# Using Algebra to Solve Problems (1)

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.

Find three consecutive numbers whose sum is 33.

Using x for the smallest number, select the equation which fits the given information.

x + x + 1 + x + 2 = 33

x + x + 2 + x + 4 = 33

x + x + x = 33

Find three consecutive numbers whose sum is 33.

Use algebra to form an equation, solve it and then state the smallest of the three numbers.

Find three consecutive odd numbers whose sum is 33.

Using x for the smallest number, select the equation which fits the given information.

x + x + 1 + x + 2 = 33

x + x + 2 + x + 4 = 33

x + 3x + 5x = 33

Find three consecutive odd numbers whose sum is 33.

Use algebra to form an equation, solve it and then state the smallest of the three numbers.

Find four consecutive even numbers whose sum is 204.

Using x for the smallest number, select the equation which fits the given information.

x + x + 1 + x + 2 + x + 3= 204

x + x + 2 + x + 4 + x + 6 = 204

x + 2x + 4x + 6x = 204

Find four consecutive even numbers whose sum is 204.

Use algebra to form an equation, solve it and then state the largest of the four numbers.

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.

4x = 276

x + x + 50 = 276

x + x + 50 + x + x + 50 = 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 length of the field in m.

(just write the number)

The shortest side of an isosceles triangle is 10 cm less than the length of one of its two identical sides.

Its perimeter is 62 cm.

Using x for the length in cm of one of the identical sides, select the equation which fits the given information.

x + x + x/10 = 62

x + x + x + 10 = 62

x + x + x - 10 = 62

The shortest side of an isosceles triangle is 10 cm less than the length of one of its two identical sides.

Its perimeter is 62 cm.

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

(just write the number)

• Question 1

Find three consecutive numbers whose sum is 33.

Using x for the smallest number, select the equation which fits the given information.

x + x + 1 + x + 2 = 33
• Question 2

Find three consecutive numbers whose sum is 33.

Use algebra to form an equation, solve it and then state the smallest of the three numbers.

10
EDDIE SAYS
x + x + 1 + x + 2 = 33
3x + 3 = 33
3x = 30
x = 10
• Question 3

Find three consecutive odd numbers whose sum is 33.

Using x for the smallest number, select the equation which fits the given information.

x + x + 2 + x + 4 = 33
• Question 4

Find three consecutive odd numbers whose sum is 33.

Use algebra to form an equation, solve it and then state the smallest of the three numbers.

9
EDDIE SAYS
x + x + 2 + x + 4 = 33
3x + 6 = 33
3x = 27
x = 9
• Question 5

Find four consecutive even numbers whose sum is 204.

Using x for the smallest number, select the equation which fits the given information.

x + x + 2 + x + 4 + x + 6 = 204
• Question 6

Find four consecutive even numbers whose sum is 204.

Use algebra to form an equation, solve it and then state the largest of the four numbers.

54
EDDIE SAYS
x + x + 2 + x + 4 + x + 6 = 204
4x + 12 = 204
4x = 192
x = 192 ÷ 4
x = 48

Largest number is x + 6 = 48 + 6 = 54
• Question 7

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 + x + x + 50 = 276
• Question 8

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 length of the field in m.

(just write the number)

94
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
• Question 9

The shortest side of an isosceles triangle is 10 cm less than the length of one of its two identical sides.

Its perimeter is 62 cm.

Using x for the length in cm of one of the identical sides, select the equation which fits the given information.

x + x + x - 10 = 62
• Question 10

The shortest side of an isosceles triangle is 10 cm less than the length of one of its two identical sides.

Its perimeter is 62 cm.

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

(just write the number)

14
EDDIE SAYS
x + x + x - 10 = 62
3x - 10 = 62
3x = 72
x = 72 ÷ 3
x = 24

Shortest side length is x - 10 = 24 - 10 = 14
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