Fractions (Economics)

Fractions

An (arithmetic) fraction is an expression of the form pq where p and q are integers or algebraic expressions (see Algebraic Fractions). One way of thinking about a fraction is as a division that hasn't been done yet. The fraction pq is another way of writing p÷q.

We use fractions to express parts of a whole. For example, the fraction 34 represents 3 parts out of 4.

We call p the numerator and q the denominator.

Fractions of the form ab, where a and b0 make up the set of rational numbers Q. Such fractions are also called arithmetic fractions.

Arithmetic fractions can be evaluated by dividing a by b. For example, evaluating 12 gives 0.5.

Examples: 34,7225,1100000,5002300

Note: Arithmetic fractions are closely related to decimals, ratios and percentages.

A fraction remains the same if we multiply both the numerator and denominator by the same non-zero number.

Examples:

14=1×24×2=2823=2×(10)3×(10)=2030

Simplifying Fractions

We can simplify a fraction by cancelling all common factors of the numerator and the denominator. A fraction which has been simplified (the numerator and denominator have no more common factors) is said to be written in its simplest form or lowest form.

Arithmetic with fractions is typically easier if the fractions are in their simplest form.

Worked Example

Worked Example

Simplify: 2436

Solution

24 and 36 have a factor of 4 in common so we can cancel 4 from the top and bottom: 24÷436÷4=69 Since 6 and 9 have a factor of 3 in common, we can cancel 3 from the top and bottom: 6÷39÷3=23 Since 2 and 3 do not have any common factors, this fraction is now in its simplest form.

This is the only way that you can simplify fractions.

Adding and Subtracting Fractions

If two or more fractions have the same denominator, they are said to have a common denominator.

Adding and subtracting fractions is straightforward if they all have a common denominator: we just add or subtract the numerators.

Example: 15+35=45

If the denominators are different, just adding or subtracting the numerators together doesn't make sense - what should the denominator of the result be? The solution is to restate both fractions over a common denominator before doing the addition (or subtraction). This common denominator is equal to the product of the denominators of the fractions we are adding (or subtracting). For example, suppose we want to calculate: 35+710 The common denominator will be 5×10=50. To restate the first fraction over this denominator, we must multiply both the numerator and denominator by 10: 3×105×10=3050 To restate the second fraction over this denominator, we must multiply both the numerator and denominator by 5: 7×510×5=3550 Since the denominators of the two fractions are now the same (50), we can just add the numerators: 3050+3550=6550 The final step is to write this fraction in its simplest form by cancelling all common factors in the numerator and denominator (if there are any). The highest common factor of 65 and 50 is 5, so: 6550=65÷550÷5=1310 Thus 35+710=1310

In summary:

1) Multiply the denominators to get a common denominator.

2) Restate each fraction over the common denominator.

3) Add (or subtract) the numerators so that you now have one fraction.

4) Simplify the fraction if required.

Worked Examples

Worked Example 1

Calculate 35+14.

Solution

35+14=3×44×5+1×55×4=1220+520=12+520=1720

Worked Example 2

Calculate 5623.

Solution

5623=5×36×32×63×6=15181218=151218=318=16

Multiplying Fractions

The rule for multiplying fractions is multiply the numerators together and multiply the denominators together:

ab×cd=acbd

Worked Example

Worked Example

Calculate 37×53.

Solution

37×53=3×57×3=1521=57

Dividing Fractions

The rule for dividing fractions is: ab÷cd=ab×dc.

This is easily remembered as “flip the second fraction, then multiply”.

Sometimes this may be written as a fraction on top of a fraction, but the rule is the same: a/bc/d=ab×dc.

Worked Example

Worked Example 1

Calculate 13÷14.

Solution

13÷14=13×41=1×43×1=43.

Worked Example 2

Calculate 35÷23.

Solution

35÷23=35×32=3×35×2=910.

Worked Example 3

Simplify the fraction 2/325/7.

Solution

Start by changing the fraction to a division symbol to see both parts side-by-side. 23÷(257) Subtract 57 from 2 by putting them over a common denominator. 257=2×7757=14757=1457=97. Now, we have: 23÷97 Use the rule for dividing fractions. 23÷97=23×79=2×73×9=1427.

Video Examples

Example 1

Prof. Robin Johnson computes 23+45 and 56+14730.

Example 2

Prof. Robin Johnson calculates 2÷14 and 3÷25.

Example 3

Prof. Robin Johnson simplifies the fraction 3/424/7.

Test Yourself

Test yourself: Numbas test on fractions

Test yourself: Another Numbas test on fractions

External Resources