MicrobiologyBytes: Maths & Computers for Biologists: Manipulating Numbers Updated: January 28, 2007 Search

Manipulating Numbers

Further information on this topic can be found in Chapter 2 of:

CoverMaths from Scratch for Biologists

Numerical ability is an essential skill for everyone studying the biological sciences but many students are frightened by the 'perceived' difficulty of mathematics, and are nervous about applying mathematical skills in their chosen field of study. Maths from Scratch for Biologists is a highly instructive, informal text that explains step by step how and why you need to tackle maths within the biological sciences. (Amazon.co.UK)

Algebra (from the Arabic, al-jabr, "the reduction") is a form of maths where symbols are used to represent numbers.

Arithmetic is concerned with the effect of operations (e.g. addition, multiplication, etc) on specified numbers.

In algebra, operations are applied to variables rather than specific numbers.

Why?   Here's a classic example:

John is 10 years old. His father is 35 years old.

After how many years will the father be twice as old as the son?

You could try to find the answer by experimenting with different numbers, but this is laborious (go ahead, try it!).
The better way is to treat this an algebra problem and solve the equation:

2 (10 + x) = 35 + x

Therefore:

20 + 2x = 35 + x

Simplify by subtracting x from each side (this is OK since you treat both sides the same):

20 + x = 35

Simplify by subtracting 20 from each side:

x = 15 years   (son is 25 and father is 50)

 

  • A mathematical expression has to be simplified - it contains no equal sign.

  • A mathematical equation contains an equal sign.
    The terms (groups of numbers of symbols) on both sides of the equal sign are equivalent (hence = ).
    You can do anything you want to an equation, as long as you treat both sides equally!
    To solve an equation, you must find the values(s) of the variable(s) which make the equattion true - i.e. both terms equal.

  • A mathematical formula represents a relationship between two or more variables (symbols or terms whose values may vary) and/or constants (numbers or terms whose value is fixed).

 

Solving equations often involves simplifying the expressions they contain, i.e. getting all similar terms on the same side of the equal sign:

Solve for x:     4x = 2(6x) - 4

Expand brackets ("parentheses"):

4x = 12x - 4

Isolate the variable:

4x - 12x = -4

Combine like terms:

-8x = -4

Simplify:

8x = 4

Divide by the coefficient of the variable (8):

8x/8 = 4/8

Reduce fractions to lowest terms:

x = 1/2

Check the answer:

4(1/2) = 2(6(1/2)) - 4

2 = 2(3) - 4

2 = 6 - 4

Order of Operations in Simplifying Equations:

1. Expand brackets and combine like terms.

2. Move all terms involving the variable to one side, all others to the other side (isolate the variable).

3. Combine like terms.

4. Divide by the coefficient of the variable (e.g. 5x).

5. Reduce fractions to lowest terms (e.g. 33/1 = 3/1 = 3).

6. Check the answer by substituting back into the original equation!

CoverAlgebra For Dummies

Algebra touches everyone's lives, from calculating mortgage interest to going dutch at a restaurant - not to mention the millions of students taking algebra classes. This friendly guide covers everything from fractions to quadratic equations. It includes real-world examples and story problems that will help even the most entrenched algebra phobes appproach the subject with ease. (Amazon.co.UK)

The accuracy of your answer will be influenced by the number of significant figures you use.

If you'd like some practice in dealing with equations, try these links, courtesy of ThinkQuest:

 

Oh dear, why is he doing this to us?

Because you can't go very far in biology without encountering things like this:

In an enzyme-catalyzed reaction, the reactant (S) combines reversibly with a catalyst (E) to form a complex (ES) with forward and reverse rate constants of k1 and k-1, respectively. The complex then dissociates into product (P) with a reaction rate constant of k2 and the catalyst is regenerated:

equation

From this can be derived the Michaelis-Menten equation:

 

v = 
Vmax * [S]
Km + [S]

where:
v = reaction rate (velocity)
[S] = substrate concentration
Vmax = Maximum rate
Km = Michaelis-Menten constant = substrate concentration at half the maximal velocity (V)
Km measures enzyme/substrate affinity, a low Km indicates a strong enzyme/substrate affinity and vice versa. However, Km is not just a binding constant that measures the strength of binding between the enzyme and substrate. Its value includes the affinity of substrate for enzyme, but also the rate at which the substrate bound to the enzyme is converted to product, e.g:

Enzyme

Reaction Catalyzed

Km(M)

Chymotrypsin

Ac-Phe-Ala + (H20) -> Ac-Phe + Ala

1.5*10-2

Carbonic Anhydrase

HCO3- + H+ -> (H2O) + CO2

2.6*10-2

Ribonuclease

Cytidine 2',3' cyclic phosphate + (H20) -> Cytidine 3'-phosphate

7.9*10-3

Pepsin

Phe-Gly + (H20) -> Phe + Gly

3*10-4

Tyrosyl-tRNA synthetase

Tyrosine + tRNA -> Tyrosyl-tRNA

9*10-4

Fumarase

Fumarate + (H20) -> Malate

5*10-6

If you want to see how the Michaelis-Menten equation is derived, click here.

 

When you perform algebraic manipulations, you soon encounter fractions, i.e. parts of numbers.

All fractions have three components: a numerator, a denominator, and a division symbol:

          1     NUMERATOR
          ---
          2     DENOMINATOR

In a simple fraction, the numerator and the denominator are both integers, e.g. 1/2

A complex fraction is a fraction where the numerator, denominator, or both contain a fraction, e.g:

1/2
3

A compound fraction is also called a mixed number, contains integers and fractions, e.g:

4 - 1/2
3

 

To manipulate fractions, you need to know the rules:

Rule 1:
Order of operations:

BEDMAS

 Brackets
  Exponents
   Division
    Multiplication
     Addition (left to right)
      Subtraction (left to right)
Rule 2:
Denominator of zero in a simple fraction		 The denominator of any fraction cannot have the value zero. If the denominator of a fraction is zero, the expression is not a legal fraction because its overall value is undefined.
Rule 3:
Zero in the numerator of a fraction		 A numerator is allowed to take on the value of zero in a fraction. Any legal fraction (denominator not equal to zero) with a numerator equal to zero has an overall value of zero.
Rule 4:
Single minus sign in a fraction		 If there is a single minus sign in a simple fraction, the overall value of the fraction will be negative.
Rule 5:
Odd number and even numbers of minus signs in a fraction		 If there is an even number of minus signs in a fraction, the value of the fraction is positive.
If there is an odd number of minus signs in a simple fraction, the value of the fraction is negative.

Rule 6:
Division symbol in a fraction

 The division symbol in a simple fraction indicates that the entire expression above the division symbol is the numerator and must be treated as if it were one number, and the entire expression below the division symbol is the denominator and must be treated as if it were one number.
Rule 7:
Properties of the number 1		 Multiplying any number by 1 does not change the value of the number.
Dividing any number by 1 does not change the value of the number.
Rule 8:
Forms of the number 1		 The number 1 can take many forms. 4-3 = 1 or 10-9 = 1 can be substituted because they have a value of 1.
When the numerator of a fraction is equivalent to the denominator of a fraction, the value of the fraction is 1. This only works when you have a legal fraction; i.e. the denominator does not equal zero. You can substitute such a fraction for the number 1.
Rule 9:
Any integer can be written as a fraction		 You can express any integer as a fraction by dividing by 1, or by choosing a numerator and denominator so that the overall value is equal to the integer.
Rule 10:
Factoring integers		 To factor an integer, break the integer down into a group of numbers whose product equals the original number. Factors are separated by multiplication signs. Note that the number 1 is the factor of every number. All factors of a number can be divided evenly into that number. (See below)
Rule 11:
Reducing fractions		 To reduce a simple fraction, follow the following steps:
1. Factor the numerator.
2. Factor the denominator.
3. Find the fraction mix that equals 1. (See below)
Rule 12:
Multiplying two simple fractions		 To multiply two simple fractions, complete the following steps:
1. Multiply the numerators.
2. Multiply the denominators.
3. Reduce the results (Rule 10):
a. Factor the product of the numerators.
b. Factor the product of the denominators.
c. Look for the fractions that have a value of 1.
Rule 13:
Multiplying an integer and a simple fraction		 To multiply a whole number and a fraction, complete the following steps.
1. Convert the whole number to a fraction (Rule 8):
2. Multiply the numerators.
3. Multiply the denominators.
4. Reduce the results (Rule 10):
a. Factor the product of the numerators.
b. Factor the product of the denominators.
c. Look for the fractions that have a value of 1.
Rule 14:
Multiplying three or more simple fractions		 To multiply three or more simple fractions, complete the following steps:
1. Multiply the numerators.
2. Multiply the denominators.
3. Reduce the results (Rule 10):
a. Factor the product of the numerators.
b. Factor the product of the denominators.
c. Look for the fractions that have a value of 1.
Rule 15:
Dividing simple fractions		 To divide one fraction by a second fraction, convert the problem to multiplication and multiply the two fractions:
1. Invert the fraction to the right of the equal sign.
2. Multiply the numerators.
3. Multiply the denominators.
4. Reduce the results (Rule 10):
a. Factor the product of the numerators.
b. Factor the product of the denominators.
c. Look for the fractions that have a value of 1.
Rule 16:
Dividing a simple fraction by an integer		 To divide a fraction by a whole number, or a whole number by a fraction, convert the division process to a multiplication process, and complete the following steps:
1. Convert the whole number to a fraction.
2. Change the sign to and invert the fraction to the right of the sign.
3. Multiply the numerators.
4. Multiply the denominators.
5. Reduce the results (Rule 10):
a. Factor the product of the numerators.
b. Factor the product of the denominators.
c. Look for the fractions that have a value of 1.
Rule 17:
Dividing three or more simple fractions		 To divide three or more fractions, complete the following steps:
1. Change the signs to sign and invert the fractions to the right of the signs.
2. Multiply the numerators.
3. Multiply the denominators.
4. Reduce the results (Rule 10):
a. Factor the product of the numerators.
b. Factor the product of the denominators.
c. Look for the fractions that have a value of 1.
Rule 18:
Building fractions		 To build a fraction is the reverse of reducing the fraction. Instead of searching for the 1 in a fraction and reducing, insert a 1 (or equivalent) and build (see below).
Rule 19:
Adding two simple fractions		 To add fractions, the denominators must be equal. Complete the following steps to add two fractions:
1. Build each fraction so that both denominators are equal.
2. Add the numerators of the fractions.
3. The denominators will be the denominator of the built-up fractions.
4. Reduce the answer.
Rule 20:
Subtracting two simple fractions		 To subtract, the denominators must be equal. You essentially following the same steps as in addition.
1. Build each fraction so that both denominators are equal.
2. Combine the numerators according to the operation of subtraction or additions.
3. The denominators will be the denominator of the built-up fractions.
4. Reduce the answer.
Rule 21:
Order of operations for nested brackets		 If brackets are enclosed in other brackets, work from the inside out.
Rule 22:
Order of operations in fractions		 Brackets instruct you to simplify the expression within the bracket before you proceed. The division symbol has the same role as the bracket. It instructs you to treat the quantity above the numerator as if it were enclosed in a bracket, and to treat the quantity below the numerator as if it were enclosed in another bracket.
Rule 23:
Converting complex fractions to simple fractions		 To manipulate complex fractions, convert them to simple fractions and follow rules 1 through 23.
Rule 24:
Multiplying and dividing two complex fractions		 To multiply two complex fractions, convert the fractions to simple fractions and follow the steps you use to multiply two simple fractions.
Rule 25:
Adding and subtracting two complex fractions		 To multiply add or subtract two complex fractions, convert the fractions to simple fractions and follow the steps you use to add or subtract two simple fractions.
Rule 26:
Compound fractions		 To manipulate compound fractions, convert them to simple fractions and follow rules 1 through 23 for simple fractions.
Rule 27:
Converting decimals to fractions		 Divide the fraction by 1, then multiply the result by 1 in a form that will remove the decimal.
Rule 28:
Converting percentages to fractions		 To convert a percentage to a fraction, convert 1% to 1/100. To convert a percentage to a decimal, convert 1% to 0.01.

 

Lowest Common Multiple (LCM) & Greatest Common Factor (GCF):

To find the LCM of two numbers, make a table of multiples (e.g. 2 & 3):

Multiples of 2:
Common Multiples:
Multiples of 3:
2
-
3
4
-
6
6
6
9

 

To find the GCF, i.e. numbers or expressions by which a larger number can be divided exactly ("factoring"), make a table of factors (e.g. 8 & 12):

Factors of 8: Common Factors: Factors of 12:
1
 1
 1
2 2 2
4 - 3
8 4 4
- - 6
- - 12

 

Adding and Subtracting fractions:

1   3
- + -
2   8
cannot be added, but
4   8
- + -
8   8

 

Example:

Add:             4   2
                 - + -
                 3   5
   The LCM of 3 and 5 is 15. 
                  4     2
                 --- + ---
                 3*5   5*3
   Both denominators most equal the LCM, 
so multiply 3 by 5, and 5 by 3. Now both denominators are the same.  
                 4*5   2*3
                 --- + ---
                 15    15
   To avoid altering the problem, multiply the numerators by the same factor as
the denominators.		 
                 20   6
                 -- + --
                 15   15
   Now the denominators the same, add the fractions together.  
                 26
                 --
                 15
   You cannot reduce this fraction further, i.e. reduce numerator and denominator to their LCFs, so this is the final answer.

 

Multiplying Fractions:

Example:

3
*
6
 = 
18
 = 
18/2
 = 
9
4
7
28
28/2
14

 

Dividing Fractions:

Example:

6
/
2
 = 
6
*
3
 = 
18
 = 
9
 = 
9
1
3
1
2
1
1

 

Reducing fractions (example):

  1. Factor the numerator
  2. Factor the denominator
  3. Find the fraction mix that equals 1
    15
     = 
    5 * 3
     = 
    5
    *
    3
     = 
    5
    *
    1
     = 
    5
    6
    2 * 3
    2
    3
    2
    2

(check by substituting back into the original)
Why might you want to do this?  To reduce a fraction to its simplest form.

Want to practice?

Numerator:
Denominator:

 

Building fractions (example):

2
 = 
2
*
1
 = 
2
*
4
 = 
8
3
3
3
4
12

 

Why might you want to do this?   To add or subtract fractions!

 

 

Converting fractions to decimals:

Simply divide the numerator by the denominator.

Example: 60/150 (= 6/16  = 2/5) = 0.4

 

 

Converting decimals to fractions:

  1. Divide the fraction by 1
  2. Multiply the result by 1 in a form that will remove the decimal:
3.3 =
3.3
 = 
3.3
*
10
 = 
33
1
1
10
10

 

Converting percentages to fractions:

One percent is 1/100. To convert a percentage into a fraction, place the number over a denominator of 100, then reduce to lowest terms.

Example 1:

45% = 
45
 = 
9
100
20

Example 2:

16.33% = 
16.33
 = 
1633
100
10000

 

 

The following conversions are very common and should probably be memorized:

Percentage:
Decimal:
Fraction:
Reduced:
10%
0.1
10/100
1/10
20%
0.2
20/100
1/5
25%
0.25
25/100
1/4
30%
0.3
30/100
3/10
33.33%
0.3333
33.33/100
1/3
40%
0.4
40/100
2/5
50%
0.5
50/100
1/2
60%
0.6
60/100
3/5
66.66%
0.6666
66.66/100
2/3
70%
0.7
70/100
7/10
75%
0.75
75/100
3/4
80%
0.8
80/100
4/5
90%
0.9
90/100
9/10
100%
1.0
100/100
1

If you'd like some practice dealing with fractions, try these links, courtesy of ThinkQuest:

Or these, courtesy of Visual Fractions:

© MicrobiologyBytes 2007.