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maths.com >>>> Algebra >>>> Changing The Subject Of A Formula

Algebra
Changing The Subject
Of A Formula

The Meaning Of The Words

Level 1    Use One Operation    4 Types

Level 2    Use Two Operations    5 Types

Level 3    Use Three Operations    2 Types

Level 4    Multiply Out Brackets
                And Two Other Operations    2 Types

Level 5    Multiply To Remove Fraction
                Multiply Out Brackets
                And Two Other Operations    2 Types

Videos


Changing The Subject Of A Formula

The Meaning Of The Words

Formula means
Relationship between two or more variables
Example y = x + 5 where x and y are variables.

Subject Of A Formula means
The variable on its own, usually on the left hand side.
Example y is the subject of the formula y = x + 5

Changing The Subject Of A Formula means
Rearrange the formula so that a different variable is on its own.
Making x the subject of the formula y = x + 5 gives x = y - 5

Changing The Subject Of A Formula

Level 1    Use One Operation

Level 1    Type 1    Use subtraction BECAUSE we have addition

Level 1    Type 2    Use addition BECAUSE we have subtraction

Level 1    Type 3    Use division BECAUSE we have multiplication

Level 1    Type 4    Use multiplication BECAUSE we have division


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 1     Use subtraction BECAUSE we have addition

Example 1    with full explanation

\( \text{Make}  x  \text{the subject of} \)
\(y = x + 3 \)

We require x to be the subject of the formula
The subject is written on the left
So we switch the sides to get x on the left

Switch sides
\(x + 3 = y \)

We require x by itself on the left hand side
But we have x + 3
The inverse of addition is subtraction
We need to subtract 3 from the left side
But, to keep the equality true,
we need to subtract 3 from the right side as well
So subtract 3 from both sides

Subtract 3 from both sides
\(x + 3 - 3 = y - 3 \)

Simplify
\( x = y - 3 \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 1     Use subtraction BECAUSE we have addition

Example 1    with key points only

\( \text{Make}  x  \text{the subject of} \)
\( y = x + 3 \)

Switch sides
\( x + 3 = y \)

Subtract 3 from both sides
\( x + 3 - 3 = y - 3 \)

Simplify
\( x = y - 3 \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 1     Use subtraction BECAUSE we have addition

Example 2    with key points only

\( \text{Make}  x  \text{the subject of} \)
\( y = x + m \)

Switch sides
\( x + m = y \)

Subtract m from both sides
\( x + m - m = y - m \)

Simplify
\( x = y - m \)


When you do a question yourself
it is often helpful to write in these key points
before you do the actual algebra.

It gets you to think of the logic of the process
AND
in an examination
you might get a method mark
even if the actual working is not quite correct
because you have shown the method in words.


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 2     Use addition BECAUSE we have subtraction

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = x - 5 \)

Switch sides
\( x - 5 = y \)

Add 5 to both sides
\( x - 5 + 5 = y + 5 \)

Simplify
\( x = y + 5 \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 2     Use addition BECAUSE we have subtraction

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = x - m \)

Switch sides
\( x - m = y \)

Add m to both sides
\( x - m + m = y + m \)

Simplify
\( x = y + m \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 3     Use division BECAUSE we have multiplication

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = 8x \)

Switch sides
\( 8x= y \)

Divide both sides by 8
\( \Large \frac{8x}{8} \normalsize = \Large \frac{y}{8} \)

Simplify
\( x = \Large \frac{y}{8} \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 3     Use division BECAUSE we have multiplication

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = mx \)

Switch sides
\( mx= y \)

Divide both sides by m
\( \Large \frac{mx}{m} \normalsize = \Large \frac{y}{m} \)

Simplify
\( x = \Large \frac{y}{m} \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 4     Use multiplication BECAUSE we have division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{x}{8} \)

Switch sides
\( \Large \frac{x}{8} \normalsize = y \)

Multiply both sides by 8
\( \Large \frac{8x}{8} \normalsize = 8y \)

Simplify
\( x = 8y \)


Changing The Subject Of A Formula

Level 1    Use one of addition, subtraction, multiplication or division

Type 4     Use multiplication BECAUSE we have division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{x}{m} \)

Switch sides
\( \Large \frac{x}{m} \normalsize = y \)

Multiply both sides by m
\( \Large \frac{mx}{m} \normalsize = my \)

Simplify
\( x = my \)


Changing The Subject Of A Formula

Level 2    Use Two Operations

Level 2    Type 1    Use subtraction and then use division

Level 2    Type 2    Use addition and then use division

Level 2    Type 3    Use multiplication and then use subtraction

Level 2    Type 4    Use multiplication and then use addition

Level 2    Type 5    Use multiplication and then use division


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 1     Use subtraction and then use division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = 2x + 5 \)

Switch sides
\( 2x + 5 = y \)

Subtract 5 from both sides
\( 2x + 5 - 5 = y - 5 \)

Simplify
\( 2x = y - 5 \)

Divide both sides by 2
\( \Large \frac{2x}{2} \normalsize = \Large \frac{y  -  5}{2} \)

Simplify
\( x = \Large \frac{y  -  5}{2} \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 1     Use subtraction and then use division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = mx + c \)

Switch sides
\( mx + c = y \)

Subtract c from both sides
\( mx + c - c = y - c \)

Simplify
\( mx = y - c \)

Divide both sides by m
\( \Large \frac{mx}{m} \normalsize = \Large \frac{y  -  c}{m} \)

Simplify
\( x = \Large \frac{y  -  c}{m} \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 2     Use addition and then use division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = 3x - 7 \)

Switch sides
\( 3x - 7 = y \)

Add 7 to both sides
\( 3x - 7 + 7 = y + 7 \)

Simplify
\( 3x = y + 7 \)

Divide both sides by 3
\( \Large \frac{3x}{3} \normalsize = \Large \frac{y  +  7}{3} \)

Simplify
\( x = \Large \frac{y  +  7}{3} \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 2     Use addition and then use division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = mx - c \)

Switch sides
\( mx - c = y \)

Add c to both sides
\( mx - c + c = y + c \)

Simplify
\( mx = y + c \)

Divide both sides by m
\( \Large \frac{mx}{m} \normalsize = \Large \frac{y  +  c}{m} \)

Simplify
\( x = \Large \frac{y  +  c}{m} \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 3     Use multiplication and then use subtraction

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{x}{2} \normalsize + 5 \)

Switch sides
\( \Large \frac{x}{2} \normalsize + 5 = y \)

Fractions are more difficult to work with
SO to make work easier and errors less likely
GET RID OF THE FRACTION FIRST

Multiply EVERYTHING on both sides by 2
\( 2 \Large ( \frac{x}{2} ) \normalsize + 2(5) = 2(y) \)

Simplify
\( x + 10 = 2y \)

Subtract 10 from both sides
\( x + 10 - 10 = 2y - 10 \)

Simplify
\( x = 2y - 10 \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 3     Use multiplication and then use subtraction

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{x}{m} \normalsize + c \)

Switch sides
\( \Large \frac{x}{m} \normalsize + c = y \)

Fractions are more difficult to work with
SO to make work easier and errors less likely
GET RID OF THE FRACTION FIRST

Multiply EVERYTHING on both sides by m
\( m \Large ( \frac{x}{m} ) \normalsize + m(c) = m(y) \)

Simplify
\( x + cm = my \)

Subtract cm from both sides
\( x + cm - cm = my - cm \)

Simplify
\( x = my - cm \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 4     Use multiplication and then use addition

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{x}{4} \normalsize - 7 \)

Switch sides
\( \Large \frac{x}{4} \normalsize - 7 = y \)

Fractions are more difficult to work with
SO to make work easier and errors less likely
GET RID OF THE FRACTION FIRST

Multiply EVERYTHING on both sides by 4
\( 4 \Large ( \frac{x}{4} ) \normalsize - 4(7) = 4(y) \)

Simplify
\( x - 28 = 4y \)

Add 28 to both sides
\( x - 28 + 28 = 4y + 28 \)

Simplify
\( x = 4y + 28 \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 4     Use multiplication and then use addition

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{x}{m} \normalsize - c \)

Switch sides
\( \Large \frac{x}{m} \normalsize - c = y \)

Fractions are more difficult to work with
SO to make work easier and errors less likely
GET RID OF THE FRACTION FIRST

Multiply EVERYTHING on both sides by m
\( m \Large ( \frac{x}{m} ) \normalsize - m(c) = m(y) \)

Simplify
\( x - cm = my \)

Add cm to both sides
\( x - cm + cm = my + cm \)

Simplify
\( x = my + cm \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 5     Use multiplication and then use division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{2x}{3} \)

Switch sides
\( \Large \frac{2x}{3} \normalsize = y \)

Multiply both sides by 3
\( 3 \Large ( \frac{2x}{3} ) \normalsize = 3(y) \)

Simplify
\( 2x = 3y \)

Divide both sides by 2
\( \Large \frac{2x}{2} \normalsize = \Large \frac{3y}{2} \)

Simplify
\( x = \Large \frac{3y}{2} \)


Changing The Subject Of A Formula

Level 2    Use two of addition, subtraction, multiplication or division

Type 5     Use multiplication and then use division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{ax}{b} \)

Switch sides
\( \Large \frac{ax}{b} \normalsize = y \)

Multiply both sides by b
\( b \Large ( \frac{ax}{b} ) \normalsize = b(y) \)

Simplify
\( ax = by \)

Divide both sides by a
\( \Large \frac{ax}{a} \normalsize = \Large \frac{by}{a} \)

Simplify
\( x = \Large \frac{by}{a} \)


Changing The Subject Of A Formula

Level 3    Use Three Operations

Level 3    Type 1    Use multiplication subtraction division

Level 3    Type 2    Use multiplication addition division


Changing The Subject Of A Formula

Level 3    Use 3 of addition, subtraction, multiplication or division

Type 1     Use multiplication then subtraction and then division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{3x}{4} \normalsize + 5 \)

Switch sides
\( \Large \frac{3x}{4} \normalsize + 5 = y \)

1.   Remove fractions first
2.   Work without fractions
3.   Create fractions last

Multiply EVERYTHING on both sides by 4
\( 4 \Large ( \frac{3x}{4} ) \normalsize + 4(5) = 4(y) \)

Simplify
\( 3x + 20 = 4y \)

Subtract 20 from both sides
\( 3x + 20 - 20 = 4y - 20 \)

Simplify
\( 3x = 4y - 20 \)

Divide both sides by 3
\( \Large \frac{3x}{3} \normalsize = \Large \frac{4y  -  20}{3} \)

Simplify
\( x = \Large \frac{4y  -  20}{3} \)


Changing The Subject Of A Formula

Level 3    Use 3 of addition, subtraction, multiplication or division

Type 1     Use multiplication then subtraction and then division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{xw}{t} \normalsize + p \)

Switch sides
\( \Large \frac{xw}{t} \normalsize + p = y \)

1.   Remove fractions first
2.   Work without fractions
3.   Create fractions last

Multiply EVERYTHING on both sides by t
\( t \Large (\frac{xw}{t}) \normalsize + t(p) = t(y) \)

Simplify
\( xw + pt = ty \)

Subtract pt from both sides
\( xw + pt - pt = ty - pt \)

Simplify
\( xw = ty - pt \)

Divide both sides by w
\( \Large \frac{xw}{w} \normalsize = \Large \frac{ty  -  pt}{w} \)

Simplify
\( x = \Large \frac{ty  -  pt}{w} \)


Changing The Subject Of A Formula

Level 3    Use 3 of addition, subtraction, multiplication or division

Type 2     Use multiplication then addition and then division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{5x}{6} \normalsize - 7 \)

Switch sides
\( \Large \frac{5x}{6} \normalsize - 7 = y \)

1.   Remove fractions first
2.   Work without fractions
3.   Create fractions last

Multiply EVERYTHING on both sides by 6
\( 6 \Large (\frac{5x}{6}) \normalsize - 6(7) = 6(y) \)

Simplify
\( 5x - 42 = 6y \)

Add 42 to both sides
\( 5x - 42 + 42 = 6y + 42 \)

Simplify
\( 5x = 6y + 42 \)

Divide both sides by 5
\( \Large \frac{5x}{5} \normalsize = \Large \frac{6y  +  42}{5} \)

Simplify
\( x = \Large \frac{6y  +  42}{5} \)


Changing The Subject Of A Formula

Level 3    Use 3 of addition, subtraction, multiplication or division

Type 2     Use multiplication then addition and then division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac{ex}{s} \normalsize - h \)

Switch sides
\( \Large \frac{ew}{s} \normalsize - h = y \)

1.   Remove fractions first
2.   Work without fractions
3.   Create fractions last

Multiply EVERYTHING on both sides by s
\( s \Large (\frac{ew}{s}) \normalsize - s(h) = s(y) \)

Simplify
\( ex - sh = sy \)

Add sh to both sides
\( ex - sh + sh = sy + sh \)

Simplify
\( ex = sy + sh \)

Divide both sides by e
\( \Large \frac{ex}{e} \normalsize = \Large \frac{sy  +  sh}{e} \)

Simplify
\( x = \Large \frac{sy  +  sh}{e} \)


Changing The Subject Of A Formula

Level 4    Multiply Out Brackets
                And Two Other Operations

Level 4    Type 1   
Multiply out brackets then subtraction and then division

Level 4    Type 2   
Multiply out brackets then addition and then division


Changing The Subject Of A Formula

Level 4    Multiply out brackets and two other operations

Type 1     Multiply out brackets then subtraction and then division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = 2(x + 5) \)

Switch sides
\( 2(x + 5) = y \)

1.   Multiply out brackets
2.   Work without fractions
3.   Create fractions last

Multiply out brackets
\( 2(x) + 2(5) = y \)

Simplify
\( 2x + 10 = y \)

Subtract 10 from both sides
\( 2x + 10 - 10 = y - 10 \)

Simplify
\( 2x = y - 10 \)

Divide both sides by 2
\( \Large \frac{2x}{2} \normalsize = \Large \frac{y  -  10}{2} \)

Simplify
\( x = \Large \frac{y  -  10}{2} \)


Changing The Subject Of A Formula

Level 4    Multiply out brackets and two other operations

Type 1     Multiply out brackets then subtraction and then division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = w(x + t) \)

Switch sides
\( w(x + t) = y \)

1.   Multiply out brackets
2.   Work without fractions
3.   Create fractions last

Multiply out brackets
\( w(x) + w(t) = y \)

Simplify
\( wx + tw = y \)

Subtract tw from both sides
\( wx + tw - tw = y - tw \)

Simplify
\( wx = y - tw \)

Divide both sides by w
\( \Large \frac{wx}{w} \normalsize = \Large \frac{y  -  tw}{w} \)

Simplify
\( x = \Large \frac{y  -  tw}{w} \)


Changing The Subject Of A Formula

Level 4    Multiply out brackets and two other operations

Type 2     Multiply out brackets then addition and then division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = 4(x - 3) \)

Switch sides
\( 4(x - 3) = y \)

1.   Multiply out brackets
2.   Work without fractions
3.   Create fractions last

Multiply out brackets
\( 4(x) - 4(3) = y \)

Simplify
\( 4x - 12 = y \)

Add 12 to both sides
\( 4x - 12 + 12 = y + 12 \)

Simplify
\( 4x = y + 12 \)

Divide both sides by 4
\( \Large \frac{4x}{4} \normalsize = \Large \frac{y  +  12}{4} \)

Simplify
\( x = \Large \frac{y  +  12}{4} \)


Changing The Subject Of A Formula

Level 4    Multiply out brackets and two other operations

Type 2     Multiply out brackets then addition and then division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = a(x - b) \)

Switch sides
\( a(x - b) = y \)

1.   Multiply out brackets
2.   Work without fractions
3.   Create fractions last

Multiply out brackets
\( a(x) - a(b) = y \)

Simplify
\( ax - ab = y \)

Add ab to both sides
\( ax - ab + ab = y + ab \)

Simplify
\( ax = y + ab \)

Divide both sides by 4
\( \Large \frac{ax}{a} \normalsize = \Large \frac{y  +  ab}{a} \)

Simplify
\( x = \Large \frac{y  +  ab}{a} \)


Changing The Subject Of A Formula

Level 5    Multiply Out Fraction
                Multiply Out Brackets
                And Two Other Operations

Level 5    Type 1   
Multiply out fraction   
multiply out brackets then subtraction and then division

Level 5    Type 2   
Multiply out fraction   
multiply out brackets then subtraction and then division


Changing The Subject Of A Formula

Level 5    Multiply Out Fraction
                Multiply Out Brackets
                And Two Other Operations

Type 1    Multiply out fraction
               multiply out brackets then subtraction and then division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac {5(x  +  2)}{7} \)

Switch sides
\( \Large \frac {5(x  +  2)}{7} \normalsize = y \)

1.   Multiply out fraction
2.   Multiply out brackets
3.   Work without fractions
4.   Create fractions last

Multiply out fraction    Multiply both sides by 7
\( 7 \LARGE [ \Large \frac {5(x  +  2)}{7} \LARGE ] \normalsize = 7[y] \)

Simplify
\( 5(x  +  2) = 7y \)

Multiply out brackets
\( 5(x) + 5(2) = 7y \)

Simplify
\( 5x + 10 = 7y \)

Subtract 10 from both sides
\( 5x + 10 - 10 = 7y - 10 \)

Simplify
\( 5x = 7y - 10 \)

Divide both sides by 5
\( \Large \frac{5x}{5} \normalsize = \Large \frac{7y  -  10}{5} \)

Simplify
\( x = \Large \frac{7y  -  10}{5} \)


Changing The Subject Of A Formula

Level 5    Multiply Out Fraction
                Multiply Out Brackets
                And Two Other Operations

Type 1    Multiply out fraction
               multiply out brackets then subtraction and then division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac {a(x  +  b)}{c} \)

Switch sides
\( \Large \frac {a(x  +  b)}{c} \normalsize = y \)

1.   Multiply out fraction
2.   Multiply out brackets
3.   Work without fractions
4.   Create fractions last

Multiply out fraction    Multiply both sides by c
\( c \LARGE [ \Large \frac {a(x  +  b)}{c} \LARGE ] \normalsize = c[y] \)

Simplify
\( a(x  +  b) = cy \)

Multiply out brackets
\( a(x) + a(b) = cy \)

Simplify
\( ax + ab = cy \)

Subtract ab from both sides
\( ax + ab - ab = cy - ab \)

Simplify
\( ax = cy - ab \)

Divide both sides by a
\( \Large \frac{ax}{a} \normalsize = \Large \frac{cy  -  ab}{a} \)

Simplify
\( x = \Large \frac{cy  -  ab}{a} \)


Changing The Subject Of A Formula

Level 5    Multiply Out Fraction
                Multiply Out Brackets
                And Two Other Operations

Type 2    Multiply out fraction
               multiply out brackets then addition and then division

Example 1

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac {4(x  -  3)}{5} \)

Switch sides
\( \Large \frac {4(x  -  3)}{5} \normalsize = y \)

1.   Multiply out fraction
2.   Multiply out brackets
3.   Work without fractions
4.   Create fractions last

Multiply out fraction    Multiply both sides by 5
\( 5 \LARGE [ \Large \frac {4(x  -  3)}{5} \LARGE ] \normalsize = 5[y] \)

Simplify
\( 4(x  -  3) = 5y \)

Multiply out brackets
\( 4(x) - 4(3) = 5y \)

Simplify
\( 4x - 12 = 5y \)

Add 12 to both sides
\( 4x - 12 + 12 = 5y + 12 \)

Simplify
\( 4x = 5y + 12 \)

Divide both sides by 4
\( \Large \frac{4x}{4} \normalsize = \Large \frac{5y  +  12}{4} \)

Simplify
\( x = \Large \frac{5y  +  12}{4} \)


Changing The Subject Of A Formula

Level 5    Multiply Out Fraction
                Multiply Out Brackets
                And Two Other Operations

Type 2    Multiply out fraction
               multiply out brackets then addition and then division

Example 2

\( \text{Make}  x  \text{the subject of} \)
\( y = \Large \frac {w(x  -  t)}{h} \)

Switch sides
\( \Large \frac {w(x  -  t)}{h} \normalsize = y \)

1.   Multiply out fraction
2.   Multiply out brackets
3.   Work without fractions
4.   Create fractions last

Multiply out fraction    Multiply both sides by h
\( h \LARGE [ \Large \frac {w(x  -  t)}{h} \LARGE ] \normalsize = h[y] \)

Simplify
\( w(x  -  t) = hy \)

Multiply out brackets
\( w(x) - w(t) = hy \)

Simplify
\( wx - tw = hy \)

Add tw to both sides
\( wx - tw + tw = hy + tw \)

Simplify
\( wx = hy + tw \)

Divide both sides by w
\( \Large \frac{wx}{w} \normalsize = \Large \frac{hy  +  tw}{w} \)

Simplify
\( x = \Large \frac{hy  +  tw}{w} \)


Changing The Subject Of A Formula

Videos





 

 

 

Edited by Dr David Cornelius an Independent Private Maths Tutor with over 25 years of experience and The Secretary of The Association of Tutors in the UK for 15 years.