lower area = 14.15 × 6.45 = 91.2675 [1]; upper area = 14.25 × 6.55 = 93.3375 [1]. (c) Lower perimeter = 2(14.15 + 6.45) = 41.2 [1]; upper perimeter = 2(14.25 + 6.55) = 41.6 [1]. (d) Both bounds round to 90 at 1 significant figure, so the area is 90 cm² to 1 s.f. [1]. [7] Examiner insight: Match each bound to the right pair of values. A claim of accuracy is supported by showing that the lower and upper bounds round to the same figure.

12. (a) The fifth root of 32 is 2 [1]; 2³ = 8 [1]. (b) 5³ = 125, so 3 [1]. (c) log 8 + log 125 = log 1000 [1] = 3 [1]. (d) Take logs: x log 3 = log 40 [1]; x = log 40 / log 3 [1] = 3.36 [1]. [8] Examiner insight: In (d) write the exact form log 40 / log 3 before the decimal. Check by testing that 3 raised to 3.36 is close to 40.

Where marks are usually lost

• Leaving a recurring decimal rounded instead of converting it exactly.
• Leaving an LCM answer as a bare number when the question asks for a time.
• Combining unlike surds, such as treating 3√2 + 2√5 as 5√7.
• Reading 5⁻² as -25 instead of 1/25.
• Giving standard form with a first factor of 10 or more, or below 1.
• Using the wrong open or closed circle, or “and” for a two-ray inequality.
• Taking 1.035⁴ as 4 × 1.035, or rounding the multiplier too early.
• Stating a bound without strict and non-strict inequality symbols.
• Failing to comment on whether a result is reasonable in context.

Next steps

• Review the revision notes and the study guide.
• Go to the MYP Mathematics hub and the printable checklist.
• Try all free 10-minute diagnostics.
• Book a free trial class: /trial/.

Official syllabus

IB Mathematics guide, Middle Years Programme (for use from September 2020/January 2021, first assessment 2022), International Baccalaureate Organization: mathematics framework, numerical and abstract reasoning.