Number and Operations — Free IB MYP4 Mathematics (Standard) Questions

1QuestionClassifying Rational and Irrational NumbersConcept Practice

4 marks~6 minCriterion A

A structural engineer models the natural frequency of a suspension bridge cable using the function

f(x) = \sqrt{x}

, where

x

represents cable tension measured in kilonewtons (kN).

The engineer notes that only rational frequency values allow synchronisation with the bridge's monitoring system.

a

State the definition of a rational number. [1]

b

Explain why

f(x) = \sqrt{x}

produces an irrational value for most positive values of

x

. [1]

c

Identify two values of

x

for which

f(x)

is rational, and justify whether the engineer can always find a suitable tension value. [2]

Solutions

2QuestionSimple and Compound InterestConcept Practice

2 marks~3 minCriterion A

A property developer is fencing a triangular plot of land. The plot is right-angled, and surveyors have confirmed that the two shorter boundary lengths satisfy

a^2 = 441 \text{ m}^2

and

b^2 = 1\,600 \text{ m}^2

.

a

Calculate the area of the square on the longest boundary. [1]

b

The developer claims the longest boundary exceeds 65 m, so a more expensive fencing grade is required. Justify whether this claim is correct, using your result from part (a). [1]

Solutions

3QuestionMultiplication and Division of FractionsConcept Practice

2 marks~3 minCriterion A

A nutritionist is scaling down a granola recipe. The full batch requires

\dfrac{3}{4}

cup of rolled oats, but she is making

\dfrac{1}{3}

of the batch.

a

Calculate the amount of rolled oats needed. [1]

b

Standard dry measuring cups come in sizes

\dfrac{1}{4}

cup and

\dfrac{1}{8}

cup. Justify whether the calculated amount can be measured exactly using these cups. [1]

Solutions

4QuestionLaws of Exponents for Integer PowersConcept Practice

2 marks~3 minCriterion B

Explain why

a^m \cdot a^n = a^{m+n}

holds true, using repeated multiplication in your reasoning. [1]

Support your explanation with the example

2^3 \cdot 2^2

, showing each repeated-multiplication step clearly. [1]

Solutions

5QuestionConverting Between FormsConcept Practice

3 marks~5 minCriterion B

A spacecraft's distance from Earth is recorded using powers of

10

.

Power of $10$	$10^3$	$10^2$	$10^1$	$10^0$
Decimal equivalent	$1000$	$100$	$10$	$1$

a

State the number of zeros in the decimal equivalent of

10^3

and of

10^0

. [1]

b

Explain the relationship between the exponent in a power of

10

and the number of zeros in its decimal equivalent, using at least two values from the table to support your explanation. [1]

c

A scientist states: "Because $10^6$ has 6 zeros, a distance of $10^6$ km is exactly one million kilometres — large enough to matter in space navigation." Justify whether the scientist's reasoning about the number of zeros is mathematically correct, and explain what this means for expressing large distances in standard form. [1]

Solutions

6QuestionMultiplying and Dividing Numbers in Scientific NotationConcept Practice

2 marks~3 minCriterion A

A satellite imaging system captures two circular coverage zones. The large zone has a radius of

9 \times 10^4

km and the small zone has a radius of

3 \times 10^2

km.

Using

A = \pi r^2

, calculate the ratio of the area of the large zone to the area of the small zone. Express your answer in scientific notation. [1]

A minimum coverage ratio of

1 \times 10^4

is required for the large zone to be classified as a primary zone. Justify whether the large zone meets this classification. [1]

Solutions

7QuestionSolving Word Problems Involving PercentagesConcept Practice

3 marks~5 minCriterion A

A consumer electronics retailer is offering the following discounts:

TV — original price	450 dollars	discount: 20%
Soundbar — original price	200 dollars	discount: 15%

A customer has a budget target of 115 dollars in savings from this purchase.

a

Calculate the discount amount, in dollars, on the TV. [1]

b

Calculate the total amount saved, in dollars, when buying both items. [1]

c

Advise the customer whether their savings target of 115 dollars will be met. Justify your answer. [1]

Solutions

8QuestionUnderstanding Absolute and Percentage ErrorAssessment Practice

4 marks~6 minCriterion C

A student measures the time for a ball to fall from a fixed height, repeating the experiment four times. The accepted value, determined by a precision instrument, is

3.0

seconds.

Trial 1:

3.0

s
Trial 2:

3.2

s
Trial 3:

2.8

s
Trial 4:

3.1

s

\text{Percentage error} = \frac{|\text{Approximate value} - \text{Exact value}|}{|\text{Exact value}|} \times 100\%

a

Calculate the percentage error for Trials 2, 3, and 4. [2]

b

The student claims their method is reliable because the mean of all four trials equals the accepted value. Calculate the mean of the four trials, then justify whether this claim is valid. [2]

Solutions

9QuestionCalculations Using Rounded ValuesAssessment Practice

2 marks~3 minCriterion B

When two numbers are each rounded to

n

decimal places, the sum is also expressed to

n

decimal places.

The four sums below illustrate this:

3 + 4 = 7

(rounded to 0 decimal places)

3.2 + 4.3 = 7.5

(rounded to 1 decimal place)

3.25 + 4.31 = 7.56

(rounded to 2 decimal places)

3.253 + 4.314 = 7.567

(rounded to 3 decimal places)

Analyse the relationship between the number of decimal places in the addends and in their sum. Hence, formulate a conjecture for the maximum possible error in the sum when both addends are rounded to

n

decimal places. [2]

Solutions

10QuestionUnderstanding Absolute and Percentage ErrorAssessment Practice

8 marks~12 minCriterion A

A manufacturing engineer measures two components of a mechanical assembly. Component

a = 5.0 \pm 0.1

cm and component

b = 3.0 \pm 0.1

cm.

Use: absolute error of a product

= a_0\,\Delta b + b_0\,\Delta a + \Delta a\,\Delta b

, where

a_0

,

b_0

are measured values and

\Delta a

,

\Delta b

are absolute errors.

a

Calculate the absolute error and percentage error for the sum

a + b

and the product

a \times b

. [2]

b

A second engineer measures

c = 6.0 \pm 0.2

cm and

d = 4.0 \pm 0.2

cm. Deduce whether the percentage error in the product

c \times d

is greater than, equal to, or less than the percentage error in

a \times b

, and explain why. [3]

c

Prove algebraically that when two measured quantities are added, the maximum absolute error of the sum equals the sum of the individual absolute errors. Then advise the quality control team whether the assembly, with total length

a + b

, should be approved for a specification requiring the total to lie within

\pm 0.15

cm of 8.0 cm. [3]

Solutions

11QuestionUnderstanding Absolute and Percentage ErrorAssessment Practice

4 marks~6 minCriterion D

Biologists use the capture-recapture method to estimate the population size of fish in a lake. They capture, mark, and release 50 fish. One week later, they capture 80 fish and find that 10 are marked. The estimated population size

N

is given by:

N = \frac{\text{number marked in first catch} \times \text{total in second catch}}{\text{number marked in second catch}}

a

Calculate

N

. [1]

b

State two assumptions the model requires for

N

to be a reliable estimate. [1]

c

A biologist suspects that marked fish are more visible to predators, reducing their survival rate before the second catch. Analyse how this would affect the value of

N

calculated using the formula, and justify whether the estimate of 400 fish should be considered an overestimate or underestimate of the true population. [2]

Solutions

12QuestionNumber Line Representation of Real NumbersAssessment Practice

8 marks~12 minCriterion B

A structural engineer checks whether cable anchor points fall between integer-metre marks on a measuring rod. The relevant cable lengths (in metres) are

\sqrt{2}

,

\sqrt{3}

,

\sqrt{5}

,

\sqrt{6}

,

\sqrt{7}

, and

\sqrt{8}

.

a

Construct a number line from 1 to 3 and plot each cable length, labelling every point clearly. [2]

b

Deduce the consecutive integers between which

\sqrt{10}

and

\sqrt{11}

lie, showing the reasoning that supports each deduction. [2]

c

Formulate a general rule: for any positive integer

n

, identify the consecutive integers between which

\sqrt{n}

lies, and express your rule as a complete inequality. [2]

d

A cable of length

\sqrt{n}

must measure strictly between 4 m and 5 m to fit a specific anchor bracket. Justify whether

n = 18

satisfies this requirement, using your general rule and the properties of inequalities. [2]

Solutions

13QuestionClassifying Rational and Irrational NumbersAssessment Practice

6 marks~9 minCriterion C

A music producer stores a 3-minute song as a WAV (lossless) file of 30 MB and as an MP3 (lossy) file of 3 MB.

a

Calculate the compression ratio

\dfrac{\text{MP3 size}}{\text{WAV size}}

and express your answer as a simplified fraction. [2]

b

A second algorithm targets a compression ratio of

\dfrac{1}{\sqrt{10}}

. Explain why applying this ratio to the 30 MB WAV file produces a compressed file size that is irrational. [2]

c

The producer must choose one format to archive all future recordings. Using the compression ratios from parts (a) and (b), advise the producer which format to select for long-term archiving, and explain what is permanently lost by choosing the smaller file size. [2]

Solutions

14QuestionUnderstanding Integers Fractions and DecimalsAssessment Practice

4 marks~6 minCriterion D

A civil engineer designs a circular fountain with a radius of

3.5

m. To estimate the area of the base, she uses the rational approximation

\pi \approx \dfrac{22}{7}

.

a

Calculate the area of the fountain's base using

\pi = \dfrac{22}{7}

. Give your answer in m

^2

. [2]

b

The engineer's supplier sells concrete in whole m

^2

units only. Using

\pi \approx 3.14159

, calculate the more precise area, then advise the engineer whether the approximation

\dfrac{22}{7}

is sufficient for placing her concrete order. [2]

Solutions

15QuestionProfit Loss and Discount CalculationsAssessment Practice

6 marks~9 minCriterion B

A retail store runs a "Buy 1 Get 1" promotion in which the second item is discounted. Three schemes are available:

Scheme A: 50% off the second item
Scheme B: 40% off the second item
Scheme C: 30% off the second item

A storewide flat 10% discount also applies and may be used either before or after the scheme discount.

a

Calculate the overall discount percentage on the total cost of two items for all six combinations (three schemes × two orders of application). [2]

b

Show that, for any scheme discount fraction

x

and flat discount fraction

y

, the overall discount fraction on two items is

\dfrac{x + 2y - xy}{2}

, regardless of the order in which the discounts are applied. [3]

c

A consumer watchdog claims that stores deliberately advertise the order of discounts to make savings appear larger than they are. Using your result from (b), assess whether this claim has mathematical merit. [1]

Solutions

16QuestionProfit Loss and Discount CalculationsAssessment Practice

4 marks~6 minCriterion C

A retailer purchases a jacket at a cost price of 45 dollars and sells it for 72 dollars.

a

Calculate the markup amount on the jacket. [1]

b

Deduce the percentage markup based on the cost price. [1]

c

The retailer claims: "A markup of over 50% on cost price means I am making more than 50 cents profit on every dollar of revenue I receive."

Assess whether this claim is correct, supporting your answer with a calculation. [2]

Solutions

17QuestionSolving Real-World Finance ProblemsAssessment Practice

4 marks~6 minCriterion D

A retailer purchases goods for 350 dollars and sells them at a marked price.

a

Write the formula for percentage profit. [1]

b

Determine the selling price if the retailer targets a 40% profit on the cost price. [1]

c

The retailer's supplier raises the cost price by 15%, but the selling price remains at 490 dollars. Advise the retailer whether the selling price should be renegotiated, using your calculated percentage profit to support your recommendation. [2]

Solutions

18QuestionAddition and Subtraction of FractionsAssessment Practice

5 marks~8 minCriterion B

A pharmacist mixes two liquid medicines in equal parts. Medicine A is administered at

\frac{1}{n}

of a standard dose per millilitre, and Medicine B at

\frac{1}{n+1}

of a standard dose per millilitre, where

n

is a positive integer. The combined concentration per millilitre is

\frac{1}{n} + \frac{1}{n+1}

.

a

Calculate each sum for

n = 2, 3, 4, 5, 6

, expressing each result as a single fraction in simplest form.

\frac{1}{2}+\frac{1}{3}, \quad \frac{1}{3}+\frac{1}{4}, \quad \frac{1}{4}+\frac{1}{5}, \quad \frac{1}{5}+\frac{1}{6}, \quad \frac{1}{6}+\frac{1}{7}

[2]

b

Explain the pattern you observe in the numerators and denominators of your results. [1]

c

Deduce the combined concentration when

n = 9

, without performing the full calculation. [1]

d

A safe combined concentration must remain below

\frac{1}{4}

of a standard dose per millilitre. Using the general rule

\frac{1}{n}+\frac{1}{n+1} = \frac{2n+1}{n(n+1)}

, advise the pharmacist whether the mixture is safe for all values of

n \geq 4

, and state the smallest value of

n

for which it becomes safe. [1]

Solutions

19QuestionOrder of Operations with Mixed Numbers and DecimalsAssessment Practice

2 marks~3 minCriterion C

A student purchases 4 cinema tickets at

7.25

dollars each and applies a

6.00

dollar voucher to the total bill.

a

Calculate the amount the student pays, showing the correct order of operations. [1]

b

The student's friend argues that the voucher should be subtracted from one ticket price before multiplying. Justify whether the friend's interpretation is mathematically valid, referring to the real-world transaction. [1]

Solutions

20QuestionMultiplication and Division of FractionsAssessment Practice

4 marks~6 minCriterion D

A baker is halving a recipe that requires

2\dfrac{1}{3}

cups of flour. The baker mistakenly calculates

2\dfrac{1}{3} \times 2 = 4\dfrac{2}{3}

cups instead of halving correctly.

a

Show that the correct amount of flour needed is

1\dfrac{1}{6}

cups. [2]

b

The baker measures

1\dfrac{1}{6}

cups using a set of cups marked in increments of

\dfrac{1}{8}

cup. Advise the baker whether fraction arithmetic remains a reliable method for scaling this recipe, given the measurement tools available. [2]

Solutions

21QuestionRounding Large and Small NumbersAssessment Practice

2 marks~3 minCriterion B

A scientist records a measurement as

12345.67

metres.

The table below shows this value rounded to increasing numbers of significant figures.

1

sig. fig.:

10\,000

2

sig. fig.:

12\,000

3

sig. fig.:

12\,300

4

sig. fig.:

12\,350

5

sig. fig.:

12\,346

6

sig. fig.:

12\,345.7

Analyse the pattern in the table and describe the general rule for rounding a number to

n

significant figures. [2]

Solutions

22QuestionRounding Large and Small NumbersAssessment Practice

5 marks~8 minCriterion A

A spherical water storage tank has a radius of

1.234 \times 10^2

metres. A civil engineer must verify whether the tank can hold at least

7.90 \times 10^6 \ \text{m}^3

of water to meet regional supply requirements.

The volume of a sphere is given by

V = \frac{4}{3}\pi r^3

a

Calculate the volume of the tank in

\text{m}^3

. Express your answer in scientific notation. [2]

b

Round your answer from part (a) to three significant figures, expressing your result in scientific notation. [1]

c

Advise the engineer whether this tank is suitable for the regional supply requirement, justifying your advice with a numerical comparison. [2]

Solutions

23QuestionSignificant Figures in MeasurementAssessment Practice

8 marks~12 minCriterion D

A pharmacist must administer a dose of

2.4875 \, \text{mL}

using a syringe graduated to the nearest

0.1 \, \text{mL}

. The patient's safe dosage range is

2.4 \, \text{mL}

to

2.6 \, \text{mL}

.

a

State the value of

2.4875 \, \text{mL}

rounded to one, two, and three significant figures. [3]

b

Analyse whether each rounded value from part (a) falls within the safe dosage range, and explain why one of the three values is impractical to measure with this syringe. [2]

c

Advise the pharmacist which rounded value to administer, justifying your recommendation in terms of both the syringe's precision and patient safety. [3]

Solutions

24QuestionUse of Estimation in Checking ResultsAssessment Practice

2 marks~3 minCriterion C

A surveyor is mapping a triangular plot of land, triangle

ABC

. The angle at vertex

A

measures

\angle BAC = 110°

. The diagram shows that

\angle ABC

is visibly smaller than

\angle ACB

.

a

Deduce the combined size of

\angle ABC

and

\angle ACB

, showing your working. [1]

b

Estimate the size of

\angle ABC

, justifying your answer using the diagram and your result from part (a). [1]

Solutions

25QuestionSimplifying Expressions with ExponentsAssessment Practice

6 marks~9 minCriterion A

A materials scientist models the thermal conductivity of a composite alloy using the expression

\left( \frac{16a^{-4}b^{6}}{81c^{-8}d^{12}} \right)^{-\frac{3}{4}}

where

a

,

b

,

c

,

d

represent measurable physical parameters, all positive and non-zero.

a

Show that the expression simplifies to

\dfrac{27a^{3}d^{9}}{8b^{\frac{9}{2}}c^{6}}

, stating each law of exponents you apply. [4]

b

The scientist states: "Because the outer exponent is negative, the simplified expression must always be less than 1." Critique this claim, justifying your reasoning with a specific numerical counterexample. [2]

Solutions

26QuestionLaws of Exponents for Integer PowersAssessment Practice

4 marks~6 minCriterion C

A biologist models two bacterial populations. Population A grows according to

y = 2^x

, where

y

is the population size (thousands) and

x

is time in hours. Population B decays and is modelled by

y = \dfrac{1}{2^x}

.

a

Show that

\dfrac{1}{2^x} = 2^{-x}

, stating the law of exponents used. [1]

b

Describe the geometric transformation that maps the graph of

y = 2^x

onto the graph of

y = 2^{-x}

, explaining how the change in exponent produces this transformation. [2]

c

At

x = 3

hours, Population A has reached 8 thousand. Calculate the size of Population B at the same time, then justify whether this value is a realistic prediction for a decaying bacterial culture. [1]

Solutions

27QuestionSquare Roots and Cube RootsAssessment Practice

2 marks~3 minCriterion D

A square concrete slab has an area of

150 \, \text{m}^2

. A foundation measures

12.2 \, \text{m}

wide.

a

Calculate the side length of the slab, giving your answer to one decimal place. [1]

b

The construction manager states: "An estimate is good enough — we don't need the exact value." Using your result from part (a), advise the manager whether a one-decimal-place estimate is sufficient to decide if the slab fits the foundation. [1]

Solutions

28QuestionApplications in Science and Engineering ContextsAssessment Practice

6 marks~9 minCriterion D

An astronomer uses the distance modulus formula

d = 10^{\frac{m - M + 5}{5}}

to estimate the distance

d

(in parsecs) to a star, where

m

is the apparent magnitude and

M

is the absolute magnitude.

For a particular star,

M = 4.0

and the measured apparent magnitude is

m = 8.5 \pm 0.2

due to atmospheric distortion.

a

Calculate the distance to the star using

m = 8.5

. [2]

b

Calculate the distances corresponding to

m = 8.3

and

m = 8.7

. Express both answers in scientific notation to 3 significant figures. [2]

c

The astronomer claims the distance modulus formula gives a reliable estimate of this star's distance. Justify whether this claim is valid, using your results from parts (a) and (b) and considering that interstellar dust dims starlight. [2]

Solutions

29QuestionMultiplying and Dividing Numbers in Scientific NotationAssessment Practice

6 marks~9 minCriterion C

The mass of Earth is

5.97 \times 10^{24}

kg and the mass of the Moon is

7.35 \times 10^{22}

kg.

a

Calculate the ratio of the mass of Earth to the mass of the Moon. Express your answer in scientific notation to three significant figures. [2]

b

Explain why scientific notation is appropriate when comparing astronomical masses, and explain how rounding to three significant figures affects the reliability of the ratio calculated in part (a). [2]

c

A scientist states: "This ratio means Earth is approximately 81 times more massive than the Moon, so the Moon has a negligible gravitational influence on Earth." Critique this statement, using your result from part (a) and your knowledge of real-world gravitational effects. [2]

Solutions

30QuestionSimplifying Ratios and Comparing QuantitiesAssessment Practice

6 marks~9 minCriterion B

Rectangles are designed so that the ratio of length to width is always

3:5

. The

n

th rectangle has side lengths

3n

cm and

5n

cm, so the first rectangle has sides

3

cm and

5

cm, the second has sides

6

cm and

10

cm, and so on.

a

Calculate the perimeter and area of each of the first three rectangles. Hence find the ratio of perimeter to area for each rectangle, simplifying fully. [2]

b

Deduce a general formula for the ratio of perimeter to area of the

n

th rectangle, expressing your answer in simplest form in terms of

n

. [2]

c

A designer claims that once a rectangle in this sequence has a perimeter-to-area ratio below

1:10

, it is no longer suitable for a tiling pattern because the border becomes negligible compared to the surface. Use your formula to find the smallest value of

n

for which this condition is met, and advise the designer on which rectangles in the sequence should be excluded from the tiling pattern. [2]

Solutions

31QuestionPercentage Increase and DecreaseAssessment Practice

5 marks~8 minCriterion D

The table below shows the annual revenue of a technology company over five years.

Year	1	2	3	4	5
Revenue (millions of dollars)	2.0	3.0	5.0	4.5	4.0

a

Calculate the percentage increase in revenue for each year in which revenue rose, and deduce which year had the greatest percentage increase. [2]

b

Calculate the percentage decrease in revenue from Year 3 to Year 5. [1]

c

The company's board set a target of 10% revenue growth each year. Using your results, advise the board whether maintaining this target for the following year is justified. [2]

Solutions

32QuestionPercentage Increase and DecreaseAssessment Practice

2 marks~3 minCriterion C

A clothing store buys shirts for 20 dollars each and sells them for 35 dollars each. The store applies a 15% discount to the selling price.

a

Calculate the profit per shirt after the discount is applied. [1]

b

Justify whether the manager's claim — that a 15% discount on the selling price causes a greater than 15% reduction in profit per shirt — is correct. [1]

Solutions

Number and Operations — Free MYP4 Mathematics (Standard) Practice Questions