 # Using Algebra to Solve Problems (2)

In this worksheet, students use algebra to solve numerical problems. Key stage:  KS 4

Curriculum topic:  Algebra

Difficulty level:   ### QUESTION 1 of 10

The main purpose of algebra is to solve problems.

But first, we must "translate" the problems into algebra.

The procedure for translation is generally as follows:

(a)   Let a letter stand for the quantity to be found in the question.  (e.g.  Let the missing number be x)

(b)   State the units of measurement if necessary.  (e.g.  Let the distance be x metres)

(c)   Write statements involving the missing quantity and form an equation to connect them.

(d)   Solve the equation algebraically.

(e)   Translate the answer back into English.   (e.g.  the distance is 2.5 km)

(f)    Check the numerical answer with the facts given in the original question.

Example

Find four consecutive odd numbers whose sum is 120.

(a)    Let the first odd number be x.

(c)    This means that the next odd number is x + 2, and the next is x + 4 and the next is x + 6.

Their sum is 120.

The equation is:

x + x + 2 + x + 4 + x + 6  = 120

(d)    Solve to get:

4x + 12 = 120

4x = 120 - 12 = 108

x = 108 ÷ 4 = 27

(e)    The numbers are x, x + 2, x + 4 and x + 6.

Using x = 27, we get the odd numbers 27, 29, 31 and 33.

(f)    27 + 29 + 31 + 33 = 120 so it works.

A rectangular field is 50 m longer than it is wide.  Its perimeter is 276 m.

Using x for the width in metres, select the equation which fits the given information.

x + x + 50 = 276

x + x + 50 + x + x + 50 = 276

4x = 276

A rectangular field is 50 m longer than it is wide.  Its perimeter is 276 m.

Use algebra to form an equation, solve it and then state the area of the field in m2.

(just write the number)

A rectangular tray is 25 cm longer than it is wide.  Its perimeter is 170 cm.

Using x for the width in metres, select the equation which fits the given information.

x + 25 + x + 25 = 170

x + x + 50 + x + x + 50 = 170

x + x + 25 + x + x + 25 = 170

A rectangular tray is 25 cm longer than it is wide.  Its perimeter is 170 cm.

Use algebra to form an equation, solve it and then state the area of the tray in cm2.

(just write the number)

A rectangular window is twice as long as it is wide.  Its perimeter is 366 cm.

Using x for the width in metres, select the equation which fits the given information.

2x + 2x = 366

x + 2x + x + 2x = 366

x + x + 2 + x + x + 2 = 366

A rectangular window is twice as long as it is wide.  Its perimeter is 366 cm.

Use algebra to form an equation, solve it and then state the area of the window in cm2.

(just write the number)

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 1 cm longer than the width.

The total length of all the edges is 36 cm.

Using x for the width in metres, select the equation which fits the given information.

4x + 4(x - 1) + 4(x + 1) = 36

12x - 1 = 36

x + x - 1 + x + 1 = 36

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 1 cm longer than the width.

The total length of all the edges is 36 cm.

Use algebra to form an equation, solve it and then state the length of the cuboid in cm.

(just write the number)

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 2 cm longer than the width.

The total length of all the edges is 88 cm.

Using x for the width in metres, select the equation which fits the given information.

4x + 4(x - 1) + 4(x + 1) = 88

12x - 1 + 2 = 88

4x + 4(x - 1) + 4(x + 2) = 88

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 2 cm longer than the width.

The total length of all the edges is 88 cm.

Use algebra to form an equation, solve it and then state the volume of the cuboid in cm3.

(just write the number)

• Question 1

A rectangular field is 50 m longer than it is wide.  Its perimeter is 276 m.

Using x for the width in metres, select the equation which fits the given information.

CORRECT ANSWER
x + x + 50 + x + x + 50 = 276
• Question 2

A rectangular field is 50 m longer than it is wide.  Its perimeter is 276 m.

Use algebra to form an equation, solve it and then state the area of the field in m2.

(just write the number)

CORRECT ANSWER
4136
EDDIE SAYS
x + x + 50 + x + x + 50 = 276
4x + 100 = 276
4x = 176
x = 176 ÷ 4
x = 44

Length is x + 50 = 44 + 50 = 94 m

Area = 44 × 94 = 4136 m²
• Question 3

A rectangular tray is 25 cm longer than it is wide.  Its perimeter is 170 cm.

Using x for the width in metres, select the equation which fits the given information.

CORRECT ANSWER
x + x + 25 + x + x + 25 = 170
• Question 4

A rectangular tray is 25 cm longer than it is wide.  Its perimeter is 170 cm.

Use algebra to form an equation, solve it and then state the area of the tray in cm2.

(just write the number)

CORRECT ANSWER
1650
EDDIE SAYS
x + x + 25 + x + x + 25 = 170
4x + 50 = 170
4x = 120
x = 120 ÷ 4
x = 30

Length is x + 25 = 30 + 25 = 55 cm

Area = 30 × 55 = 1650 cm²
• Question 5

A rectangular window is twice as long as it is wide.  Its perimeter is 366 cm.

Using x for the width in metres, select the equation which fits the given information.

CORRECT ANSWER
x + 2x + x + 2x = 366
• Question 6

A rectangular window is twice as long as it is wide.  Its perimeter is 366 cm.

Use algebra to form an equation, solve it and then state the area of the window in cm2.

(just write the number)

CORRECT ANSWER
7442
EDDIE SAYS
x + 2x + x + 2x = 366
6x = 366
x = 61

Length is 2x = 2 × 61 = 122 cm

Area = 61 × 122 = 7442 cm²
• Question 7

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 1 cm longer than the width.

The total length of all the edges is 36 cm.

Using x for the width in metres, select the equation which fits the given information.

CORRECT ANSWER
4x + 4(x - 1) + 4(x + 1) = 36
EDDIE SAYS
Width = x
Height = x - 1
Length = x + 1
There are 12 edges.
• Question 8

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 1 cm longer than the width.

The total length of all the edges is 36 cm.

Use algebra to form an equation, solve it and then state the length of the cuboid in cm.

(just write the number)

CORRECT ANSWER
4
EDDIE SAYS
4x + 4(x - 1) + 4(x + 1) = 36
4x + 4x - 4 + 4x + 4 = 36
12x = 36
x = 3

Length = x + 1 = 3 + 1 = 4 cm
• Question 9

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 2 cm longer than the width.

The total length of all the edges is 88 cm.

Using x for the width in metres, select the equation which fits the given information.

CORRECT ANSWER
4x + 4(x - 1) + 4(x + 2) = 88
EDDIE SAYS
Width = x
Height = x - 1
Length = x + 2
There are 12 edges.
• Question 10

A cuboid has a width, which is 1 cm longer than the height, and a length, which is 2 cm longer than the width.

The total length of all the edges is 88 cm.

Use algebra to form an equation, solve it and then state the volume of the cuboid in cm3.

(just write the number)

CORRECT ANSWER
378
EDDIE SAYS
4x + 4(x - 1) + 4(x + 2) = 88
4x + 4x - 4 + 4x + 8 = 88
12x + 4 = 88
12x = 84
x = 7

Width = x = 7 cm
Height = x - 1 = 7 - 1 = 6 cm
Length = x + 2 = 7 + 2 = 9 cm

Volume = 7 × 6 × 9 = 378 cm³
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