Fractions, Percentages and Ratios

Common Fractions

In the fraction a/b, a is the numerator and b is the denominator (b ≠ 0). Operations with fractions:

Operation	Rule	Example
Addition	bring to a common denominator	1/2 + 1/3 = 5/6
Multiplication	numerator by numerator, denominator by denominator	2/3 · 3/4 = 1/2
Division	the second fraction is inverted	2/3 : 4/9 = 3/2

• General rule: a/b + c/d = (ad + bc) / bd.
• When adding, denominators are not added: 1/2 + 1/3 ≠ 2/5.
• Mixed number 2 1/3 = (2·3 + 1)/3 = 7/3.

Percentages

1 percent = 0,01. The three main types of problems:

Problem	Formula
Finding p% of N	N · p / 100
Finding the whole when p% of it is a given part	N = part · 100 / p
What percent of the whole is the part	part / whole · 100%

• When increased by p%, the new value is N · (1 + p/100); when decreased by p%, it is N · (1 − p/100).
• In successive changes, the coefficients are multiplied. Successive percentages are not added: 1,2 · 0,8 = 0,96, that is, it decreases by 4%.

Ratios and Proportions

The ratio a : b is the quotient of a divided by b. Equal ratios form a proportion: a : b = c : d.

• Proportion: product of the means = product of the extremes: ad = bc.
• Dividing a number in the ratio m : n: one part = number / (m + n), and the parts are taken m times and n times.
• Combining two ratios: if a : b = 2 : 3 and b : c = 4 : 5, then we make b equal in both: 8 : 12 and 12 : 15, so a : b : c = 8 : 12 : 15.

1. Identify the "whole" and the "part" from the problem statement
2. Convert the percentage to a fraction by multiplying by 0,01
3. Apply the appropriate formula
4. Check the answer against the problem statement

Examples

• 15% of 240: 240 · 0,15 = 36.
• The price was 500 som, increased by 20%, and then decreased by 20%: 500 · 1,2 · 0,8 = 480.
• Dividing 90 in the ratio 2 : 3: 90 / 5 = 18, so the parts are 36 and 54.