Important Formulas Fractions and Decimals - (Maths) Class 7 (Old NCERT)

Important Formulas

Fractions

• Definition: A fraction is a number representing a part of a whole and is written in the form a/b, where a and b are whole numbers and b = 0. In the fraction a/b, a is the numerator and b is the denominator.
• Proper fraction: A fraction whose numerator is less than the denominator (for example, 3/5).
• Improper fraction: A fraction whose numerator is greater than or equal to the denominator (for example, 7/4 or 5/5).
• Mixed fraction (mixed number): A combination of a whole number and a proper fraction (for example, 2 1/3).
• Equivalent fractions: Fractions that express the same value. Multiply or divide the numerator and denominator of a fraction by the same non-zero number to get an equivalent fraction (for example, 1/2 = 2/4 = 5/10).
• Like and unlike fractions: Fractions with the same denominator are called like fractions. Fractions with different denominators are called unlike fractions.
• Lowest terms (simplest form): A fraction is in its lowest terms if the numerator and denominator have no common factor other than 1. To simplify, divide numerator and denominator by their highest common factor (HCF).
• Reciprocal: Two fractions are reciprocals if their product is 1. The reciprocal of a non-zero fraction a/b is b/a.
• Product of two fractions:Product of two fractions equals the product of their numerators divided by the product of their denominators.
  Formula: (a/b) × (c/d) = (a × c)/(b × d)
• Division of fractions:Division by a non-zero fraction is multiplication by its reciprocal.
  Formula: (a/b) ÷ (c/d) = (a/b) × (d/c)
• Converting a mixed fraction to an improper fraction:Multiply the whole number by the denominator and add the numerator; place this sum over the original denominator.
  Formula: m n/p = (m × p + n)/p
• Converting an improper fraction to a mixed number: Divide the numerator by the denominator to get a whole number and a remainder; write the remainder over the denominator.
• Comparing fractions:To compare two or more fractions:
  1. Find the LCM of their denominators.
  2. Convert each fraction to an equivalent fraction with denominator equal to this LCM.
  3. Compare the numerators. Larger numerator → larger fraction (if denominators same).
• Converting unlike fractions into like fractions (for addition/subtraction): Find the LCM of denominators and convert each fraction to an equivalent fraction with that LCM as denominator.
• Adding and subtracting fractions: After converting to like fractions, add or subtract their numerators and keep the common denominator, then simplify if possible.

Decimals

• Definition: A decimal is another way to represent numbers, especially parts less than one, and can be seen as a fraction with denominator 10, 100, 1000, etc.
• Parts of a decimal: A decimal number has two parts: the whole number part (left of the decimal point) and the decimal part (right of the decimal point).
• Decimal places: The number of digits in the decimal part is the number of decimal places (for example, 0.875 has three decimal places).
• Like and unlike decimals: Decimals with the same number of decimal places are like decimals; otherwise they are unlike decimals.
• Annexing zeros: Adding zeros at the right end of the decimal part does not change the value. For example, 0.1 = 0.10 = 0.100 and 0.5 = 0.50.
• Converting unlike decimals into like decimals: Add zeros to the shorter decimal so both have the same number of decimal places.
• Comparing decimals: Compare whole parts first. If whole parts are equal, compare digits of decimal part from left to right until a difference is found; the number with the larger digit at the first difference is greater.
• Converting a decimal to a fraction: Remove the decimal point to form the numerator. The denominator is 10, 100, 1000, etc., depending on number of decimal places, then simplify.
• Converting a fraction to a decimal: Convert the fraction to an equivalent fraction with denominator 10, 100, 1000, etc., if possible; otherwise perform division numerator ÷ denominator (use terminating or recurring decimals as result).
• Adding and subtracting decimals: Convert to like decimals, write numbers in columns with decimal points vertically aligned, add or subtract as whole numbers and place the decimal point in the result directly under the other decimal points.
• Multiplying decimals by 10, 100, 1000, ...: Multiply by 10 → shift decimal point right by 1 place; by 100 → shift right by 2 places; by 1000 → shift right by 3 places, and so on.
• Multiplying a decimal by a whole number: Multiply ignoring the decimal point, then place the decimal point so that the product has the same number of decimal places as the original decimal.
• Multiplying two decimals: Multiply as whole numbers; then place the decimal point in the product so that the number of decimal places equals the sum of decimal places in the two factors.
• Dividing a decimal by 10, 100, 1000, ...: Divide by 10 → shift decimal point left by 1 place; by 100 → shift left by 2 places; by 1000 → shift left by 3 places, and so on.
• Dividing a decimal by a whole number: Divide the whole-number part first; place decimal point in quotient and continue division into decimal places; append zeros to dividend if needed to continue.
• Dividing a decimal by a decimal: Multiply both dividend and divisor by the same power of 10 to make the divisor a whole number, then divide as with whole-number divisor.