Problem-solving Pack

Mathematics

Year 9 · Algebra, Equations and Lines

Problem-solving Pack

Name: _________________________________

Date: _________________ Class: ___________

These problems are designed to challenge you. Read each question carefully. Show all your reasoning — a correct answer without working receives no credit.

1Problem 1 of 48

Line through two points. The line AB passes through

A(-2, 1)

and

B(7, 4)

.

(a) Find the gradient as a fraction in simplest form.
(b) Find the equation in

y = mx + c

form.
(c) State the equation of a line parallel to AB through the origin.

Working space

2Problem 2 of 48

Dividing a segment. Point C divides AB in ratio 2:1, where

A(-2, 1)

B(7, 4)

.

Find C.

Working space

3Problem 3 of 48

Distance and midpoint together. Two villages at

A(2, 5)

and

B(10, 11)

on a map (km).

(a) Find the straight-line distance.
(b) Find the midpoint.
(c) A new road perpendicular to AB through the midpoint. Find its equation.

Working space

4Problem 4 of 48

Mid-segment investigation. A quadrilateral with vertices A(0, 0), B(6, 0), C(8, 4), D(2, 6).

(a) Find the midpoints P, Q, R, S of sides AB, BC, CD, DA.
(b) Show PQRS is a parallelogram by comparing gradients.

Working space

5Problem 5 of 48

Perpendicular bisector. Find the equation of the perpendicular bisector of A(1, 2) and B(5, 6).

Working space

6Problem 6 of 48

Triangle on coordinate plane. Find the perimeter of the triangle with vertices A(0, 0), B(4, 3), C(8, 0).

Working space

7Problem 7 of 48

Distance and perpendicular distance. A point

P(5, 1)

and a line

y = 2x - 1

.

(a) Find the equation of the line through P perpendicular to the given line.
(b) Find the foot of the perpendicular (i.e. the point on the line closest to P).
(c) Find the perpendicular distance from P to the line.

Working space

8Problem 8 of 48

Investigating parallel/perpendicular. Lines

\ell_1: y = mx + 2

and

\ell_2: y = (3 - m)x + 5

.

(a) For what value of

m

are

\ell_1

and

\ell_2

parallel?
(b) For what value(s) of

m

are they perpendicular?

Working space

9Problem 9 of 48

Triangle bounded by three lines. Sketch the triangle bounded by

y = x + 1

y = -x + 5

y = 0

.

(a) Find the three vertices.
(b) Find the area.

Working space

10Problem 10 of 48

Identifying a parallelogram. Show that the points A(1, 1), B(4, 1), C(6, 4), D(3, 4) form a parallelogram.

Working space

11Problem 11 of 48

Centroid of a triangle. A triangle has vertices A(0, 0), B(6, 0), C(3, 6).

(a) Find the centroid (average of vertices).
(b) Find the medians and verify they all pass through the centroid.

Working space

12Problem 12 of 48

Reflection of a line. The line

y = 2x + 1

is reflected in the

x

-axis.

(a) Find the equation of the image.
(b) Find the equation when reflected in the

y

-axis.

Working space

13Problem 13 of 48

Solve simultaneously. Solve the system

y = 2x + 1

and

3x + y = 11

, and verify by substitution into both equations.

Working space

14Problem 14 of 48

Man and son. A man is currently 4× his son's age. In 4 years, he will be 3× as old.

(a) Let son's age be

s

. Write an equation.
(b) Solve to find both ages.

Working space

15Problem 15 of 48

Concert ticket pricing. Friday: 50 adult + 20 child = £790. Saturday: 80 adult + 40 child = £1320.

(a) Write 2 simultaneous equations.
(b) Solve them.

Working space

16Problem 16 of 48

Mobile phone tariffs. Tariff A: £15 + 10p/min. Tariff B: £25 + 5p/min.

(a) Write equations for total cost

C

in terms of minutes

m

.
(b) Find

m

for equal cost.
(c) Which is cheaper for 300 min/month?

Working space

17Problem 17 of 48

Three methods. Solve

2x + 3y = 19

and

4x - y = 17

using:

(a) Substitution
(b) Elimination
(c) Equating

y = y

(or graphical)

Working space

18Problem 18 of 48

Boat and current. A boat travels 36 km downstream in 2 hours and back upstream in 3 hours.

(a) Find the speeds (downstream and upstream).
(b) Set up simultaneous equations for boat speed

b

and current

c

.
(c) Solve.

Working space

19Problem 19 of 48

Mixture problem. A 10 L solution is 30% acid. How much pure water should be added to make it 20% acid?

Working space

20Problem 20 of 48

Sum and product. Two numbers have sum 14 and product 48.

(a) Set up simultaneous equations.
(b) Solve.
(c) The numbers are the roots of a quadratic. State it.

Working space

21Problem 21 of 48

No solution or infinite solutions. Consider the systems:

(a)

x + y = 5

and

2x + 2y = 12

.
(b)

x + y = 5

and

2x + 2y = 10

.

For each, decide whether it has a unique solution, no solution, or infinite. Justify.

Working space

22Problem 22 of 48

Two-digit reversal. A two-digit number has digit sum 11. When the digits are reversed, the new number is 27 less than the original.

(a) Set up two equations in

a

(tens) and

b

(units).
(b) Solve.

Working space

23Problem 23 of 48

Geometry application. A rectangle has perimeter 30 cm and area 50 cm². Find its dimensions.

Working space

24Problem 24 of 48

Three friends sharing. Alice, Bob, Charles share a sum.

A + B = 50, B + C = 60, A + C = 70

.

(a) Add all three equations and find

A + B + C

.
(b) Find each share.

Working space

25Problem 25 of 48

Index laws round-up. Simplify each:

(a)

3 \times 3 \times 3 \times 3 \times 3

as a power of 3.
(b)

\dfrac{x^8}{x^3}

.
(c)

(2^4)^3

as a power of 2.
(d)

x^5 \times x^{-3}

Working space

26Problem 26 of 48

UK population growth. 2016:

6.50 \times 10^7

. Annual growth 0.6%.

(a) Find the multiplier.
(b) Estimate 2020 population (3 s.f.).

Working space

27Problem 27 of 48

Standard form arithmetic. Work out each in standard form:

(a)

(4 \times 10^6) \times (3 \times 10^{-2})

(b)

\dfrac{6 \times 10^8}{1.5 \times 10^3}

(c)

(2 \times 10^4)^3

Working space

28Problem 28 of 48

Fractional indices. Evaluate exactly:

(a)

25^{1/2}

(b)

8^{1/3}

(c)

16^{3/4}

(d)

27^{-2/3}

Working space

29Problem 29 of 48

Standard form: bacteria growth. Starts at

2.5 \times 10^4

. Doubles per hour.

(a) After 1 hour.
(b) After 5 hours (3 s.f.).
(c) After how many hours >

10^8

Working space

30Problem 30 of 48

Astronomical distances. Distance from Sun to Pluto ≈

5.9 \times 10^9

km. Distance light travels per second ≈

3 \times 10^5

km/s.

(a) Find the time for light to reach Pluto.
(b) Convert to hours.

Working space

31Problem 31 of 48

Solving index equations. Solve each:

(a)

2^x = 32

(b)

5^x = 1/125

(c)

x^4 = 81

(all real solutions)

Working space

32Problem 32 of 48

Large vs small. Order from smallest to largest:

5 \times 10^4, 2 \times 10^5, 8 \times 10^3, 9 \times 10^4

Working space

33Problem 33 of 48

Compound percentage with standard form. A population is

4.0 \times 10^6

in 2020, growing 2.5% per year.

(a) Multiplier.
(b) Population in 2025 (3 s.f.).
(c) When does it exceed

5 \times 10^6

Working space

34Problem 34 of 48

Atomic mass. A hydrogen atom has mass ≈

1.67 \times 10^{-24}

g.

(a) How many hydrogen atoms in 1 g?
(b) Express in standard form.

Working space

35Problem 35 of 48

Algebraic indices. Simplify each:

(a)

(2x^3)^4

(b)

\dfrac{12 x^7}{4 x^3}

(c)

\sqrt[3]{27 x^6}

(d)

(a^{1/2})^4

Working space

36Problem 36 of 48

Doubling chessboard. A famous problem: a chessboard has 64 squares. Place 1 grain on the first, 2 on the second, 4 on the third, ... doubling each time.

(a) How many grains on the 10th square?
(b) The 32nd square?
(c) The 64th square — give in standard form.

Working space

37Problem 37 of 48

Triangle on coordinate plane. Triangle ABC has vertices

A(1, 1)

B(4, 1)

C(1, 5)

.

(a) Find the area.
(b) Reflect in the

x

-axis. State the image vertices.
(c) Rotate 90° anticlockwise about the origin. State the image vertices.
(d) Translate by vector

(2, -3)

. State the image vertices.

Working space

38Problem 38 of 48

Similar figures. Two similar rectangles have linear scale factor 3:2.

(a) The larger has area 81 cm². Find the smaller area.
(b) The larger has perimeter 60 cm. Find the smaller perimeter.
(c) If the larger has dimensions

a

b

, express the smaller in terms of

a

and

b

Working space

39Problem 39 of 48

Cone scaling. Two similar cones have heights 5 cm and 10 cm.

(a) Find the volume scale factor.
(b) The smaller has volume 25 cm³. Find the larger volume.
(c) The smaller has surface area 30 cm². Find the larger SA.

Working space

40Problem 40 of 48

Triangle congruence. Triangles ABC and PQR have

AB = PQ = 6

BC = QR = 8

, and the included angle

B = Q = 50°

.

(a) State the congruence rule.
(b) State the relationship between the third sides.
(c) Are these triangles similar?

Working space

41Problem 41 of 48

Transformation composition. A point

P(2, 3)

is reflected in the

y

-axis, then rotated 90° anticlockwise about the origin, then translated by

(1, -2)

.

(a) Find the position after each step.
(b) Find the final image.

Working space

42Problem 42 of 48

Similar triangles in a real-world scenario. A tree of height 12 m casts a shadow 4 m long, while a flagpole of unknown height casts a shadow 7 m long.

(a) Why are the shadow triangles similar?
(b) Find the flagpole's height.
(c) Find the angle of elevation of the sun.

Working space

43Problem 43 of 48

Volume scaling. Two similar pyramids have linear scale factor 4:7.

(a) Find the volume scale factor.
(b) The smaller pyramid has volume 64 cm³. Find the larger.
(c) The larger pyramid has surface area 245 cm². Find the smaller SA.

Working space

44Problem 44 of 48

Two-step transformation challenge. A triangle has vertices

A(2, 1), B(5, 1), C(2, 4)

.

(a) Reflect in the

x

-axis: state image vertices.
(b) Then rotate 180° about the origin.
(c) Identify the single transformation that combines both into one.

Working space

45Problem 45 of 48

Map scale and similarity. A 1:50 000 map shows a park of area 8 cm². Find the real area in (a) m², (b) km².

Working space

46Problem 46 of 48

Congruence test puzzle. Two triangles have:

A: sides 5, 6, 7
B: sides 5, 6, included angle of 40°

(a) Which uniqueness rules might apply?
(b) Are they necessarily congruent?
(c) Compute the third side of triangle B using the cosine rule (or Pythagoras-like reasoning), and decide whether it could equal 7.

Working space

47Problem 47 of 48

Enlargement from a centre. Triangle ABC has

A(2, 1), B(5, 1), C(3, 4)

. Enlarge by sf 3 from centre

P(1, 1)

.

(a) Find the image vertices.
(b) Find the image area, given the original area is 4.5.

Working space

48Problem 48 of 48

Symmetry investigation. A regular hexagon has rotational and reflective symmetries.

(a) State the order of rotational symmetry.
(b) State the number of lines of reflective symmetry.
(c) Compare with an equilateral triangle.

Working space