Forming & Solving Equations

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.

Here is a rectangle.

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.

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

 is a rectangle.
is a trapezium.

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

Work out the area of the rectangle.

Here is a rectangle and a right-angled triangle.

All measurements are in centimetres.
The area of the rectangle is greater than the area of triangle.

Find the set of possible values of .

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.

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.

The diagram shows a trapezium.

All the measurements are in centimetres.

The area of the trapezium is 351 cm².

Show that 

Work out the value of .

The diagram shows the plan of a floor.

The area of the floor is 138 m².

Work out the value of .

 is a trapezium.
 is a rectangle.

All measurements are in centimetres.

The two shapes have the same perimeter.

Work out the length of  .

Given that

find the possible values of .

In a chess club, there are  boys and  girls.

If 5 more boys and 8 more girls join, there would be half as many boys as girls.

Show that 

If instead,

10 more boys and 1 more girl join, there would be the same number of boys and girls.

Work out  and .

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

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

Three identical isosceles triangles are joined to make this trapezium.

Each triangle has base cm and perpendicular height cm

Work out an expression, in terms of and , for the area of the trapezium.

Give your answer in its simplest form.

......................................cm²

This diagram shows the same trapezium.

Work out an expression, in terms of , for the perimeter of the trapezium.

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

Theo starts with savings of £18
James starts with no savings.

Each week from now,

   Theo will save £4.50 and James will save £4

In how many weeks will Theo and James have savings in the ratio 15 : 8 ?

Rana sells 192 cakes in the ratio    small : medium : large    =    7 : 6 : 11

The profit for one medium cake is twice the profit for one small cake.
The profit for one large cake is three times the profit for one small cake.

Her total profit is £532.48

Work out the profit for one small cake.

£.....................

Sundip and Emma have some money.
The ratio of Sundip’s money to Emma’s money is 3 : 5.
Emma spends £450 of her money.
The ratio of Sundip’s money to Emma’s money is now 2 : 3.

Find how much money Sundip has.

£ ...........................................................

A pattern is made using identical rectangular tiles.

Find the total area of the pattern.

The diagram shows a trapezium.

 = cm.
is the same length as .
 is twice the length of .
 is 4 cm longer than .

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

Gemma has the same number of sweets as Betty.

Gemma gives 24 of her sweets to Betty.
Betty now has 5 times as many sweets as Gemma.

Work out the total number of sweets that Gemma and Betty have.

Here is a cuboid.

All measurements are in centimetres.
 is an integer.
The total volume of the cuboid is less than 900 cm³

Show that 5

3 kg of potatoes and 4 kg of carrots have a total cost of 440p.
4 kg of potatoes and 3 kg of carrots have a total cost of 470p.

Work out the total cost of 1 kg of potatoes and 1 kg of carrots.

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

Work out the value of .

The diagram shows a garden in the shape of a rectangle.

All measurements are in metres.
The perimeter of the garden is 32 metres.

Work out the value of .

The diagram shows a right-angled triangle.

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

is a triangle.

Angle =  angle .

The length of side is  cm.
The length of side is  cm.
The length of side is  cm.

Work out the perimeter of the triangle.
Give your answer as a number of centimetres.

Redlands School sent  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.

Here is a rectangle.

All measurements are in centimetres.
The area of the rectangle is 48 cm².
Show that 

Ali is  years old.

Bhavara is twice as old as Ali.

Ceris is 3 years younger than Ali.

The total of their ages is 125 years.

Work out the age of each person.

Kate has the following question for homework.

Show that Kate can form the equation   

Kate correctly factorises the equation to get

Her answer to the homework question is

Is her answer correct?
Tick a box.

Yes          No

Give a reason for your answer.

and are three points on a circle.

The radii from and are shown.

Is a diameter of the circle?
You must show your working.

Here are two right-angled triangles.

Given that

find the value of .

You must show all your working.

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.

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

A counter is taken at random from the bag.

The probability that the counter is green is 

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 
Find the number of red counters and the number of green counters that were in the bag originally.

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

Here is a right-angled triangle.

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.

is a five-sided shape.

is a rectangle.
is an equilateral triangle.



The perimeter of is 100 cm.
The area of is cm²

Show that 

The diagram shows a sector of a circle with centre and radius .

is the point on and is the point on such that

Angle

Given that

the perimeter of sector the perimeter of triangle

find the value of .
Give your answer correct to 3 significant figures.

The total surface area of a solid hemisphere is equal to the curved surface area of a cylinder.

The radius of the hemisphere is  cm.
The radius of the cylinder is twice the radius of the hemisphere.

Given that

find the value of .

A bowl contains  pieces of fruit.
Of these,  are oranges and the rest are apples.

Two pieces of fruit are going to be taken at random from the bowl.
The probability that the bowl will then contain  apples is 

Work out the value of 
Show your working clearly.

A sphere has radius 

A cone has

   radius 
   perpendicular height 

The sphere and the cone have the same volume.

Work out    radius of cone : perpendicular height of cone

Give your answer in the form where  and  are integers.