Forming and solving equations

The area of the rectangle is 120 cm²

Work out the value of x.

x =

The area of the rectangle is 120 cm²

Work out the value of x.

There are n counters in a box. 6 of the counters are black. Two counters are chosen at random without replacement. The probability that both counters are black is

Use an algebraic method to work out the value of n.

n =

Here is an equilateral triangle. All measurements are in centimetres.

Work out the value of the perimeter of the triangle.

Here is an equilateral triangle. All measurements are in centimetres.

Work out the value of the perimeter of the triangle.

Here is a rectangle. All measurements are in centimetres.

AB: BC = 1: 3

Work out the value of the area.

The total cost of a taxi ride is calculated by adding a fixed charge of £4 and a charge of £2 per mile.

Write a formula to work out the total cost, £ C, of a journey of m miles.

The total cost of a taxi ride is calculated by adding a fixed charge of £4 and a charge of £2 per mile.

Write a formula to work out the total cost, £ C, of a journey of m miles.

A window cleaner uses this formula.

C = cost, in £, for the customer W = number of windows to be cleaned

How much does it cost for 6 windows to be cleaned?

The cost for another customer was £24

Show why this cost must be incorrect.

4 chocolate bars and 3 packets of mints cost £4.70 5 chocolate bars and 1 packet of mints cost £4.50

Work out the cost of a chocolate bar and the cost of a packet of mints.

9k + 7 and 2k² + 3 are consecutive integers. 9 k + 7 is the smaller integer.

Work out the value of the next consecutive integer.

x is a square number.

AB is a straight line. Both angles are given in degrees.

By working out the value of x, work out the ratio smaller angle: larger angle

In the diagrams, all lengths are in centimetres.

The two shapes have equal areas.

Work out k: m

The Venn diagram shows information about 80 people who visited an online shop. ξ = 80 people T = people who bought trainers H = people who bought a hoodie

One person is chosen at random.

Work out the probability that they bought a hoodie.

One person who bought trainers is chosen at random.

Work out the probability that they bought a hoodie.

Multiplying y by 6 gives the same result as adding 15 to y

Write this as an equation.

Show that the value of y is not 4

A chocolate box in the shape of a prism is being designed. All lengths are in centimetres. The cross section is a regular hexagon with side x The length is 5 x

The total surface area of the box must be less than 650 cm²

Work out the largest possible integer value of x. You must show your working.

Sunita is x years old. Beth is one year younger than Sunita. Joel is double Sunita’s age. The mean of their ages is 5 How old is Joel ?

Sunita is x years old. Beth is one year younger than Sunita. Joel is double Sunita’s age. The mean of their ages is 5 How old is Joel ?

K, L and M are weights. Both of the scales balance exactly.

How many M weights are needed to balance one L weight?

A factory packs x boxes of teabags per hour. Each box contains 80 teabags. Show that the factory packs teabags per minute.

A graph has the equation y = x² + px + r where p and r are constants. The graph passes through the points (0, 4), (1, 3) and (8, w) Work out the value of w.

The length of this rectangle is 6 times the width.

Two of these rectangles are joined, with no overlap, to make this L-shape.

The perimeter of the L-shape is 98.8 cm Work out the value of the perimeter of one of the rectangles.

The length of this rectangle is 6 times the width.

Two of these rectangles are joined, with no overlap, to make this L-shape.

The perimeter of the L-shape is 98.8 cm Work out the value of the perimeter of one of the rectangles.

In a class there are n boys a total of 25 students. Two of the students are chosen at random. The probability that both students are boys is Work out the value of n.

d is 6 more than c. Circle the correct equation.

d = 6 c c = 6 d d = c + 6 c = d + 6

Here is the rule for a sequence. After the first two terms, each term is the sum of the previous two terms The 1st term is 33 The 2nd term is x The 4th term is 73 Work out the value of x.

x =

An expression for the nth term of a different sequence is n – n² Ruth says, ‘‘All the terms will be negative because n² is always greater than n.’’ Is she correct?

Give a reason for your answer.

Here is the rule for a sequence.

After the first two terms, each term is the sum of the previous two terms

The 1st term is 33 The 2nd term is x The 4th term is 73

Work out the value of x.

An expression for the nth term of a different sequence is n – n² Ruth says, ‘‘All the terms will be negative because n² is always greater than n.’’

Is she correct?

Give a reason for your answer.

Here is a number machine.

Work out the output.

Here is a different number machine.

Work out a formula for d in terms of c.

y is 3 more than x. Circle the correct equation.

The flight of a plane was in two stages. The table shows information about the flight.

	Distance (miles)	Speed (mph)	Time (hours)
1st stage	731	x	731 x
2nd stage	287	x – 24	287 x 24

In total, the flight lasted 2 hours. Work out the value of x.

b is 3 more than the square root of a. Circle the correct equation.

Bobbi has these notes.

The total value of her notes is £ T Write a formula for T in terms of x.

Here are two rectangles.

The area of the shaded rectangle is the area of the large rectangle. Work out the value of x.

The cost of a taxi journey is £3 plus £2 per mile. Circle the cost of a journey of 6 miles.

£5 £12 £15 £30

Kate has the following question for homework.

Show that Kate can form the equation x 2 + 5 x – 1400 = 0

Kate correctly factorises the equation to get (x + 40)(x – 35) = 0 Her answer to the homework question is x = – 40 or x = 35 Is her answer correct? Tick a box.

Give a reason for your answer.

Amy and Brad each have some money. Carly has no money. Amy gives £7 to Carly. Brad gives £5 to Carly. Now they all have the same amount of money. How much money did Amy have to begin with?

Toby is forming and solving equations.

The product of half of a number and three more than the number is the same as the square of the number Toby uses y to represent the number. Write an equation that Toby could form.

Toby forms another equation.

He wants to work out the values of x. Here is his working.

What error has he made in his working?

Zoe is thinking of a number.

What number is she thinking of?

In a bag there are 10p coins, 20p coins and 50p coins. There are two fewer 20p coins than 10p coins. There are five more 50p coins than 10p coins.

Complete the table.

Altogether, there are 60 coins. Work out the total value of the 20p coins.

Towns P, Q and R are connected by roads PQ, PR and QR.

PR is 10 km longer than PQ. QR is twice as long as PR. The total length of the three roads is 170 km

Work out the length of PQ.

Towns P, Q and R are connected by roads PQ, PR and QR. PR is 10 km longer than PQ. QR is twice as long as PR. The total length of the three roads is 170 km

Work out the length of PQ.

Here are two sets of numbers, A and B. Set A Set B

mean of Set A: mean of Set B = 3: 8 Work out the value of x.

In a chess club, there are x boys and y girls.

If 5 more boys and 8 more girls join, there would be half as many boys as girls. Show that y = 2 x + 2

If instead, 10 more boys and 1 more girl join, there would be the same number of boys and girls. Work out x and y.

x =

y =

A company uses this formula to work out the cost, £ A, of a taxi ride. A = 4 + 1.8 m + b £4 is a fixed charge m is the number of miles travelled £ b is a charge for booking online

Clare books a taxi online and travels 8 miles. She pays £20 altogether. How much is the charge for booking online?

A different company has a fixed charge of £3 charges £1.90 per mile has no charge for booking online. Write a formula for the cost, £ C, of a taxi ride with this company.

At a café, 2 teas and 1 coffee cost £3.40 1 tea and 4 coffees cost £7.30 Work out the cost of 1 tea and the cost of 1 coffee.

A linear sequence starts a + 2 b a + 6 b a + 10 b …….. …….. The 2nd term has value 8 The 5th term has value 44 Work out the values of a and b.

a =

b =

e is 3 more than d. f is 5 less than d.

Write an expression for e in terms of d.

Write an expression for f in terms of d.

Work out e – f Simplify your answer.

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 ?

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 ?

Amy has x beads. Billy has three more beads than Amy. Carly has four times as many beads as Billy. Circle the expression for the number of beads that Carly has.

4 x + 3 3 x + 4 4(x + 3) x + 12

Circle the expression for the range of n consecutive integers.

The value of A is double the value of B. Circle the correct formula.

Karl has twin sisters.

The sum of the ages of Karl and his twin sisters is 39 In 4 years’ time the twins will be 18

How old will Karl be in 4 years’ time?

The length of a rectangle is five times the width.

Work out the width of the rectangle.

A football team has P points. P = 3 W + D W is the number of wins D is the number of draws

A team has 6 wins and 2 draws. How many points does the team have?

After 33 games a different team has 53 points. 11 games were draws. How many games has this team lost ?

Use 8 km/h = 5 mph to convert 96 km/h to mph

x km/h = y mph Use 8 km/h = 5 mph to write a formula for y in terms of x.

x km/h = y mph

Use 8 km/h = 5 mph to write a formula for y in terms of x.