Forming & Solving Equations (Cambridge (CIE) IGCSE Maths: Extended)

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  and every large cake costs .

The number of small cherry cakes that can be bought for $4 is the same as the number of large cherry cakes that can be bought for $13.

Find the cost of a small cherry cake.

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

The diagram shows a trapezium.

Work out the value of .

.................................................. 

Naga has marbles.
Panav has three times as many marbles as Naga.
Naga loses 5 marbles and Panav buys 10 marbles.
Together they now have more than 105 marbles.

Write down and solve an inequality in .

The cost of one ruler is cents.
The cost of one protractor is cents.

The total cost of 5 rulers and 1 protractor is 245 cents.
The total cost of 2 rulers and 3 protractors is 215 cents.

Write down two equations in terms of and and solve these equations to find the cost of one protractor.

........................................... cents 

The diagram shows a triangle.

In the diagram, all the measurements are in metres.

The perimeter of the triangle is 56 m. The area of the triangle is A m².

Work out the value of A.

Here is a right-angled triangle.

Four of these triangles are joined to enclose the square as shown below.

Show that the area of the square is 

Becky has some marbles.
Chris has two times as many marbles as Becky.
Dan has seven more marbles than Chris.

They have a total of 57 marbles.

Dan says,
"If I give some marbles to Becky, each of us will have the same number of marbles."

Is Dan correct?
You must show how you get your answer.

The diagram shows the plan of a floor.

The area of the floor is 138 m².

Work out the value of .

In this question, all measurements are in centimetres. 

The height of the triangle is  and the height of the rectangle is ().
The length of the base of the triangle is and the length of the rectangle is ().
The area of the triangle is cm² and the area of the rectangle is cm².

Write down an expression, in terms of , for the height of the rectangle. 

Show that 

By factorising and solving  , find the two possible heights of the triangle.

is inversely proportional to the square of . When 
Find when .

..................................................

Carol walks 12 km at km/h and then a further 6 km at km/h.
The total time taken is 5 hours.

i) Write an equation, in terms of , and show that it simplifies to  .

[3]

ii) Factorise  .

[2]

iii) Solve the equation  .

= .................... or = ................... [1]

iv) Write down Carol’s walking speed during the final 6 km.

........................................... km/h [1]

Petra records the score in each test she takes.  

The mean of the first scores is .
The mean of the first  scores is 
Find the score in terms of  and .
Give your answer in its simplest form.

Solve the equation to find the difference between the perimeters of the two rectangles.  

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

  The difference between the areas of the two rectangles is 62 cm².

i) Show that   

[3]

ii) Factorise   

[2]

The sequence of diagrams above is made up of small lines and dots.

The number of dots in Diagram is .  
Find the value of and the value of .  

= .............................................. 
= .............................................. 

,       ,  .

The exact answer to is   .  
Use algebra to find the value of .  

= ................................................. 

Klaus buys  silver balloons and gold balloons for a party.   He buys

• more gold balloons than silver balloons

• at least 15 silver balloons

• less than 50 gold balloons

• a total of no more than 70 balloons.

Write down four inequalities, in terms of  and/or , to show this information.

is a square with a side length of 
is the midpoint of .
is the point on  where 

 is a right-angled triangle.

Find an expression, in terms of , for the area of triangle . Give your expression in its simplest form.

The lengths of the sides are in centimetres.

The area of triangle  is equal to the area of triangle .
Work out the value of , giving your answer in the form  where  and  are integers.

The diagram shows a rectangle, , and two congruent triangles, and .

area of rectangle  = area of triangle  + area of triangle 

Work out the length of .

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