Maths Essentials for Competitive Exams

Math Essentials for Competitive Exams

Understanding Percentage Increase and Decrease

A clear understanding of how percentages affect numbers is fundamental.

Percentage Increase:

• When a number is **INCREASED by P%**, the new value is the Original Value plus P% of the Original Value.
• **Formula:** New Value = Original Value * (1 + P/100)
Example: If X is increased by 100%, New Value = X * (1 + 100/100) = X * (1+1) = 2X.
Example: If X is increased by 200%, New Value = X * (1 + 200/100) = X * (1+2) = 3X.

Percentage Decrease:

• When a number is **DECREASED by P%**, the new value is the Original Value minus P% of the Original Value.
• **Formula:** New Value = Original Value * (1 - P/100)
Example: If X is decreased by 50%, New Value = X * (1 - 50/100) = X * (1 - 0.5) = 0.5X or X/2.
Common error: Do not add the percentage for a decrease.

Converting Repeating Decimals to Fractions & Addition Strategy

Converting repeating decimals accurately and efficiently is vital. Simplifying fractions is key for speed.

Method for Mixed Repeating Decimals (e.g., 0.a.bc where only 'bc' repeats):

• Identify non-repeating and repeating parts.
• Multiply the decimal by powers of 10 to isolate repeating blocks.
• Subtract the equations to eliminate the repeating decimal.
• Simplify the resulting fraction.
Example: Convert 0.3(24) (where 24 repeats)
Let X = 0.32424...
10X = 3.2424... (Eq 1)
1000X = 324.2424... (Eq 2)
Subtracting (Eq 1 from Eq 2): 990X = 321
X = 321/990 = 107/330.

Method for Pure Repeating Decimals (e.g., 0.(abc) where 'abc' repeats):

• For 0.(ab), it is ab/99. For 0.(abc), it is abc/999.
• The denominator consists of '9's, matching the number of repeating digits.
Example: Convert 0.(54) (where 54 repeats)
Let Y = 0.5454...
100Y = 54.5454...
Subtracting: 99Y = 54
Y = 54/99 = 6/11.

Strategy for Adding/Subtracting Fractions:

• Always simplify fractions to their lowest terms FIRST before performing addition or subtraction. This significantly reduces calculation complexity and time.
Example: Adding 0.3(36) + 0.(54)
Convert 0.3(36) to 333/990, then simplify to 37/110.
Convert 0.(54) to 54/99, then simplify to 6/11.
Add simplified fractions: 37/110 + 6/11 = 37/110 + (6 * 10)/(11 * 10) = 37/110 + 60/110 = 97/110.

Additional Helpful Tips for Competitive Exams

Ratios and Proportions (A Powerful Shortcut):

• If a ratio is given in the form A/(A+B) = X/Y, it is often beneficial to quickly deduce A/B from it.
• Componendo and Dividendo Rule: This is a significant time-saver. If A/B = C/D, then (A+B)/(A-B) = (C+D)/(C-D). This rule can directly solve problems that involve expressions like (A+B)/(A-B).

Common Percentage-to-Fraction Equivalents (Memorize These!):

• 1/2 = 50%
• 1/3 = 33.33...% (~33.3%)
• 1/4 = 25%
• 1/5 = 20%
• 1/6 = 16.66...% (~16.7% or 16 2/3%)
• 1/7 = 14.28...% (~14.3%)
• 1/8 = 12.5%
• 1/9 = 11.11...% (~11.1%)
• 1/10 = 10%
• 1/11 = 9.09...%
• 1/12 = 8.33...% (~8.3%)

Approximation Strategies:

• **Rounding:** Round numbers to the nearest convenient value (whole number, tens, hundreds) based on the precision needed for the options.
• **Compatible Numbers:** Look for numbers that are easy to multiply or divide.
• **Fraction Equivalents:** Utilize the common percentage-to-fraction equivalents for quicker mental calculations (e.g., 12.5% of X is simply X/8).

General Math Tips:

• **Practice Mental Math:** Regularly engage in mental calculations to improve speed and accuracy.
• **Memorize Tables:** Strong command of multiplication tables (up to 20-25).
• **Squares and Cubes:** Know squares up to 25-30 and cubes up to 15-20.
• **Read Carefully:** Pay close attention to keywords (e.g., "increased by," "increased to," "approximate").