MYP year 4 maths practice

Pick a topic and have a go. When you're done, press Show me how to check your working step by step.

Topics marked Extended are for extended mathematics. Not sure which you take? Your teacher will know.

Number systems, recurring decimals and irrational numbers2 questions

Here is a list of numbers: $π$ , $5$ , $\sqrt{5}$ , $- 6$ , $1.2$ , $\sqrt{49}$ .

Write down all the irrational numbers in the list. [3]

Decide for each number whether it is impossible to write as a fraction of two integers.
Simplify first: for example a square root of a square number is a whole number.
The irrational numbers are: $π$ , $\sqrt{5}$ .
Answer: $π$ , $\sqrt{5}$

4 steps

Write the recurring decimal $0. \dot{5} \dot{6}$ as a fraction in its simplest form.

[3]

Let $x = 0.5656 \dots$ .
Two digits repeat, so multiply by 100: $100 x = 56.56 \dots$ .
Subtract: $99 x = 56$ , so $x = \frac{56}{99}$ .
Simplify: $\frac{56}{99}$ .
Answer: $\frac{56}{99}$

5 steps · 1 to fill in

Fractions, percentages, ratio and proportion2 questions

Work out 17.5% of 40.

[1]

17.5% means 17.5 out of 100, so multiply by $\frac{17.5}{100}$ .
$\frac{17.5}{100} \times 40 = 7$
Answer: $7$

3 steps · 1 to fill in

Elif and Femi share 63 minutes of screen time in the ratio $5 : 2$ .

How many minutes of screen time does Elif get? [1]
How many more does Elif get than Femi? [1]

Total parts: $5 + 2 = 7$ .
One part: $63 \div 7 = 9$ .
Elif gets $5 \times 9 = 45$ ; Femi gets $2 \times 9 = 18$ .
Difference: 27.
Answers: (a) 45; (b) 27

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Powers, roots and standard form2 questions

Simplify $p^{2} \times p^{8}$ , writing your answer as a single power.

[1]

Same base, so add the powers: $p^{2} \times p^{8} = p^{10}$ .
Answer: $p^{10}$

2 steps

Write $0.00807$ in standard form.

[1]

Move the decimal point so the number is between 1 and 10: 8.07.
The point moved 3 places right, so the power is -3: $8.07 \times 10^{- 3}$ .
Answer: $8.07 \times 10^{- 3}$

3 steps

Estimation, rounding and bounds2 questions

Round $5780.509$ to 1 decimal place.

[1]

Look at the next digit after the last one you keep: 5 or more rounds up.
5780.509 → 5780.5
Answer: $5780.5$

3 steps

Round every number in $\frac{27.8 \times 6.1}{0.39}$ to 1 significant figure, then use the rounded values to estimate its value.

[2]

Round: $27.8 \approx 30$ , $6.1 \approx 6$ , $0.39 \approx 0.4$ .
Work out: $\frac{30 \times 6}{0.4} = 450$ .
Answer: $450$

3 steps · 1 to fill in

Surds and fractional indicesExtended2 questions

Simplify $\sqrt{96}$ .

[1]

Find the largest square factor: $96 = 16 \times 6$ .
$\sqrt{96} = \sqrt{16} \sqrt{6} = 4 \sqrt{6}$
Answer: $4 \sqrt{6}$

3 steps

Without a calculator, find the exact value of $32^{- \frac{1}{5}}$ .

[2]

The denominator 5 means a fifth root: $\sqrt[5]{32} = 2$ .
Raise to the power 1: $2^{1} = 2$ .
A negative power means one over it: $\frac{1}{2}$ .
Answer: $\frac{1}{2}$

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Financial mathematics: interest and depreciation2 questions

Work out the simple interest on €3200 invested for 7 years at 5% per year.

[1]

Interest each year: $3200 \times 5 % = 160.00$ .
For 7 years: $160.00 \times 7 = 1120.00$ .
Answer: €1120.00

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€1300 is invested at 2% per year compound interest. Work out its value after 3 years, to the nearest cent.

[2]

Multiplier: $1 + \frac{2}{100} = 1.02$ .
Value: $1300 \times {1.02}^{3} = 1379.57$ .
Answer: €1379.57

3 steps · 1 to fill in

Expressions, expanding and factorising2 questions

Expand and simplify $(x + 3) (x + 6)$ .

[1]

Multiply every term in the first bracket by every term in the second.
$x^{2} + 6 x + 3 x + 18$
Collect like terms: $x^{2} + 9 x + 18$ .
Answer: $x^{2} + 9 x + 18$

4 steps

Factorise $x^{2} + 3 x - 54$ .

[2]

Find two numbers that multiply to -54 and add to 3: they are -6 and 9.
So $x^{2} + 3 x - 54 = (x - 6) (x + 9)$ .
Answer: $(x - 6) (x + 9)$

3 steps

Linear equations, inequalities and formulae2 questions

Solve $9 x + 14 = - 40$ .

[1]

Subtract 14 from both sides: $9 x = - 54$ .
Divide by 9: $x = - 6$ .
Answer: $x = - 6$

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Solve $3 (6 x + 4) = 5 x - 92$ .

[2]

Expand the bracket first. Collect the x terms on one side and the numbers on the other.
Divide to find $x = - 8$ .
Answer: $x = - 8$

3 steps · 1 to fill in

Simultaneous equations2 questions

Solve the simultaneous equations $4 x + 4 y = - 16$ and $4 x - 2 y = - 4$ .

[3]

Add or subtract the equations so one letter disappears.
Solve for the other letter, then substitute back: $x = - 2, y = - 2$ .
Check in both equations.
Answer: $x = - 2, y = - 2$

4 steps

Solve the simultaneous equations $6 x - 4 y = 36$ and $4 x - 5 y = 17$ .

[4]

Multiply one or both equations so the coefficients of one letter match.
Eliminate it, solve, and substitute back: $x = 8, y = 3$ .
Answer: $x = 8, y = 3$

3 steps

Quadratic equations by factorising2 questions

Solve $(x + 3) (x - 3) = 0$ .

[1]

If two numbers multiply to give 0, one of them is 0.
$x + 3 = 0$ or $x - 3 = 0$ , so $x = - 3$ or $x = 3$ .
Answer: $x = - 3 or x = 3$

3 steps · 1 to fill in

Solve $x^{2} - 11 x + 30 = 0$ by factorising.

[3]

Two numbers that multiply to 30 and add to -11: -5 and -6.
$(x - 5) (x - 6) = 0$ , so $x = 5$ or $x = 6$ .
Answer: $x = 5 or x = 6$

3 steps · 1 to fill in

Quadratic formula, completing the square and the discriminantExtended2 questions

Work out the discriminant of $2 x^{2} - x - 3$ and say how many real roots the equation $2 x^{2} - x - 3 = 0$ has.

[2]

$Δ = b^{2} - 4 a c = (- 1)^{2} - 4 (2) (- 3) = 25$
Positive: two roots; zero: one; negative: none.
Answer: $Δ = 25$ ; two real roots

3 steps

Use the quadratic formula to solve $4 x^{2} + 3 x - 3 = 0$ . Give your answers to 2 decimal places.

[3]

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- 3 \pm \sqrt{57}}{8}$
So $x \approx - 1.32$ or $x \approx 0.57$ .
Answer: $x = - 1.32 or x = 0.57$

3 steps

Sequences and general rules2 questions

A sequence starts $- 15, - 19, - 23, - 27$ . Continue it for two more terms and describe, in words, how to get from one term to the next.

[2]

The difference between terms is -4.
Next terms: $- 27 - 4 = - 31$ and $- 31 - 4 = - 35$ .
Answer: $- 31, - 35$ ; subtract 4 each time

3 steps · 1 to fill in

Find an expression for the $n$ th term of the sequence $12, 6, 0, - 6, - 12, \dots$ .

[2]

The common difference is -6, so the rule starts $- 6 n$ .
When $n = 1$ , $- 6 n = - 6$ , so add 18: $- 6 n + 18$ .
Answer: $- 6 n + 18$

3 steps

Functions: notation, graphs, domain and range2 questions

Given $f (x) = x^{2} - 4 x + 3$ , work out $f (3)$ .

[1]

Substitute $x = 3$ : $1 (3)^{2} - 4 (3) + 3 = 0$ .
Answer: $0$

2 steps

Given $g (x) = - 4 x - 6$ , solve $g (x) = - 30$ .

[2]

Set up $- 4 x - 6 = - 30$ and solve.
Answer: $x = 6$

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Transformations, composite and inverse functionsExtended2 questions

The functions are $f (x) = x^{2} + 3$ and $g (x) = - 3 x - 4$ . Evaluate the composite $f (g (2))$ by working from the inside out.

[2]

Work out the inside first: $g (2) = - 10$ .
Then $f (- 10) = (- 10)^{2} + 3 = 103$ .
Answer: $103$

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Find the inverse function $f^{- 1} (x)$ of $f (x) = 3 x + 6$ .

[2]

Write $y = 3 x + 6$ and swap x and y: $x = 3 y + 6$ .
Make y the subject: $y = \frac{x - 6}{3}$ .
Answer: $f^{- 1} (x) = \frac{x - 6}{3}$

3 steps

Exponential and logarithmic functionsExtended2 questions

Find $x$ when $3^{x} = 729$ . Use your answer to state the value of $\log_{3} 729$ .

[2]

Write 729 as a power of 3: $3^{6}$ .
So $x = 6$ , and the logarithm asks the same question.
Answer: $x = 6$ ; $\log_{3} 729 = 6$

3 steps

Solve $3 \times {1.25}^{x} = 41$ . Give your answer to 2 decimal places.

[3]

Divide by 3: ${1.25}^{x} = \frac{41}{3}$ .
Take logs: $x = \frac{\log (41 / 3)}{\log 1.25} \approx 11.72$ .
Answer: $x \approx 11.72$

3 steps

Rational expressions and equationsExtended2 questions

Simplify $\frac{x^{2} - 25}{x^{2} - 7 x + 10}$ .

[2]

Factorise top and bottom: $\frac{(x - 5) (x + 5)}{(x - 5) (x - 2)}$ .
Cancel the common factor $(x - 5)$ .
Answer: $\frac{x + 5}{x - 2}$

3 steps

Write $\frac{5}{x + 3} + \frac{1}{x + 5}$ as a single fraction.

[3]

Common denominator $(x + 3) (x + 5)$ .
Top: $5 (x + 5) + 1 (x + 3) = 6 x + 28$ .
Answer: $\frac{6 x + 28}{(x + 3) (x + 5)}$

3 steps

Quadratic inequalitiesExtended2 questions

Solve $(x + 1) (x - 2) < 0$ .

[2]

The graph is a U-shaped parabola crossing the x-axis at -1 and 2.
It is below the axis (negative) between the roots.
Answer: $- 1 < x < 2$

3 steps

Solve $x^{2} - 8 x + 7 \leq 0$ .

[3]

Factorise: $(x - 1) (x - 7)$ .
Sketch the parabola and read off where it is on or below the axis.
Answer: $1 \leq x \leq 7$

3 steps

Angles and polygons2 questions

Four angles meet at a point. Three of them are $67^{\circ}$ , $45^{\circ}$ and $80^{\circ}$ . Find the fourth angle.

[1]

Angles around a point add up to $360^{\circ}$ .
$360 - 67 - 45 - 80 = 168$
Answer: $168^{\circ}$

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Work out the size of each interior angle of a regular pentagon.

[2]

Sum of interior angles: $(5 - 2) \times 180 = 540$ .
Regular, so all 5 are equal: $540 \div 5 = 108$ .
Answer: $108^{\circ}$

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Perimeter, area, surface area and volume2 questions

A circle has radius 3 cm. Work out its circumference, to 1 decimal place.

[1]

$C = 2 π r$ : $2 \times π \times 3$ ≈ 18.8
Answer: 18.8 cm

2 steps

A cylinder has radius 5 cm and height 4 cm. Work out its volume, to 3 significant figures.

[2]

$V = π r^{2} h = π \times 5^{2} \times 4 \approx 314$
Answer: 314 cm³

2 steps

Pythagoras' theorem2 questions

The two sides next to the right angle of a triangle measure 8 cm and 11 cm. How long is the longest side, to 1 decimal place?

[1]

$c^{2} = 8^{2} + 11^{2} = 185$
$c = \sqrt{185} \approx 13.6$
Answer: 13.6 cm

3 steps

A right-angled triangle has hypotenuse 10 cm and one shorter side 6 cm. Work out the length of the third side, to 1 decimal place.

[2]

The hypotenuse is the longest side, so subtract: $10^{2} - 6^{2} = 64$ .
$\sqrt{64} \approx 8.0$
Answer: 8.0 cm

3 steps

Coordinate geometry of straight lines2 questions

Find the gradient of the line through $(- 5, 2)$ and $(- 1, - 6)$ .

[1]

$m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{- 6 - (2)}{- 1 - (- 5)} = - 2$
Answer: $- 2$

2 steps

The points $(- 5, 23)$ and $(7, - 25)$ both lie on one straight line. Work out its gradient $m$ and its $y$ -intercept $c$ , and use them to write the equation of the line.

[3]

Gradient: $\frac{- 25 - (23)}{7 - (- 5)} = - 4$ .
Substitute a point to find c: $23 = - 4 (- 5) + c, c = 3$ .
Answer: $y = - 4 x + 3$

3 steps

Right-angled trigonometry2 questions

A right-angled triangle has a hypotenuse of length 11 cm and an angle of $62^{\circ}$ . Calculate the length of the side next to (not the hypotenuse) this angle, correct to 3 significant figures.

[2]

$\cos θ = \frac{adj}{hyp}$ , so $x = 11 \cos 62^{\circ} \approx 5.16$ .
Answer: 5.16 cm

2 steps

In a right-angled triangle, the side facing the angle $θ$ measures 12 cm and the shorter side next to it measures 3 cm. Calculate $θ$ correct to 1 decimal place.

[2]

$\tan θ = \frac{12}{3}$
$θ = \tan^{- 1} (\frac{12}{3}) \approx {76.0}^{\circ}$
Answer: ${76.0}^{\circ}$

3 steps

Transformations, congruence and similarity2 questions

Reflect the point $P (- 2, - 5)$ in the y-axis. Where does it end up?

[1]

Reflecting in the y-axis keeps y and changes the sign of x.
Answer: $(2, - 5)$

2 steps

Triangle T has corners at $(3, 2)$ , $(3, 4)$ , $(3, 3)$ . Enlarge T by a scale factor of 3 about the origin and give the corners of the new triangle.

[2]

Multiply every coordinate by 3.
Answer: $(9, 6)$ , $(9, 12)$ , $(9, 9)$

2 steps

Arcs and sectors2 questions

A slice of a circle (a sector) has radius 4 cm and an angle of $30^{\circ}$ at the centre. Calculate the length of its curved edge, correct to 3 significant figures.

[2]

$\frac{30}{360} \times 2 π \times 4 \approx 2.09$
Answer: 2.09 cm

2 steps

A slice of a circle (a sector) has radius 20 cm and an angle of $135^{\circ}$ at the centre. Calculate how much area it covers, correct to 3 significant figures.

[2]

$\frac{135}{360} \times π \times 20^{2} \approx 471$
Answer: 471 cm²

2 steps

Circle theoremsExtended2 questions

Points A, B and C lie on a circle with centre O. Angle ACB at the circumference is $29^{\circ}$ , and C is on the major arc. Find angle AOB.

[1]

The angle at the centre is twice the angle at the circumference standing on the same arc: $2 \times 29 = 58$ .
Answer: $58^{\circ}$

2 steps

ABCD is a cyclic quadrilateral. Angle A is $85^{\circ}$ and angle B is $93^{\circ}$ . Find angles C and D.

[2]

Opposite angles of a cyclic quadrilateral add up to $180^{\circ}$ .
C is opposite A, D is opposite B.
Answer: C = $95^{\circ}$ , D = $87^{\circ}$

3 steps

Sine rule, cosine rule and area of a triangleExtended2 questions

Triangle ABC has $a = 18$ , $b = 14$ and angle $C = 34^{\circ}$ . Find its area, to 3 significant figures.

[2]

$Area = \frac{1}{2} a b \sin C = \frac{1}{2} (18) (14) \sin 34^{\circ} \approx 70.5$
Answer: 70.5 units²

2 steps

In triangle ABC, $a = 4$ , $b = 8$ and $C = 39^{\circ}$ . Find $c$ , to 3 significant figures.

[3]

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$
$c = \sqrt{4^{2} + 8^{2} - 2 (4) (8) \cos 39^{\circ}} \approx 5.50$
Answer: $c \approx 5.50$

3 steps

Trigonometric functions and the unit circleExtended2 questions

Write down the exact value of $\tan 60^{\circ}$ .

[1]

Use the special triangles (half an equilateral triangle, or half a square).
Answer: $\sqrt{3}$

2 steps

Solve $\cos x = - 0.6$ for $0^{\circ} \leq x \leq 360^{\circ}$ , giving your answers to 1 decimal place.

[3]

Calculator value: $x = \cos^{- 1} (- 0.6) \approx {126.9}^{\circ}$ .
Use the symmetry of the graph (or the unit circle) for the second solution in the range.
Answer: $x \approx {126.9}^{\circ} or {233.1}^{\circ}$

3 steps

Collecting data, sampling and bias2 questions

A class survey asked 34 students one question. The frequencies are: apple 9, banana 7, orange 3, grapes ?.

Find the missing frequency. [1]
What fraction of the students chose grapes? Simplify it. [1]

The frequencies add up to the number of students: $34 - 19 = 15$ .
Answers: (a) 15; (b) $\frac{15}{34}$

2 steps

A group of 240 people is made up of 120 junior members, 30 adult members, 90 senior members. A stratified sample of 40 is taken.

How many from each group should be in the sample? [3]

Each group's share of the sample matches its share of the population.
$\frac{120}{240} \times 40 = 20$ ; $\frac{30}{240} \times 40 = 5$ ; $\frac{90}{240} \times 40 = 15$
Answer: 20 junior members, 5 adult members, 15 senior members

3 steps · 1 to fill in

Averages and spread2 questions

Here are some data: $25, 14, 25, 2, 37, 52, 18, 43$ . Find the mean, median, mode and range.

Mean [1]
Median [1]
Mode [1]
Range [1]

Mean: add them up and divide by 8: $216 \div 8 = 27$ .
Median: put them in order (2, 14, 18, 25, 25, 37, 43, 52) and take the middle.
Range = largest − smallest.
Answers: (a) $27$ ; (b) $25$ ; (c) $25$ ; (d) $50$

4 steps

The mean of 6 numbers is 11. 5 of them are $7, 8, 5, 19, 18$ . Find the missing number.

[2]

Total of all 6 numbers: $11 \times 6 = 66$ .
Subtract the ones you know: $66 - 57 = 9$ .
Answer: $9$

3 steps · 1 to fill in

Statistical graphs: box plots, cumulative frequency and histograms2 questions

Here are 11 test scores in order: $13, 17, 19, 37, 53, 67, 70, 75, 83, 85, 88$ . Find the median, the quartiles and the interquartile range.

Median, lower quartile, upper quartile [2]
Interquartile range [1]

The median is the 6th value: 67.
The lower quartile is the middle of the lower five values (19); the upper quartile the middle of the upper five (83).
IQR = $83 - 19 = 64$ .
Answers: (a) 67, 19, 83; (b) 64

4 steps · 1 to fill in

Times, in minutes, are grouped into classes $0 < t \leq 5$ , $5 < t \leq 15$ , $15 < t \leq 25$ , $25 < t \leq 45$ . The class $5 < t \leq 15$ has frequency 10.

Find the frequency density of this class for a histogram. [2]

Frequency density = frequency ÷ class width.
$10 \div 10$
Answer: $1$

3 steps

Scatter graphs, correlation and regression2 questions

A line of best fit drawn through some data has equation $y = 2.2 x - 3$ . Predict the value of $y$ for $x = 20$ .

[1]

Substitute: $2.2 \times 20 - 3 = 41$ .
Answer: $41$

2 steps

Find the mean point $(\bar{x}, \bar{y})$ of the data $x : 16, 9, 4, 5, 6$ and $y : 30, 39, 25, 57, 59$ . The line of best fit should pass through it.

[2]

$\bar{x} = 40 \div 5$ , $\bar{y} = 210 \div 5$
Answer: $(8, 42)$

2 steps

Probability of single and combined events2 questions

There are 9 red, 1 blue, 6 green counters in a bag. Without looking, Zoe pulls one out.

Find the chance that Zoe's counter is blue. [1]
Find the chance that it is NOT blue. [1]

There are 16 counters altogether and 1 are blue.
P(not A) = 1 − P(A).
Answers: (a) $\frac{1}{16}$ ; (b) $\frac{15}{16}$

3 steps

The probability that a spinner lands on blue is $0.8$ . It is spun 200 times. How many times would you expect it to land on blue?

[1]

Expected frequency = probability × number of trials: $0.8 \times 200 = 160$ .
Answer: 160

2 steps

Conditional probability and independenceExtended2 questions

A survey of students: 37 girls and 27 boys walk to school; 38 girls and 34 boys do not walk.

A student who walks is chosen at random. Find the probability that the student is a girl. [2]
A girl is chosen at random. Find the probability that she walks. [2]

Given that the student walks, only the 64 walkers count.
Given a girl, only the 75 girls count.
Answers: (a) $\frac{37}{64}$ ; (b) $\frac{37}{75}$

3 steps

A box has 6 red and 4 green pens. Two pens are taken at random without replacement. Find the probability that both are red.

[3]

$P (red then red) = \frac{6}{10} \times \frac{5}{9} = \frac{1}{3}$
The second fraction changes because the first pen is not put back.
Answer: $\frac{1}{3}$

3 steps

Standard deviationExtended2 questions

Find the mean and the (population) standard deviation of $16, 13, 14, 20, 5, 9$ , to 2 decimal places.

Mean [1]
Standard deviation [2]

Use the statistics mode of a calculator, or find the mean squared deviation and take its square root.
Answers: (a) $12.83$ ; (b) $4.81$

2 steps

A set of marks has mean 29 and standard deviation 3.9. Every mark is increased by 10. Find the new mean and standard deviation.

[2]

Adding a constant moves every mark, so the mean changes but the spread does not.
Answer: mean 39, standard deviation 3.9

2 steps

Sets and Venn diagrams2 questions

$A$ is the set of multiples of 2 and $B$ the set of multiples of 6, using the whole numbers from 1 to 30.

List the elements of $A \cap B$ . [2]
How many elements are in $A \cup B$ ? [2]

$A = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30}$ , $B = {6, 12, 18, 24, 30}$
Intersection: in both. Union: in either (count each number once).
Answers: (a) ${6, 12, 18, 24, 30}$ ; (b) 15

3 steps

In a class of 36 students, 19 play football, 18 play an instrument and 7 do both.

How many students play football only? [1]
How many students do neither? [2]

Fill in the middle of the Venn diagram first: 7.
Then play football only: $19 - 7 = 12$ ; play an instrument only: $18 - 7 = 11$ .
Neither: $36 - 12 - 11 - 7 = 6$ .
Answers: (a) 12; (b) 6

4 steps · 1 to fill in

Algorithms and flowcharts2 questions

Follow these instructions. Start with 4. Multiply by 2, then subtract 4. Repeat with the answer until you have done this 3 times.

What number do you finish with? [2]

The numbers are 4, 4, 4.
Answer: 4

2 steps

A flowchart says: start with 11. If the number is even, halve it; if it is odd, multiply it by 3 and add 1. Stop when you reach 1. How many steps does it take?

[2]

Write each number down as you go and count the steps until you reach 1.
Answer: 14 steps

2 steps

Systematic counting2 questions

A café offers 2 starters, 3 main courses and 4 desserts. How many different three-course meals can be chosen?

[1]

Multiply the number of choices for each course: $2 \times 3 \times 4 = 24$ .
Answer: 24

2 steps

A 3-digit code uses the digits 0 to 9, and the first digit cannot be 0 and no digit is repeated. How many different codes are possible?

[2]

Count the choices for each position in turn: $9 \times 9 \times 8$ = 648.
Answer: 648

2 steps

Networks and graphsExtended2 questions

A network has vertices A, B, C, D, E with degrees A: 4, B: 2, C: 2, D: 3, E: 1. How many edges does it have?

[2]

Each edge adds 1 to the degree of each of its two ends, so the degrees add up to twice the number of edges.
$12 \div 2 = 6$
Answer: 6

3 steps · 1 to fill in

Roads between five villages have these lengths (km): B–A 4, C–A 11, D–A 9, E–B 7, B–C 9, A–E 6. Find the length of the shortest route from A to E.

[3]

Work outwards from A, keeping the shortest distance found so far to each village (Dijkstra's method), or list the possible routes.
Shortest A to E: 6 km.
Answer: 6 km

3 steps

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