MYP year 5 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

Write $0.1 \dot{7}$ as a fraction in its simplest form.

[3]

Let $x = 0.177 \dots$ . Then $10 x = 1.77 \dots$ and $100 x = 17.7 \dots$ .
Subtract: $90 x = 17 - 1 = 16$ .
$x = \frac{16}{90} = \frac{8}{45}$
Answer: $\frac{8}{45}$

4 steps · 1 to fill in

Write the recurring decimal $0. \dot{9} \dot{0}$ as a fraction in its simplest form.

[3]

Let $x = 0.9090 \dots$ .
Two digits repeat, so multiply by 100: $100 x = 90.90 \dots$ .
Subtract: $99 x = 90$ , so $x = \frac{90}{99}$ .
Simplify: $\frac{10}{11}$ .
Answer: $\frac{10}{11}$

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Fractions, percentages, ratio and proportion2 questions

After a 25% rise, the price of a games console is €106.25. What was the price before the rise?

[3]

The new price is 125% of the original, so the multiplier is 1.25.
Original = new price ÷ multiplier: $106.25 \div 1.25 = 85.00$ .
Answer: €85.00

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Gita and Hugo share 56 euros in the ratio $4 : 3$ .

How many euros does Gita get? [1]
How many more does Gita get than Hugo? [1]

Total parts: $4 + 3 = 7$ .
One part: $56 \div 7 = 8$ .
Gita gets $4 \times 8 = 32$ ; Hugo gets $3 \times 8 = 24$ .
Difference: 8.
Answers: (a) 32; (b) 8

5 steps · 1 to fill in

Powers, roots and standard form2 questions

Calculate $(5.1 \times 10^{8}) \times (5.4 \times 10^{- 5})$ without changing to ordinary numbers first. Write the result in standard form, correct to 3 significant figures.

[2]

Multiply the numbers: $5.1 \times 5.4 = 27.5$ .
Add the powers of 10: $10^{3}$ .
Adjust so the first number is between 1 and 10: $2.75 \times 10^{4}$ .
Answer: $2.75 \times 10^{4}$

4 steps

Write $402000000$ in standard form.

[1]

Move the decimal point so the number is between 1 and 10: 4.02.
The point moved 8 places left, so the power is 8: $4.02 \times 10^{8}$ .
Answer: $4.02 \times 10^{8}$

3 steps

Estimation, rounding and bounds2 questions

A rectangle measures 27 m by 6 m, each to the nearest m.

Write down the lower and upper bounds of the length. [1]
Work out the upper bound of the area. [2]
Work out the lower bound of the perimeter. [2]

To the nearest 1: the bounds are 0.5 below and 0.5 above.
Upper area: $27.5 \times 6.5 = 178.75$ .
Lower perimeter: $2 (26.5 + 5.5) = 64$ .
Answers: (a) 26.5 m and 27.5 m; (b) 178.75 m²; (c) 64 m

4 steps · 1 to fill in

Round every number in $\frac{78.3 \times 2.1}{0.48}$ to 1 significant figure, then use the rounded values to estimate its value.

[2]

Round: $78.3 \approx 80$ , $2.1 \approx 2$ , $0.48 \approx 0.5$ .
Work out: $\frac{80 \times 2}{0.5} = 320$ .
Answer: $320$

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Surds and fractional indicesExtended2 questions

Rationalise the denominator of $\frac{5}{3 + \sqrt{7}}$ and simplify.

[3]

Multiply top and bottom by $3 - \sqrt{7}$ .
Bottom: $(3)^{2} - (\sqrt{7})^{2} = 9 - 7 = 2$ .
Top: $5 (3 - \sqrt{7})$ , so the answer is $\frac{15}{2} - \frac{5 \sqrt{7}}{2}$ .
Answer: $\frac{15}{2} - \frac{5 \sqrt{7}}{2}$

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Without a calculator, find the exact value of $64^{- \frac{2}{3}}$ .

[2]

The denominator 3 means a cube root: $\sqrt[3]{64} = 4$ .
Raise to the power 2: $4^{2} = 16$ .
A negative power means one over it: $\frac{1}{16}$ .
Answer: $\frac{1}{16}$

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Financial mathematics: interest and depreciation2 questions

A car is bought for €38000. Its value drops by 18% each year. Find the first whole number of years after which it is worth under €15200.

[3]

Each year multiply by $0.82$ .
Year by year (or with a table): after 5 years the value is €14088.11, the first value below €15200.
Answer: 5 years

3 steps

€7000 is invested at 5% per year compound interest. Work out its value after 4 years, to the nearest cent.

[2]

Multiplier: $1 + \frac{5}{100} = 1.05$ .
Value: $7000 \times {1.05}^{4} = 8508.54$ .
Answer: €8508.54

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Expressions, expanding and factorising2 questions

Factorise $49 x^{2} - 64$ .

[2]

This is a difference of two squares: $(7 x)^{2} - 8^{2}$ .
So it factorises as $(7 x - 8) (7 x + 8)$ .
Answer: $(7 x - 8) (7 x + 8)$

3 steps

Factorise $x^{2} - 2 x - 48$ .

[2]

Find two numbers that multiply to -48 and add to -2: they are -8 and 6.
So $x^{2} - 2 x - 48 = (x - 8) (x + 6)$ .
Answer: $(x - 8) (x + 6)$

3 steps

Linear equations, inequalities and formulae2 questions

Solve the inequality $6 x - 4 \leq 14$ .

[2]

Add 4: $6 x \leq 18$ .
Divide by 6: $x \leq 3$ .
Answer: $x \leq 3$

3 steps

Solve $2 (5 x - 2) = 7 x + 26$ .

[2]

Expand the bracket first. Collect the x terms on one side and the numbers on the other.
Divide to find $x = 10$ .
Answer: $x = 10$

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Simultaneous equations2 questions

Solve the simultaneous equations $y = 3 x - 6$ and $6 x + 5 y = 75$ by substitution.

[4]

Substitute $y = 3 x - 6$ into the second equation.
Solve the linear equation for x, then work out y: $x = 5, y = 9$ .
Answer: $x = 5, y = 9$

3 steps

Solve the simultaneous equations $4 x - 5 y = - 25$ and $6 x - 6 y = - 24$ .

[4]

Multiply one or both equations so the coefficients of one letter match.
Eliminate it, solve, and substitute back: $x = 5, y = 9$ .
Answer: $x = 5, y = 9$

3 steps

Quadratic equations by factorising2 questions

Solve $2 x^{2} + 5 x = - 3$ .

[3]

Rearrange so one side is 0: $2 x^{2} + 5 x + 3 = 0$ .
Factorise: $(2 x + 3) (x + 1) = 0$ .
So $x = - \frac{3}{2}$ or $x = - 1$ .
Answer: $x = - \frac{3}{2} or x = - 1$

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Solve $x^{2} - 12 x + 27 = 0$ by factorising.

[3]

Two numbers that multiply to 27 and add to -12: -3 and -9.
$(x - 3) (x - 9) = 0$ , so $x = 3$ or $x = 9$ .
Answer: $x = 3 or x = 9$

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Quadratic formula, completing the square and the discriminantExtended2 questions

Write $x^{2} - 12 x + 3$ in the form $(x + p)^{2} + q$ , and state the coordinates of the turning point of $y = x^{2} - 12 x + 3$ .

[2]
Turning point: [1]
What is the minimum value of $y$ ? [1]

Halve the x coefficient: $p = - 6$ .
$x^{2} - 12 x + 3 = (x - 6)^{2} - 36 + 3 = (x - 6)^{2} - 33$
The turning point is $(6, - 33)$ .
Answers: (a) $(x - 6)^{2} - 33$ ; (b) $(6, - 33)$ ; (c) $- 33$

4 steps

Use the quadratic formula to solve $x^{2} + 3 x - 12 = 0$ . Give your answers to 2 decimal places.

[3]

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a} = \frac{- 3 \pm \sqrt{57}}{2}$
So $x \approx - 5.27$ or $x \approx 2.27$ .
Answer: $x = - 5.27 or x = 2.27$

3 steps

Sequences and general rules2 questions

Find the $n$ th term of the quadratic sequence $- 5, - 1, 9, 25, 47, \dots$ .

[3]

Second difference 6, so the rule starts $3 n^{2}$ .
Subtract $3 n^{2}$ from each term and find the linear rule that is left: $- 5 n - 3$ .
So the nth term is $3 n^{2} - 5 n - 3$ .
Answer: $3 n^{2} - 5 n - 3$

4 steps

Find an expression for the $n$ th term of the sequence $- 13, - 7, - 1, 5, 11, \dots$ .

[2]

The common difference is 6, so the rule starts $6 n$ .
When $n = 1$ , $6 n = 6$ , so add -19: $6 n - 19$ .
Answer: $6 n - 19$

3 steps

Functions: notation, graphs, domain and range2 questions

Find the range of $f (x) = x^{2} - 6 x + 8$ for $2 \leq x \leq 5$ .

[3]

Complete the square: $f (x) = (x - 3)^{2} - 1$ , so the minimum is -1 at $x = 3$ , inside the domain.
The maximum is at the end of the domain further from 3: $3$ .
Answer: $- 1 \leq f (x) \leq 3$

3 steps

Given $g (x) = 5 x - 1$ , solve $g (x) = - 16$ .

[2]

Set up $5 x - 1 = - 16$ and solve.
Answer: $x = - 3$

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Transformations, composite and inverse functionsExtended2 questions

Given $f (x) = x^{2} - 2$ and $g (x) = 2 x - 4$ , find and simplify $f (g (x))$ .

[3]

Replace x in f by $g (x)$ : $(2 x - 4)^{2} - 2$ .
Expand: $4 x^{2} - 16 x + 14$ .
Answer: $4 x^{2} - 16 x + 14$

3 steps

Find the inverse function $f^{- 1} (x)$ of $f (x) = - 2 x + 5$ .

[2]

Write $y = - 2 x + 5$ and swap x and y: $x = - 2 y + 5$ .
Make y the subject: $y = \frac{x - 5}{- 2}$ .
Answer: $f^{- 1} (x) = \frac{x - 5}{- 2}$

3 steps

Exponential and logarithmic functionsExtended2 questions

Write $\log_{2} 4 + \log_{2} 7 - \log_{2} 7$ as a single logarithm and evaluate it if you can.

[2]

Adding logs multiplies, subtracting divides: $\log_{2} \frac{4 \times 7}{7} = \log_{2} 4$ .
Answer: $\log_{2} 4$ = $2$

2 steps

Solve $2 \times {1.1}^{x} = 53$ . Give your answer to 2 decimal places.

[3]

Divide by 2: ${1.1}^{x} = \frac{53}{2}$ .
Take logs: $x = \frac{\log (53 / 2)}{\log 1.1} \approx 34.38$ .
Answer: $x \approx 34.38$

3 steps

Rational expressions and equationsExtended2 questions

Solve $\frac{- 10}{x + 3} = \frac{- 55}{x - 6}$ .

[3]

Cross-multiply: $- 10 (x - 6) = - 55 (x + 3)$ .
Solve the linear equation: $x = - 5$ , and check it does not make a denominator zero.
Answer: $x = - 5$

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Write $\frac{4}{x - 1} + \frac{4}{x - 3}$ as a single fraction.

[3]

Common denominator $(x - 1) (x - 3)$ .
Top: $4 (x - 3) + 4 (x - 1) = 8 x - 16$ .
Answer: $\frac{8 x - 16}{(x - 1) (x - 3)}$

3 steps

Quadratic inequalitiesExtended2 questions

For which values of $k$ does the equation $x^{2} + k x + 9 = 0$ have no real solutions? Give your answer as an inequality.

[3]

No real roots when the discriminant is negative: $k^{2} - 4 (1) (9) < 0$ .
So $k^{2} < 36$ , giving $- 6 < k < 6$ .
Answer: $- 6 < k < 6$

3 steps

Solve $x^{2} - 9 x + 20 > 0$ .

[3]

Factorise: $(x - 4) (x - 5)$ .
Sketch the parabola and read off where it is above the axis.
Answer: $x < 4 or x > 5$

3 steps

Angles and polygons2 questions

Each exterior angle of a regular polygon is $45^{\circ}$ . How many sides does it have?

[2]

Exterior angles add up to $360^{\circ}$ .
$360 \div 45 = 8$
Answer: 8 sides

3 steps · 1 to fill in

Work out the size of each interior angle of a regular heptagon, to 1 decimal place.

[2]

Sum of interior angles: $(7 - 2) \times 180 = 900$ .
Regular, so all 7 are equal: $900 \div 7 = 128.6$ .
Answer: ${128.6}^{\circ}$

3 steps · 1 to fill in

Perimeter, area, surface area and volume2 questions

A food tin is a cylinder with a lid and a base. Its radius is 5 cm and it is 10 cm tall. How much metal sheet covers its whole outside (base, lid and curved side)? Answer to 3 significant figures.

[3]

Two circles: $2 π r^{2}$ .
The curved surface unrolls into a rectangle $2 π r \times h$ .
Total $\approx 471$ .
Answer: 471 cm²

4 steps

A cuboid is 6 cm long, 9 cm wide and 6 cm high. Work out its volume and its surface area.

Volume [1]
Surface area [2]

Volume = length × width × height = $6 \times 9 \times 6 = 324$ .
Surface area: three pairs of rectangles, $2 (54 + 36 + 54) = 288$ .
Answers: (a) 324 cm³; (b) 288 cm²

3 steps · 1 to fill in

Pythagoras' theorem2 questions

A box is 14 cm long, 6 cm wide and 10 cm high. Find the length of the longest straight stick that fits inside it (corner to opposite corner), to 1 decimal place.

[3]

Diagonal of the base: $\sqrt{14^{2} + 6^{2}}$ .
Then with the height: $\sqrt{14^{2} + 6^{2} + 10^{2}} = \sqrt{332} \approx 18.2$ .
Answer: 18.2 cm

3 steps

A right-angled triangle has hypotenuse 18 cm and one shorter side 13 cm. Work out the length of the third side, to 1 decimal place.

[2]

The hypotenuse is the longest side, so subtract: $18^{2} - 13^{2} = 155$ .
$\sqrt{155} \approx 12.4$
Answer: 12.4 cm

3 steps

Coordinate geometry of straight lines2 questions

A line has gradient $1$ . Find the equation of the line perpendicular to it that passes through $(6, - 4)$ , in the form $y = m x + c$ .

[3]

Perpendicular gradients multiply to −1: $m = - 1$ .
Substitute the point: $c = - 4 - (- 1) (6) = 2$ .
Answer: $y = - 1 x + 2$

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The points $(- 2, 4)$ and $(7, 22)$ 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: $\frac{22 - (4)}{7 - (- 2)} = 2$ .
Substitute a point to find c: $4 = 2 (- 2) + c, c = 8$ .
Answer: $y = 2 x + 8$

3 steps

Right-angled trigonometry2 questions

From a point 51 m from the base of a tower, the angle of elevation of a window is $36^{\circ}$ and of the top of the tower is $48^{\circ}$ . How far is the window below the top? Give your answer to 3 significant figures.

[4]

Top: $51 \tan 48^{\circ}$ .
Window: $51 \tan 36^{\circ}$ .
Subtract: 19.6 m.
Answer: 19.6 m

4 steps

In a right-angled triangle, the side facing the angle $θ$ measures 8 cm and the hypotenuse measures 10 cm. Calculate $θ$ correct to 1 decimal place.

[2]

$\sin θ = \frac{8}{10}$
$θ = \sin^{- 1} (\frac{8}{10}) \approx {53.1}^{\circ}$
Answer: ${53.1}^{\circ}$

3 steps

Transformations, congruence and similarity2 questions

Triangle Q is a scaled-up copy of triangle P. A 7 cm side on P corresponds to a 10.5 cm side on Q.

What is the scale factor from P to Q? [1]
Another side on P is 13 cm. How long is the corresponding side on Q? [1]
P covers an area of 24 cm². What area does Q cover? [2]

Scale factor $10.5 \div 7 = 1.5$ .
Lengths multiply by the scale factor; areas by its square, $2.25$ .
Answers: (a) $1.5$ ; (b) 19.5 cm; (c) 54 cm²

3 steps

Triangle T has corners at $(1, 2)$ , $(3, 3)$ , $(4, 1)$ . Turn T through $90^{\circ}$ anticlockwise about the origin and give the corners of the new triangle.

[2]

A quarter turn anticlockwise about the origin sends $(x, y)$ to $(- y, x)$ .
Answer: $(- 2, 1)$ , $(- 3, 3)$ , $(- 1, 4)$

2 steps

Arcs and sectors2 questions

The curved edge of a sector is 21.50 cm long and its radius is 14 cm.

Calculate the angle at the centre, correct to 1 decimal place. [2]
How long is the whole boundary of the sector? [1]

Solve $\frac{θ}{360} \times 2 π (14) = 21.50$ .
Perimeter = two radii + arc.
Answers: (a) ${88.0}^{\circ}$ ; (b) 49.50 cm

3 steps

A slice of a circle (a sector) has radius 19 cm and an angle of $30^{\circ}$ at the centre. Calculate how much area it covers, correct to 3 significant figures.

[2]

$\frac{30}{360} \times π \times 19^{2} \approx 94.5$
Answer: 94.5 cm²

2 steps

Circle theoremsExtended2 questions

AB is a diameter of a circle and C is a point on the circle. Angle CAB is $65^{\circ}$ . A tangent to the circle at B meets AC extended at T. Find angle ABC and angle ATB.

Angle ABC [2]
Angle ATB [2]

The angle in a semicircle is $90^{\circ}$ , so angle ACB = $90^{\circ}$ and ABC = $180 - 90 - 65 = 25^{\circ}$ .
The tangent at B is perpendicular to the radius, so angle ABT = $90^{\circ}$ ; in triangle ABT, $∠ A T B = 180 - 90 - 65 = 25^{\circ}$ .
Answers: (a) $25^{\circ}$ ; (b) $25^{\circ}$

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ABCD is a cyclic quadrilateral. Angle A is $66^{\circ}$ and angle B is $99^{\circ}$ . 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 = $114^{\circ}$ , D = $81^{\circ}$

3 steps

Sine rule, cosine rule and area of a triangleExtended2 questions

In triangle ABC, $a = 10$ , $b = 9$ and $A = 70^{\circ}$ . Find angle B, to 1 decimal place. (Angle B is acute.)

[3]

$\frac{\sin B}{b} = \frac{\sin A}{a}$
$\sin B = \frac{9 \sin 70^{\circ}}{10}$ , so $B \approx {57.7}^{\circ}$
Answer: ${57.7}^{\circ}$

3 steps

In triangle ABC, $a = 8$ , $b = 11$ and $C = 147^{\circ}$ . Find $c$ , to 3 significant figures.

[3]

$c^{2} = a^{2} + b^{2} - 2 a b \cos C$
$c = \sqrt{8^{2} + 11^{2} - 2 (8) (11) \cos 147^{\circ}} \approx 18.2$
Answer: $c \approx 18.2$

3 steps

Trigonometric functions and the unit circleExtended2 questions

For the graph of $y = 6 \sin (3 x) + 5$ , write down the amplitude, the period in degrees, and the maximum and minimum values of y.

Amplitude and period [2]
Maximum and minimum [2]

Amplitude = 6; period = $360 \div 3 = 120^{\circ}$ .
Max = $5 + 6 = 11$ , min = $5 - 6 = - 1$ .
Answers: (a) amplitude 6, period $120^{\circ}$ ; (b) max 11, min -1

3 steps · 1 to fill in

Solve $\cos x = 0.7$ for $0^{\circ} \leq x \leq 360^{\circ}$ , giving your answers to 1 decimal place.

[3]

Calculator value: $x = \cos^{- 1} (0.7) \approx {45.6}^{\circ}$ .
Use the symmetry of the graph (or the unit circle) for the second solution in the range.
Answer: $x \approx {45.6}^{\circ} or {314.4}^{\circ}$

3 steps

Collecting data, sampling and bias2 questions

A group of 600 people is made up of 120 junior members, 180 adult members, 300 senior members. A stratified sample of 50 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.
$\frac{120}{600} \times 50 = 10$ ; $\frac{180}{600} \times 50 = 15$ ; $\frac{300}{600} \times 50 = 25$
Answer: 10 junior members, 15 adult members, 25 senior members

3 steps · 1 to fill in

A class survey asked 30 students one question. The frequencies are: walk 12, bus 8, car ?, bike 2.

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

The frequencies add up to the number of students: $30 - 22 = 8$ .
Answers: (a) 8; (b) $\frac{2}{5}$

2 steps

Averages and spread2 questions

The table shows the number of goals scored in some matches. Goals: $0, 2, 3, 4, 7$ . Frequency: $5, 7, 11, 2, 12$ . Find the mean number of goals per match, to 2 decimal places.

[3]

Multiply each value by its frequency and add: $139$ .
Divide by the total frequency 37.
Answer: $3.76$

3 steps

The mean of 6 numbers is 13. 5 of them are $24, 19, 7, 6, 3$ . Find the missing number.

[2]

Total of all 6 numbers: $13 \times 6 = 78$ .
Subtract the ones you know: $78 - 59 = 19$ .
Answer: $19$

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Statistical graphs: box plots, cumulative frequency and histograms2 questions

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 $25 < t \leq 45$ has frequency 28.

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

Frequency density = frequency ÷ class width.
$28 \div 20$
Answer: $1.4$

3 steps

Here are 11 test scores in order: $11, 15, 21, 26, 30, 41, 49, 70, 72, 77, 86$ . Find the median, the quartiles and the interquartile range.

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

The median is the 6th value: 41.
The lower quartile is the middle of the lower five values (21); the upper quartile the middle of the upper five (72).
IQR = $72 - 21 = 51$ .
Answers: (a) 41, 21, 72; (b) 51

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Scatter graphs, correlation and regression2 questions

Use your calculator (or the formulas) to find the equation of the regression line of $y$ on $x$ for: $x : 3, 6, 11, 15, 17, 19$ and $y : 19, 26, 34, 36, 46, 44$ . Give the gradient and intercept to 3 significant figures.

[3]

Enter the data in the statistics mode of the calculator and choose linear regression.
The gradient is about 1.59 and the intercept about 15.4.
Answer: $y = 1.59 x + 15.4$

3 steps

Find the mean point $(\bar{x}, \bar{y})$ of the data $x : 1, 6, 14, 1, 9, 18, 2$ and $y : 21, 50, 56, 57, 24, 18, 25$ . The line of best fit should pass through it.

[2]

$\bar{x} = 51 \div 7$ , $\bar{y} = 251 \div 7$
Answer: $(7.29, 35.86)$

2 steps

Probability of single and combined events2 questions

Two independent events A and B have $P (A) = 0.6$ and $P (B) = 0.9$ .

Find the probability that both happen. [1]
Find the probability that neither happens. [2]

Independent: multiply the probabilities.
P(neither) = $(1 - 0.6) (1 - 0.9)$ .
Answers: (a) $0.54$ ; (b) $0.04$

3 steps

The probability that a spinner lands on blue is $0.2$ . It is spun 60 times. How many times would you expect it to land on blue?

[1]

Expected frequency = probability × number of trials: $0.2 \times 60 = 12$ .
Answer: 12

2 steps

Conditional probability and independenceExtended2 questions

Each morning there is a $0.2$ chance that the school bus runs late, and mornings are independent of each other. Over 2 mornings, what is the probability that it runs late on at least one of them?

[3]

P(at least once) = 1 − P(never late).
$1 - (1 - 0.2)^{2} = 0.36$
Answer: $0.36$

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A box has 8 red and 9 green pens. Two pens are taken at random without replacement. Find the probability that both are red.

[3]

$P (red then red) = \frac{8}{17} \times \frac{7}{16} = \frac{7}{34}$
The second fraction changes because the first pen is not put back.
Answer: $\frac{7}{34}$

3 steps

Standard deviationExtended2 questions

The number of pets owned by 23 families: value $0, 1, 2, 3, 4$ , frequency $3, 6, 2, 6, 6$ . Find the mean and the standard deviation, to 2 decimal places.

Mean [1]
Standard deviation [2]

Enter the values with their frequencies in the calculator's statistics mode.
Answers: (a) $2.26$ ; (b) $1.42$

2 steps

A set of marks has mean 42 and standard deviation 1.9. Every mark is multiplied by 3. Find the new mean and standard deviation.

[2]

Multiplying scales every distance from the mean, so both the mean and the standard deviation are multiplied by 3.
Answer: mean 126, standard deviation 5.7

2 steps

Sets and Venn diagrams2 questions

56 people were asked about two apps. 20 use app P, 16 use app Q and 29 use neither. How many use both apps?

[3]
One of these people is chosen at random. Find the probability that they use exactly one app. [2]

Let $x$ be the number who use both: $(20 - x) + (16 - x) + x + 29 = 56$ .
Solve: $x = 9$ .
Answers: (a) 9; (b) $\frac{9}{28}$

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In a class of 37 students, 6 play football, 14 play an instrument and 2 do both.

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

Fill in the middle of the Venn diagram first: 2.
Then play football only: $6 - 2 = 4$ ; play an instrument only: $14 - 2 = 12$ .
Neither: $37 - 4 - 12 - 2 = 19$ .
Answers: (a) 4; (b) 19

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Algorithms and flowcharts2 questions

A function machine multiplies the input by 5 and then adds 11. The output is 21. What was the input? Describe how you undo the machine.

[2]

Undo each step in reverse order with the inverse operation.
$(21 - 11) \div 5 = 2$
Answer: 2 (subtract 11, then divide by 5)

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A flowchart says: start with 18. 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: 20 steps

2 steps

Systematic counting2 questions

At a meeting, each of the 12 people shakes hands once with every other person. How many handshakes are there? Explain why you do not just work out $12 \times 11$ .

[3]

Each person shakes 11 hands: $12 \times 11 = 132$ , but this counts every handshake twice.
Divide by 2: 66.
Answer: 66 (each handshake would be counted twice)

3 steps

A 3-digit code uses the digits 0 to 9, and no digit is repeated. How many different codes are possible?

[2]

Count the choices for each position in turn: $10 \times 9 \times 8$ = 720.
Answer: 720

2 steps

Networks and graphsExtended2 questions

A network has these edges: BA, CB, DB, AC, CD. Is it possible to travel along every edge exactly once? Count the odd vertices to decide.

[3]

Degrees: A 2, B 3, C 3, D 2.
A route along every edge once exists only when there are 0 or 2 odd vertices.
Answer: 2 odd vertices: yes, starting and finishing at the two odd vertices

3 steps

Roads between five villages have these lengths (km): B–A 14, C–A 6, D–C 15, E–A 5, C–E 13, B–D 2. 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: 5 km.
Answer: 5 km

3 steps

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