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

1QuestionUnderstanding Integers Fractions and DecimalsConcept Practice

3 marks~5 minCriterion A

A pharmacist measures three liquid volumes as fractions of a 1-litre bottle. The number line below shows the fill levels A, B, and C.

Point A is at 0.25 litres, point B at 0.5 litres, and point C at 0.75 litres.

a

Deduce the fraction of the bottle represented by point A, expressing your answer in simplest form. [1]

b

Justify why points A and C together fill exactly one complete bottle. [1]

c

The pharmacist states: "Any two of these three volumes can be combined to fill at least three-quarters of the bottle." Assess this claim by calculating all possible two-point combinations. [1]

Solutions

2QuestionProfit Loss and Discount CalculationsConcept Practice

2 marks~3 minCriterion B

A retailer offers a bulk-purchase discount scheme:

Number of items purchased	1	2	3
Discount applied	$3\%$	$6\%$	$9\%$

a

Deduce the general rule for the discount percentage when

n

items are purchased. [1]

b

A customer needs a discount of at least

12\%

to stay within budget. Justify whether purchasing 4 items meets this requirement, using your rule from part (a). [1]

Solutions

3QuestionSimple 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

4QuestionOrder of Operations with Mixed Numbers and DecimalsConcept 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

5QuestionAddition and Subtraction of FractionsConcept Practice

2 marks~3 minCriterion A

A community garden is divided into 12 equal plots. Volunteers have planted vegetables in 5 plots and flowers in 3 plots. The remaining plots are unused.

a

Calculate the fraction of the garden used for vegetables and flowers combined. Express your answer in simplest form. [1]

b

The garden committee will fund a water irrigation system only if more than $\dfrac{2}{3}$ of all plots are in use. Justify whether the committee will fund the irrigation system. [1]

Solutions

6QuestionApproximations in Scientific and Everyday ContextsConcept Practice

2 marks~3 minCriterion A

State the formula for percentage error, where

A

is the approximate value and

E

is the exact value. [2]

Solutions

7QuestionMultiplying 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

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

10QuestionCalculations Using Rounded ValuesAssessment Practice

2 marks~3 minCriterion D

A homeowner measures a bathroom floor as

2.0

m long and

1.5

m wide, rounded to the nearest

0.5

m. Square tiles measure

0.2

m by

0.2

m, rounded to the nearest

0.05

m. The homeowner estimates the number of tiles needed using

\dfrac{\text{floor area}}{\text{tile area}}

.

Advise the homeowner whether this calculation gives a reliable tile count for purchasing purposes, with reference to the direction of both rounding errors. [2]

Solutions

11QuestionUnderstanding 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

12QuestionClassifying 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

13QuestionClassifying Rational and Irrational NumbersAssessment Practice

4 marks~6 minCriterion B

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

14QuestionSet Notation and Venn Diagrams for Number SetsAssessment Practice

6 marks~9 minCriterion D

A local library holds 200 books classified into three sets:

F

(fiction),

N

(non-fiction), and

A

(award-winning). The librarian records:

|F| = 90, \quad |N| = 70, \quad |A| = 50

|F \cap N| = 30, \quad |F \cap A| = 20, \quad |N \cap A| = 15, \quad |F \cap N \cap A| = 10

a

Construct a Venn diagram showing all eight regions, including the region outside all three sets. [2]

b

Deduce the number of books belonging to exactly one category, and interpret what this result suggests about the library's classification system. [2]

c

Discuss two limitations of using this set-based model to represent a real library collection, explaining how each limitation reduces the model's usefulness. [2]

Solutions

15QuestionTaxation Budgeting and Loan PaymentsAssessment Practice

4 marks~6 minCriterion D

The graph below shows the total amount owed on a loan over a 5-year period. The loan begins at 5000 dollars and the total amount owed after 5 years is 6500 dollars.

a

Calculate the annual increase in the total amount owed. [1]

b

Explain how the shape of the graph indicates the type of interest applied to this loan. [2]

c

A second loan of 5000 dollars is offered at 5% annual compound interest over 5 years. Advise a borrower which loan to choose, justifying your recommendation with a calculation. [1]

Solutions

16QuestionProfit Loss and Discount CalculationsAssessment Practice

4 marks~6 minCriterion C

A trader sold an article for 374 USD, making a loss of 15% on the cost price.

a

State the multiplier that relates the selling price to the cost price when a 15% loss occurs. [1]

b

Calculate the cost price of the article. [2]

c

The trader claims that raising the selling price by 15% would have been enough to break even. Justify whether this claim is correct, referring to the actual percentage increase required. [1]

Solutions

17QuestionConverting Between Fractions Decimals and PercentagesAssessment Practice

12 marks~18 minCriterion B

Prime numbers can produce recurring decimals when used as the denominator of a unit fraction (numerator = 1). This question investigates the relationship between a prime denominator

p

and the length of the recurring block

L

.

a

Calculate the decimal form of each unit fraction. Show your long-division working. [4]

\frac{1}{3}, \quad \frac{1}{7}, \quad \frac{1}{11}, \quad \frac{1}{13}

b

Deduce the length of the recurring block for each decimal in part (a), and record your results.

Denominator ( $p$ )	___	___	___	___
Length of recurring block ( $L$ )	___	___	___	___ [2]

c

Analyse the values in part (b) to formulate a rule for the maximum possible length

L

of the recurring block in terms of

p

. Express your rule as an equation and justify why the actual length may sometimes be shorter than this maximum. [4]

d

A computer-science student claims that knowing the recurring-block length of

\frac{1}{p}

tells you the minimum number of digits a computer must store to represent

\frac{1}{p}

exactly. Calculate the decimal form of

\frac{1}{17}

, determine its recurring-block length, and assess whether the student's claim is supported for this fraction, referring to your rule from part (c). [2]

Solutions

18QuestionAddition and Subtraction of FractionsAssessment Practice

2 marks~3 minCriterion D

A baker's cookie recipe requires

\dfrac{2}{3}

cup of flour and

\dfrac{1}{4}

cup of sugar. She decides to make half the recipe.

Calculate the halved quantity of sugar. Justify whether this halved measurement presents a practical challenge when using standard measuring cups, which are marked at

\dfrac{1}{4}

,

\dfrac{1}{3}

, and

\dfrac{1}{2}

cup intervals, and state one consequence for the finished cookies. [2]

Solutions

19QuestionApproximations in Scientific and Everyday ContextsAssessment Practice

2 marks~3 minCriterion B

A student rounds each grocery item to the nearest dollar before totalling the bill.

Estimated price (dollars)	25	32	41	18
Actual price (dollars)	25.45	31.50	39.80	17.75
Rounding error (dollars)	−0.45	0.50	1.20	0.25
Percentage error	−1.8%	1.6%	3.0%	1.4%

Analyse the data to identify the relationship between the magnitude of the rounding error and the percentage error, and explain why this relationship exists using the percentage error formula. [2]

Solutions

20QuestionUse 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

21QuestionRounding Large and Small NumbersAssessment Practice

6 marks~9 minCriterion D

A city's population is currently estimated at 2.36 million. Planners use the exponential growth model

P(t) = P_0 \, e^{kt}

where

P(t)

is the population after

t

years,

P_0

is the initial population,

k

is the annual growth rate, and

t

is time in years. The city's annual growth rate is 3.8\%.

a

Calculate the projected population after 50 years. Round your answer to three significant figures. [2]

b

Explain how rounding

P_0

to the nearest million and

k

to the nearest whole percent each affect the 50-year projection, and describe one consequence for resource planning. [2]

c

The city council must decide whether to begin construction of three new hospitals now, each designed to serve 1.5 million people. Using your projection and the limitations of the model, advise the council whether to proceed with all three, fewer, or more hospitals. [2]

Solutions

22QuestionSimplifying 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

23QuestionLaws 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

24QuestionLaws of Exponents for Integer PowersAssessment Practice

3 marks~5 minCriterion B

The intensity of light decreases as it travels from a source. A photographer measures the relative intensity of a lamp at several distances.

Distance (m):

2, \ 4, \ 8, \ 16

Relative intensity:

16, \ 4, \ 1, \ 0.25

a

Express each relative intensity as a power of 2. [1]

b

Express each distance as a power of 2, then deduce the relationship between the exponent of the distance and the exponent of the relative intensity. [1]

c

A subject is positioned 32 m from the lamp. The photographer requires a relative intensity of at least

0.1

for a usable photograph. Using laws of exponents, calculate the relative intensity at 32 m and advise the photographer whether the subject should be moved closer. [1]

Solutions

25QuestionLaws of Exponents for Integer PowersAssessment Practice

3 marks~5 minCriterion D

In atmospheric science, particle masses are often expressed as powers of ten. The mass of a dust particle is approximately

10^{-6}

kg and the mass of a grain of sand is approximately

10^{-4}

kg.

a

Deduce how many times heavier a grain of sand is than a dust particle. Apply the quotient law of exponents and show all steps. [1]

b

A scientist collects

10^3

dust particles and

10^3

grains of sand. Deduce the total combined mass of both samples, showing your working. [1]

c

The scientist claims these power-of-ten values give a sufficiently accurate model for comparing particle masses. Critique this claim, giving one specific mathematical reason. [1]

Solutions

26QuestionAdding and Subtracting with Powers of 10Assessment Practice

2 marks~3 minCriterion B

Three sums involving scientific notation are shown below.

(2 \times 10^1) + (3 \times 10^2) = 320

(5 \times 10^3) + (4 \times 10^1) = 5040

(1 \times 10^4) + (7 \times 10^2) = 10700

Analyse the three calculations and explain the pattern connecting the exponent of each term to the position of its coefficient digit in the final sum. [2]

Solutions

27QuestionConverting Between FormsAssessment Practice

8 marks~12 minCriterion C

The Andromeda Galaxy is approximately 2.5 million light-years from Earth. One light-year is equivalent to

9.46 \times 10^{12}

km. A student claims the distance to Andromeda is

2.365 \times 10^{19}

km.

a

Write 2.5 million light-years and

9.46 \times 10^{12}

km each as an ordinary number. [2]

b

Show that the distance to Andromeda in kilometres is

2.365 \times 10^{19}

km, and deduce whether the student's claim is accurate. [2]

c

A science communicator must present the distance

2.365 \times 10^{19}

km to a general audience. Advise the communicator which format — scientific notation or ordinary number form — to use, justifying your recommendation and acknowledging the limitations of your chosen format. [4]

Solutions

28QuestionConverting Between FormsAssessment Practice

8 marks~12 minCriterion D

A biologist models the population of a bacterial colony using the formula

P = 5.0 \times 10^6 \times (1.02)^t

where

t

is time in hours. Actual population counts are recorded below.

Time (hours)	0	5	10
Actual population	$5.0 \times 10^6$	$5.8 \times 10^6$	$6.5 \times 10^6$

a

Calculate the predicted population at

t = 5

and

t = 10

. Give each answer in standard form to two significant figures. [2]

b

Convert the predicted populations from part (a) and the actual populations at

t = 5

and

t = 10

into ordinary numbers. [2]

c

Calculate the absolute difference between the predicted and actual populations at

t = 5

and

t = 10

. Express each difference in standard form to two significant figures. [2]

d

Advise the biologist whether this model should be used to predict the size of the colony beyond

t = 10

hours. Use your results from parts (a) to (c) and give one biological reason to support your advice. [2]

Solutions

29QuestionSimplifying Ratios and Comparing QuantitiesAssessment Practice

12 marks~18 minCriterion B

An architect is designing a series of similar rectangular skylights for a building. The skylights scale up uniformly, with dimensions as follows.

Width (cm)	2	4	6	8	10
Length (cm)	3	6	9	12	15

a

Calculate the area of each skylight. Record your five answers with appropriate units. [2]

b

Deduce the ratio width : length for each skylight, simplifying fully. Hence explain what these ratios reveal about the relationship between the five skylights. [3]

c

Calculate the ratio of areas for each consecutive pair (

A_1 : A_2

,

A_2 : A_3

,

A_3 : A_4

,

A_4 : A_5

), simplifying each ratio fully. Analyse the pattern in these ratios and express each in the form

m^2 : n^2

. [4]

d

The architect claims that doubling the side lengths of any skylight will always produce a new skylight with exactly four times the glazing area, regardless of the original dimensions. Justify whether this claim is correct, using a general rule connecting the ratio of side lengths to the ratio of areas for similar rectangles. [3]

Solutions

30QuestionPercentage Increase and DecreaseAssessment Practice

4 marks~6 minCriterion A

A retailer monitors the price of a wireless speaker over two successive promotional periods. The initial price is 50 dollars. In the first period, the price decreases by 20%. In the second period, the price increases by 25% from the reduced price.

a

Calculate the price of the speaker after the 20% decrease. [1]

b

Calculate the final price of the speaker after the subsequent 25% increase. [1]

c

The retailer claims the speaker has "returned to its original price." Justify whether this claim is mathematically valid, and explain what the overall percentage change reveals about applying successive percentage changes in opposite directions. [2]

Solutions

31QuestionPercentage 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

32QuestionPercentage 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

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