Fractions, Ratios, Proportions and Averages

This topic covers how to compare and combine quantities: fractions, decimals, ratios, proportions, and averages — all frequently tested in comparison-based questions.

• Fractions and decimals: A fraction (like 3/4) represents a part of a whole; the same value can be written as a decimal (0.75). Convert between the two to compare quantities quickly.
• Ratios:
• A ratio compares two or more quantities of the same unit, written as a:b.
• Always convert all quantities to the same unit before forming a ratio (e.g., comparing 2 km to 500 m — convert 2 km to 2000 m first, giving ratio 2000:500 = 4:1).
• Proportion:
• A proportion states two ratios are equal: a:b = c:d.
• Cross-multiplication rule: ad = bc. This is used to solve for an unknown value in a proportion.
• Example: If 3:4 = x:20, then 4x = 60, so x = 15.
• Mean proportional: The value x that is the mean proportional between a and b satisfies x^2 = ab, so x = √(ab). Example: mean proportional between 4 and 9 is √36 = 6.
• Dividing a quantity in a given ratio: To split a quantity Q in ratio m:n:
• First part = (m/(m+n)) x Q
• Second part = (n/(m+n)) x Q
• Example: Divide 500 in ratio 2:3 → parts are (2/5)x500=200 and (3/5)x500=300.
• Averages (mean):
• Average = sum of values / number of values.
• Combined average rule: When merging two groups with different sizes, use total sum of both groups ÷ total count — never just average the two averages, since that ignores group size.
• Example: Group A (5 students, avg 60) and Group B (5 students, avg 80) combined average = (5x60 + 5x80)/10 = 70. But if group sizes differ (say 3 and 7), you must weight the sums accordingly.

Test tip: Whenever a question mixes units (km vs m, hours vs minutes), convert everything to one unit before forming the ratio or average — this is where most errors happen.