Number and Operations — Free IB MYP5 Mathematics (Standard) Questions

1QuestionNumber Line Representation of Real NumbersConcept Practice

2 marks~3 minCriterion A

A carpenter needs to mark a length of

\sqrt{5}

cm on a straight beam. She constructs a right triangle with legs of 2 cm and 1 cm, then uses a compass to transfer the hypotenuse length onto a number line scaled in centimetres.

a

Show that the hypotenuse of the triangle equals

\sqrt{5}

cm. [1]

b

Deduce between which two consecutive integers

\sqrt{5}

lies, and justify whether the carpenter's marked length is closer to the 2 cm or 3 cm graduation. [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

3QuestionCurrency Conversion and Exchange RatesConcept Practice

4 marks~6 minCriterion A

A tourist exchanges British pounds (GBP) at a currency kiosk displaying the following rates:

1 \text{ GBP} = 1.27 \text{ USD}

1 \text{ GBP} = 1.15 \text{ EUR}

1 \text{ GBP} = 1.68 \text{ AUD}

a

State the exchange rate from GBP to USD. [1]

b

Calculate the number of USD the tourist receives for 350 GBP. [1]

c

The tourist plans to travel to both the United States and the eurozone. She decides to exchange all 350 GBP into whichever single currency gives more foreign currency units. Justify which currency she should choose, supporting your answer with calculations. [2]

Solutions

4QuestionConverting Between Fractions Decimals and PercentagesConcept Practice

2 marks~3 minCriterion B

A currency exchange app displays repeating decimals when converting amounts. The app shows these fraction-to-decimal conversions:

\frac{1}{9} = 0.\overline{1}, \quad \frac{2}{9} = 0.\overline{2}, \quad \frac{3}{9} = 0.\overline{3}, \quad \frac{4}{9} = 0.\overline{4}

a

Deduce the fraction, in simplest form, that corresponds to

0.\overline{7}

. [1]

b

A currency conversion gives the result

0.\overline{7}

of a base amount. Justify whether this is greater than, equal to, or less than

\dfrac{3}{4}

of the same base amount, using fractions. [1]

Solutions

5QuestionConverting Between Fractions Decimals and PercentagesConcept Practice

3 marks~5 minCriterion C

A factory quality-control report states that

\frac{3}{8}

of a batch of components failed inspection. The supervisor needs the failure rate expressed as a decimal to enter it into a digital monitoring system that does not accept fractions.

a

Explain what mathematical operation

\frac{3}{8}

represents, identifying the role of the numerator and denominator. [1]

b

Show that

\frac{3}{8} = 0.375

by completing the long division of

3.000 \div 8

, presenting each step clearly. [1]

c

The monitoring system flags any batch with a failure rate above

0.4

. Advise the supervisor whether this batch requires further action, justifying your answer with reference to the decimal result. [1]

Solutions

6QuestionConverting Between Fractions Decimals and PercentagesConcept Practice

2 marks~3 minCriterion A

A jacket is priced at 80 dollars. Store A offers a 25% discount; Store B sells the same jacket for 61 dollars with no discount.

a

Calculate the sale price of the jacket at Store A after the 25% discount is applied. [1]

b

Advise a customer which store to purchase from, justifying your recommendation with reference to the calculated sale price. [1]

Solutions

7QuestionApproximations 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

8QuestionApplications in Science and Engineering ContextsConcept Practice

2 marks~3 minCriterion A

Proxima Centauri, the nearest star to our solar system, is approximately

4.246

light-years from Earth. Given that

1

light-year

\approx 9.461 \times 10^{12}

km, calculate the distance to Proxima Centauri in kilometres, expressing your answer in standard form. [1]

Explain why scientists use scientific notation when working with astronomical distances such as this. Give two distinct reasons. [1]

Solutions

9QuestionPercentage Increase and DecreaseConcept Practice

2 marks~3 minCriterion A

A clothing retailer increases the price of a jacket from 85 dollars by 15%.

Calculate the new price of the jacket. [2]

Solutions

10QuestionCalculations 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

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

12QuestionUnderstanding 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

13QuestionEstimation in Problem SolvingAssessment Practice

4 marks~6 minCriterion D

A wildlife conservationist estimates the fish population of a lake using the capture-recapture method. In the first sample, 50 fish are caught, tagged, and released. One week later, a second sample of 100 fish is caught, of which 10 are tagged.

a

Show that the estimated total fish population is 500. [1]

b

Explain two assumptions required for the capture-recapture method to produce a valid estimate in this scenario. [1]

c

The conservationist needs at least 400 fish in the lake to justify a new fishing licence. Advise the conservationist whether the population estimate from part (a) is sufficient to support this decision, and identify one specific reason why the estimate may not be reliable enough to act on. [2]

Solutions

14QuestionNumber 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

15QuestionComparing and Ordering Real NumbersAssessment Practice

4 marks~6 minCriterion D

An engineer is ordering six steel rods for a bridge. Their lengths, in metres, are:

l_1 = \sqrt{2}, \quad l_2 = \pi, \quad l_3 = \tfrac{22}{7}, \quad l_4 = 1.414, \quad l_5 = 3.14, \quad l_6 = 3.14159

The engineer approximates

l_1 \approx 1.414

and

l_2 \approx 3.14

when placing the order.

a

Arrange

l_1

through

l_6

in ascending order, showing the decimal values used to justify your ordering. [2]

b

Advise the engineer whether these approximations are sufficiently precise for safe use in bridge construction, referring to specific pairs of values whose ordering is affected and to the structural consequence of any error. [2]

Solutions

16QuestionNumber Line Representation of Real NumbersAssessment Practice

8 marks~12 minCriterion C

Consider the six numbers below.

Number:

\dfrac{1}{3}

,

\sqrt{2}

,

\dfrac{22}{7}

,

\pi

,

\dfrac{5}{8}

,

\sqrt{5}

Decimal:

0.333\ldots

,

1.414\ldots

,

3.\overline{142857}

,

3.14159\ldots

,

0.625

,

2.236\ldots

a

Construct a number line from 0 to 4 and plot all six numbers, labelling each to two decimal places. [2]

b

Classify each decimal expansion as terminating, repeating, or non-repeating, and state whether each number is rational or irrational. [2]

c

Give two additional examples of each decimal type (terminating, repeating, non-repeating). For rational examples, express each number as both a decimal and a fraction. [2]

d

A classmate claims: "Because $\dfrac{22}{7}$ and $\pi$ are so close on the number line, $\dfrac{22}{7}$ is a good enough substitute for $\pi$ in any calculation." Critique this claim, referring to the nature of their decimal expansions and the consequences for exact calculation. [2]

Solutions

17QuestionCurrency Conversion and Exchange RatesAssessment Practice

3 marks~5 minCriterion D

A student in the UK plans a 10-day trip to Japan with a budget of 2000 GBP. They use the exchange rate model:

\text{Amount in JPY} = \text{Amount in GBP} \times 156.4

When converting at the bank, a fee of 2.5% and a spread of 0.5% are applied, reducing the effective amount received.

a

Calculate the total amount in JPY the student receives after the bank applies the fee and spread. [1]

b

Calculate the student's daily budget in JPY, assuming equal spending over 10 days. [1]

c

Typical daily expenses in Tokyo average 35,000 JPY. Advise the student whether their current GBP budget is sufficient for the trip, justifying your answer with reference to your calculated daily figure. [1]

Solutions

18QuestionSolving Real-World Finance ProblemsAssessment Practice

4 marks~6 minCriterion C

Maria borrows 12 000 dollars at a compound interest rate of 6% per annum, compounded annually. The loan balance at the end of each year is shown below.

Year 0: 12 000.00 dollars
Year 1: 12 720.00 dollars
Year 2: 13 483.20 dollars
Year 3: 14 292.19 dollars
Year 4: 15 149.72 dollars

a

State the values of

P

,

r

, and

n

in the formula

A = P(1 + r)^n

that represent this loan over the full 4-year period. [1]

b

Calculate the total amount owed after 4 years using the formula

A = P(1 + r)^n

. [1]

c

Maria claims she can afford this loan because "the interest charged each year stays the same." Advise Maria on whether her claim is correct, using numerical evidence from the table and explaining what the pattern of annual interest charges means for her repayment burden. [2]

Solutions

19QuestionAddition and Subtraction of FractionsAssessment Practice

6 marks~9 minCriterion D

A painter is mixing a custom shade of green for a large mural. The recipe specifies

\frac{2}{5}

of the total mixture as blue paint,

\frac{1}{3}

as yellow paint, and

\frac{1}{15}

as black paint; the remainder is white paint.

a

Calculate the fraction of the total mixture that is white paint. [2]

b

The blue paint measurement has a possible error of up to

\frac{1}{60}

of the total mixture. Analyse the effect of this error on the shade of green, considering both the case where more blue is used and the case where less blue is used. [2]

c

The painter considers measuring each colour by volume rather than as a fraction of the total mixture. Given that the paints have different densities, advise the painter whether this alternative method is appropriate for achieving consistent colour across the mural. [2]

Solutions

20QuestionApproximations 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

21QuestionSignificant Figures in MeasurementAssessment Practice

2 marks~3 minCriterion D

A construction worker measures a wooden beam as

2.3

m using a tape marked every

0.1

m.

a

State the number of significant figures in

2.3

m and write the range within which the true length must lie. [1]

b

The worker cuts eight beams, each measured as

2.3

m. Advise the site manager whether this tape measure is suitable for the project, given that the total length of all eight beams must be within

0.3

m of the target value of

18.4

m. [1]

Solutions

22QuestionEstimation Techniques in Mental MathAssessment Practice

3 marks~5 minCriterion C

A line graph shows the population of a town over 10 years. In 2010, the population was 48 700. In 2020, the population was 62 300.

a

Calculate the exact percentage increase in population from 2010 to 2020. [1]

b

Estimate the percentage increase by first rounding each population to the nearest thousand. Show your rounded values. [1]

c

A town planner claims the population grew by "roughly 27%." Assess whether the estimate from part (b) supports this claim, and justify your reasoning with reference to the size of the rounding errors relative to the populations involved. [1]

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

24QuestionSimplifying 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

25QuestionLaws 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

26QuestionLaws of Exponents for Integer PowersAssessment Practice

6 marks~9 minCriterion D

A viral social media post is shared. The number of views after

n

hours is modelled by

V(n) = 500 \cdot 3^n

A content creator claims the post will exceed 40 000 views within the first 4 hours.

a

Show that

V(4) = 40\,500

, applying the laws of exponents in your working. [2]

b

Discuss two real-world limitations that could cause the actual number of views to differ from the value predicted by this model. [2]

c

The creator uses

V(4) = 40\,500

to conclude the post is "growing without limit" and plans to scale up advertising spend indefinitely. Advise the creator whether this conclusion is justified, and suggest one mathematical modification to the model that better reflects real sharing behaviour. [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

28QuestionAdding 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

29QuestionConverting Between FormsAssessment Practice

2 marks~3 minCriterion D

A team of demographers is comparing global population data across multiple decades. The world population in 2024 is 8,100,000,000.

Justify why expressing this population as

8.1 \times 10^9

is preferable to using the full integer form when presenting findings in a published research report. Your response must reference a specific practical advantage relevant to this context. [2]

Solutions

30QuestionSimplifying 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

31QuestionPercentage Increase and DecreaseAssessment Practice

4 marks~6 minCriterion C

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

32QuestionPercentage Increase and DecreaseAssessment Practice

4 marks~6 minCriterion D

A software engineer earns an annual salary of 80 000 USD. A financial advisor models the engineer's purchasing power over five years using two assumptions: salary grows at 4% per year and the cost of a standard grocery basket (initially 200 USD) grows at 2.5% per year due to inflation.

\text{Salary}_{\text{year}} = 80000 \times (1.04)^{\text{year}} \qquad \text{Cost}_{\text{year}} = 200 \times (1.025)^{\text{year}}

a

Calculate the salary and grocery basket cost for each of the five years. [2]

Year 1	Salary (USD): ___ Grocery Basket Cost (USD): ___
Year 2	Salary (USD): ___ Grocery Basket Cost (USD): ___
Year 3	Salary (USD): ___ Grocery Basket Cost (USD): ___
Year 4	Salary (USD): ___ Grocery Basket Cost (USD): ___
Year 5	Salary (USD): ___ Grocery Basket Cost (USD): ___

b

The engineer's purchasing power in a given year is defined as:

\text{Purchasing Power}_{\text{year}} = \frac{\text{Salary}_{\text{year}}}{\text{Cost}_{\text{year}}}

Deduce, using your results from part (a), whether the engineer's purchasing power increases or decreases over the five years, and calculate the overall percentage change in purchasing power from Year 1 to Year 5. [1]

c

Assess the extent to which this model provides a reliable prediction of the engineer's actual purchasing power over five years. In your response, identify at least one strength and two limitations of the model. [1]

Solutions

Number and Operations — Free MYP5 Mathematics (Standard) Practice Questions