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Forming & Solving Equations (Cambridge O Level Maths)

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Forming & Solving Equations

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Forming & Solving Equations

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1
Sme Calculator3 marks

Rangan buys 3.6kg of potatoes and 2.8kg of leeks.
The total cost is $13.72.
Leeks cost $2.65 per kilogram.

Find the cost of 1 kg of potatoes.

$ ................................................ 

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2a
Sme Calculator3 marks

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

The total cost of 3 small lemon cakes and 2 large lemon cakes is $12.36.

Find the cost of a small lemon cake.

$ ................................................ 

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2b
Sme Calculator3 marks

The cost of 18 small chocolate cakes is the same as the cost of 7 large chocolate cakes.

Find the cost of a small chocolate cake.

$ ................................................ [3]

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3
Sme Calculator3 marks

The angles of a quadrilateral are x degree, left parenthesis x plus 5 right parenthesis degree, left parenthesis 2 x minus 25 right parenthesis degree and left parenthesis x plus 10 right parenthesis degree.

Find the value of x.

x = ................................................... 

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4
Sme Calculator3 marks

Oranges cost 21 cents each.
Alex buys x oranges and Bobbie buys left parenthesis x plus 2 right parenthesis oranges.
The total cost of these oranges is $4.20.

Find the value of x.

x = ................................................... 

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5a
Sme Calculator2 marks

Esme buys x magazines at $2.45 each and y cards at $3.15 each.

Write down an expression, in terms of x and y, for the total cost, in dollars, of the magazines and the cards.

$ ................................................... 

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5b
Sme Calculator2 marks

Esme spends $60.55 in total.
She buys 8 magazines.

How many cards does she buy?

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6
Sme Calculator5 marks
i)
Factorise  y squared plus 5 y minus 84.

[2]
ii)
cie-igcse-2018-may-jun-3-p4-q5bii
The area of the rectangle is 84 cm2.

Find the perimeter.

.......................................... cm [3]

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7
Sme Calculator4 marks

The diagram shows a trapezium.

xyopd_Z~_q3-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

A D = x cm.
B C spaceis the same length as A D.
A B is twice the length of A D.
D C is 4 cm longer than A B.

The perimeter of the trapezium is 38 cm.
Work out the length of A D.

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8
Sme Calculator3 marks

q7-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

The diagram shows a parallelogram.
The sizes of the angles, in degrees, are

2 x 3 x space minus 15 2 x 2 x space plus space 24

Work out the value of x.

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9
Sme Calculator3 marks

The diagram shows a right-angled triangle.

q9-meduim-2-13-forming-and-solving-equations-edexcel-gcse-maths

All the angles are in degrees.
Work out the size of the smallest angle of the triangle.

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10
Sme Calculator5 marks

Redlands School sent x students to a revision day.

   St Samuel's School sent twice as many students as Redlands School.
   Francis Long School sent 7 fewer students than Redlands School.

   Each student paid £ 15 for the revision day.
   The students paid a total of £1155

   Work out how many students were sent by each school to the revision day.
   You must show all your working.

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11
Sme Calculator6 marks

q3e-june-2021-4024-21-cambridge-o-level-maths-d

A rectangular piece of card measures 30 cm by 24 cm.

The net of an open box is made by removing a square from each corner of this piece of card.
Each square that is removed has side xcm.
The area of the net is 576 cm2.

i)
Form an equation in x and solve it to find the value of x.

x=..................................[3]

ii)
The net is made into an open box.
1000 cm3 of sand is placed inside the box.

Find the fraction of the box that is filled with sand.
Give your answer in its simplest form.
[3]

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1
Sme Calculator3 marks

Solve.

fraction numerator 3 y over denominator 2 y minus 1 end fraction space equals space 3 over 4

y = ..............................................

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2
Sme Calculator10 marks

cie-igcse-spec-p4-q5a-diagram

The perimeter of the rectangle is 80 cm.
The area of the rectangle is Acm2.

i)
Show that  x squared space space – space 40 x space plus space A space equals space 0.

[3]
ii)
When A space equals space 300, solve the equation space x squared space space – space 40 x space plus space A space equals space 0 space by factorising.
  
x = .................. or x = .................  [3]
iii)
When A space equals space 200, solve the equation space space x squared space space – space 40 x space plus space A space equals space 0 space using the quadratic formula.
Show all your working and give your answers correct to 2 decimal places.
  
x= .................. or x = .................  [4]

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3
Sme Calculator9 marks

cie-igcse-2019-oct-nov-p4-tz2-q6b

The area of triangle DEF is 70 m2.

i)
Show that  4 x squared plus 11 x minus 300 equals 0.

 

[4]

ii)
Use the quadratic formula to solve 4 x squared plus 11 x minus 300 equals 0.
Show all your working and give your answers correct to 2 decimal places.

 

x = .......................... or x = .......................... [4]

iii)
Find the length of DE.

 

DE = ............................................... m [1]

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4
Sme Calculator3 marks

A cone with height 14.8 cm has volume 275 cm3.
 
Calculate the radius of the cone.
[The volume, V, of a cone with radius r and height h is V equals 1 third straight pi r squared h.]
 

............................................... cm

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5a
Sme Calculator2 marks

Paulo and Jim each buy sacks of rice but from different shops.
Paulo pays $72 for sacks costing $m each.
Jim pays $72 for sacks costing $left parenthesis m plus 0.9 right parenthesis each.

i)
Find an expression, in terms of m, for the number of sacks Paulo buys.
 
 

[1]

ii)
Find an expression, in terms of m, for the number of sacks Jim buys.
 
 

[1]

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5b
Sme Calculator4 marks

Paulo buys 4 more sacks than Jim.
 
Write down an equation, in terms of m, and show that it simplifies to  10 m squared plus 9 m minus 162 equals 0.

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5c
Sme Calculator4 marks
i)
Solve  10 m squared plus 9 m minus 162 equals 0. 
 
 

m = .................... or m = .................... [3]

ii)
Find the number of sacks of rice that Paulo buys.
 
 

[1]

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6a
Sme Calculator3 marks

The diagram shows a right-angled triangle ABC.

cie-igcse-2018-oct-nov-p4-tz2-q4

The area of this triangle is 30 cm2.

Show that  2 x squared plus 3 x minus 20 equals 0.

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6b
Sme Calculator3 marks

Use factorisation to solve the equation  2 x squared plus 3 x minus 20 equals 0.
  

x = ...................... or x = ...................... 

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6c
Sme Calculator3 marks

Calculate BC.
 

BC = .......................................... cm 

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7a
Sme Calculator3 marks

In this question, all measurements are in metres.

cie-igcse-2018-may-jun-2-p4-q7
The diagram shows a right-angled triangle.

Show that 5 x squared minus 12 x minus 27 equals 0.

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7b
Sme Calculator4 marks

Solve 5 x squared minus 12 x minus 27 equals 0.
Show all your working and give your answers correct to 2 decimal places.

x equals......................... or x equals......................... 

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7c
Sme Calculator2 marks

Calculate the perimeter of the triangle.

............................................ m 

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7d
Sme Calculator2 marks

Calculate the smallest angle of the triangle.

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8
Sme Calculator4 marks

At a football match, the price of an adult ticket is $x and the price of a child ticket is $left parenthesis x minus 2.50 right parenthesis.
There are 18 500 adults and 2400 children attending the football match.
The total amount paid for the tickets is $320 040.

Find the price of an adult ticket.

$ ................................................ 

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9
Sme Calculator5 marks

Here is a rectangle.

q3-1-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

The length of the rectangle is 7 cm longer than the width of the rectangle.

4 of these rectangles are used to make this 8-sided shape.

q3-2-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

The perimeter of the 8-sided shape is 70 cm.
Work out the area of the 8-sided shape.

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10
Sme Calculator5 marks

A B C D is a rectangle.
E F G H spaceis a trapezium.

q4-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in centimetres.
The perimeters of these two shapes are the same.

Work out the area of the rectangle.

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11
Sme Calculator4 marks

3 teas and 2 coffees have a total cost of £7.80
5 teas and 4 coffees have a total cost of £14.20

Work out the cost of one tea and the cost of one coffee.

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12
Sme Calculator4 marks

A cinema sells adult tickets and child tickets.

   The total cost of 3 adult tickets and 1 child ticket is £30
   The total cost of 1 adult ticket and  3 child tickets is £22

   Work out the cost of an adult ticket and the cost of a child ticket.

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13a
Sme Calculator2 marks

The diagram shows a trapezium.

q9a-hard-2-13-forming-and-solving-equations-edexcel-gcse-maths

All the measurements are in centimetres.

The area of the trapezium is 351 cm2.

Show that  2 x squared space plus space x space long dash space 351 equals space 0

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13b
Sme Calculator3 marks

Work out the value of x.

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1
Sme Calculator6 marks

Alan invests $200 at a rate of r percent sign per year compound interest.
After 2 years the value of his investment is $206.46.

i)
Show that     r to the power of 2 space end exponent plus space 200 r space minus space 323 space equals space 0.
[3]
ii)
Solve the equation   r to the power of 2 space end exponent plus space 200 r space minus space 323 space equals space 0  to find the rate of interest.
Show all your working and give your answer correct to 2 decimal places.
 
r space equals space................................................ [3]

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2
Sme Calculator7 marks

Ahmed sells different types of cake in his shop.
The cost of each cake depends on its type and its size.

Every small cake costs $ x and every large cake costs $ left parenthesis 2 x space plus space 1 right parenthesis.

Petra spends $20 on small coffee cakes and $10 on large coffee cakes.
The total number of cakes is 45.

Write an equation in terms of x.
Solve this equation to find the cost of a small coffee cake.
Show all your working.
  

$ ................................................ 

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3
Sme Calculator3 marks

he total perimeter of a semicircle is 19.02 cm.
 
Calculate the radius of the semicircle.
 

................................................ cm

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4
Sme Calculator5 marks

Paddy and Anna each invest $2000 for 5 years.
Paddy earns simple interest at a rate of 1.25% per year.
Anna earns compound interest at a rate of r% per year.
At the end of 5 years, Paddy’s investment is worth the same as Anna’s investment.
 
Calculate the value of r.
 

r equals................................................ 

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5a
Sme Calculator5 marks

All the lengths in this question are in centimetres.

cie-igcse-2020-may-jun-p4-tz3-q5

The diagram shows a shape ABCDEF made from two rectangles.
The total area of the shape is 342 cm2.

Show that x squared plus x minus 72 equals 0.

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5b
Sme Calculator3 marks

Solve  space x squared plus x minus 72 equals 0 space by factorisation.
 

x equals .................... or  x equals.................... 

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5c
Sme Calculator2 marks

Work out the perimeter of the shape ABCDEF.
 

 ............................................ cm 

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5d
Sme Calculator2 marks

Calculate angle DBC.
 

Angle DBC = ................................................ 

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6
Sme Calculator4 marks

cie-igcse-2019-oct-nov-1-q1e

A, B and C lie on the circle, centre O.
Angle AOC = left parenthesis 3 x plus 22 right parenthesis degree and angle ABC = 5 x degree.

Find the value of x.

x = ................................................... 

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7
Sme Calculator6 marks

In a shop, the price of a monthly magazine is $m and the price of a weekly magazine is $left parenthesis m minus 0.75 right parenthesis.
One day, the shop receives
•  $168 from selling monthly magazines
•  $207 from selling weekly magazines.
The total number of these magazines sold during this day is 100.

i)
Show that  50 m squared minus 225 m plus 63 equals 0.

[3]
ii)
Find the price of a monthly magazine.
Show all your working.

$ ............................................... [3]

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8
Sme Calculator5 marks

Here are two right-angled triangles.

q2-veryhard-2-13-forming-and-solving-equations-edexcel-gcse-maths

Given that

      tan space e space equals space tan space f

find the value of x.

You must show all your working.

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9
Sme Calculator6 marks

Here is a right-angled triangle.

q6-veryhard-2-13-forming-and-solving-equations-edexcel-gcse-maths

All measurements are in centimetres.
The area of the triangle is 2.5 cm2 .

Find the perimeter of the triangle.
Give your answer correct to 3 significant figures.
You must show all of your working.

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10
Sme Calculator5 marks

There are only r red counters and g green counters in a bag.

A counter is taken at random from the bag.

The probability that the counter is green is 3 over 7

The counter is put back in the bag.

2 more red counters and 3 more green counters are put in the bag.
A counter is taken at random from the bag.

The probability that the counter is green is 6 over 13
Find the number of red counters and the number of green counters that were in the bag originally.

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11a
Sme Calculator1 mark

Shanti travels 12km from home to school.
She travels at an average speed of xkm/h.

Write down an expression, in terms of x, for the time taken in minutes for Shanti to travel from home to school.

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11b
Sme Calculator1 mark

Felix travels 12km from home to school.
He travels at an average speed that is 20km/h slower than Shanti’s average speed.

Write down an expression, in terms of x, for the time taken in minutes for Felix to travel from home to school.

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11c
Sme Calculator4 marks

Felix takes 15 minutes longer to travel from home to school than Shanti.

Use your answers from part (a) and part (b) to form an equation in x and show that it simplifies to space x squared space – space 20 x space – space 960 space equals space 0.

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11d
Sme Calculator3 marks

Solve the equation space x squared space – space 20 x space – space 960 space equals space 0.

Show your working and give your answers correct to 2 decimal places.

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11e
Sme Calculator3 marks

Calculate the time Felix takes to travel from home to school.
Give your answer in minutes and seconds.

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