Chapter 7 — Fractions

Meaning: A fraction is a number that represents a part of a whole or part of a set. It is written with two numbers separated by a line: the numerator (top) and the denominator (bottom). The denominator shows into how many equal parts the whole is divided and the numerator shows how many of those parts we consider.

Everyday examples: If a roti is cut into 4 equal pieces and you eat 1 piece, you have eaten 1/4 of the roti. If there are 10 mangoes and 3 are ripe, the ripe ones form 3/10 of the collection. A fraction can also represent a division: 3/4 means 3 divided by 4 which equals 0.75.

Visualising fractions: Use shapes and objects to make fractions clear. Draw a rectangle and divide it into equal parts. Shade some parts and write the fraction. For instance, a rectangle divided into 8 equal parts with 5 shaded is 5/8. You can use fraction strips, cut paper, or share sweets to see fractions physically. This helps understand that the size of the parts depends on the denominator: 1/8 is smaller than 1/4 because the whole is cut into more pieces.

Fractions on a number line: Place fractions on a number line between 0 and 1 by dividing the segment into equal parts. For example, to place 3/5, divide the segment from 0 to 1 into 5 equal parts and mark the third point. Fractions can be greater than 1 too — place them beyond 1 on the same number line.

Key ideas to remember: Fractions compare how large a part is relative to the whole; denominator must not be zero; fractions can be simplified, converted to equivalent forms, and used in addition, subtraction, multiplication and division. Practising with drawings and real objects makes these ideas clear and useful for daily situations such as sharing, measuring and cooking.

Definitions: A proper fraction is a fraction where the numerator is less than the denominator. Proper fractions represent amounts less than one whole, for example 2/5, 3/8, 7/9. An improper fraction is a fraction where the numerator is equal to or greater than the denominator; such fractions represent quantities equal to or greater than one whole, for example 5/5, 7/4, 9/3.

How to check: Look at the two numbers. If the top number (numerator) is smaller, it is proper. If it is equal or larger, it is improper. For instance, 4/7 is proper because 4 < 7. But 8/3 is improper because 8 > 3. 6/6 equals 1 and is considered an improper fraction by definition because numerator equals denominator.

Why both forms are useful: Proper fractions are easy to understand as parts of a whole. Improper fractions keep calculations simpler in algebra and arithmetic because they stay as single fractions during multiplication or division. Mixed numbers, which combine a whole number with a proper fraction, are another way to show quantities greater than one and are often easier to read in daily life.

Visual models: Use diagrams to show the difference. If one cake is divided into 4 parts, then 3 shaded parts is 3/4 (proper). If you collect 5 such pieces from different cakes you have 5/4 which is an improper fraction equal to 1 1/4. On a number line, proper fractions fall between 0 and 1 while improper fractions are at 1 or beyond.

Practice tips: Convert between improper fractions and mixed numbers (next topic) to gain flexibility. Use improper fractions when multiplying or dividing fractions, and convert to mixed numbers when giving final answers in everyday contexts. Working with drawings and objects helps to see how parts add up to wholes.

What is a mixed number? A mixed number shows a value greater than or equal to one as a combination of a whole number and a proper fraction. For example, 2 1/3 means two whole units and one out of three equal parts of another unit. Mixed numbers are widely used in everyday speech: we say "one and a half" instead of speaking an improper fraction.

Writing and reading: Write the whole number first, then the fractional part. Use a small space between them, for example 3 2/5. When reading aloud, say the whole number followed by the fraction: "three and two-fifths." This form helps in measurement, cooking and shopping where people think in whole units plus parts.

Conversion methods: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Place the result over the same denominator. Example: 2 1/4 = (2×4 + 1)/4 = 9/4. To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number and the remainder is the numerator of the fractional part. Example: 11/3 = 3 remainder 2, so 3 2/3.

Use in operations: In many calculations it is simpler to convert mixed numbers to improper fractions first, perform multiplication or division, and then convert the result back to a mixed number. For addition and subtraction you can work with mixed numbers directly by adding whole parts and fractional parts separately, but be careful to borrow or carry if needed.

Visual idea and practice: Draw whole shapes and extra divided shapes to demonstrate. For 2 1/3 draw two full rectangles and a third rectangle divided into three with one part shaded. Practice converting back and forth until it becomes quick. Understanding both forms gives flexibility for solving problems and presenting answers in the most suitable form.

Concept: Equivalent fractions are different fractions that represent the same amount. For example, 1/2, 2/4 and 4/8 all name the same part of a whole. Equivalent fractions are useful because they let us change the form of a fraction while keeping its value, which helps in adding, subtracting and comparing fractions.

How to create equivalent fractions: Multiply or divide both numerator and denominator by the same non-zero number. For instance, multiply 2/3 by 2/2 to get 4/6. If both numerator and denominator are divisible by a common factor, divide them to get an equivalent simpler fraction. For example 6/8 can be divided by 2/2 to give 3/4.

Visual methods: Use fraction strips or paper folding. Fold a paper into 3 equal parts to show 1/3. Then fold each third into two; the whole is now divided into 6 equal parts and 1/3 equals 2/6. Drawing pies, bars or number lines helps students see that shading the same area can be described by different fractions.

Testing equivalence: Use cross multiplication: a/b and c/d are equivalent if a×d = b×c. For example, to check if 2/7 equals 6/21 compute 2×21 = 42 and 7×6 = 42; because products match, they are equivalent.

Uses of equivalent fractions: They are key to finding a common denominator when adding or subtracting fractions, and to simplify fractions for final answers. Encourage practice creating equivalent forms and using visual models so students can choose the easiest denominator for a calculation.

Definition: A fraction is in simplest form (or lowest terms) when the numerator and the denominator have no common factor other than 1. For instance, 4/9 is in simplest form, but 6/8 is not because both 6 and 8 are divisible by 2 and can be reduced to 3/4.

Why simplify: Simplifying makes fractions easier to understand, compare and use in calculations. Textbook answers and final results are normally expected in simplest form. Simplest form also reveals the true ratio between numerator and denominator.

Methods to simplify: 1) Trial division: try dividing numerator and denominator by small primes like 2, 3, 5 until no more common divisors remain. 2) Prime factorisation: write both numbers as products of prime factors and cancel common primes. 3) Euclid's algorithm: find GCD (greatest common divisor) by repeated subtraction or division steps and then divide both by the GCD.

Step-by-step example: Simplify 18/24. Find GCD of 18 and 24. Using prime factors, 18 = 2 × 3 × 3, 24 = 2 × 2 × 2 × 3. Cancel one 2 and one 3 leaving 3/4. Using GCD method, GCD(18,24)=6 so 18/24 = (18÷6)/(24÷6)=3/4.

Practice tips: Always check a fraction after operations and simplify the result. For mixed numbers, simplify the fractional part. For answers in competitions or exams, write simplest form unless instructions say otherwise. Use factor trees and mental checks to speed up simplification during exercises.

Comparing with same denominators: If two fractions have the same denominator, simply compare the numerators. The larger numerator means the larger fraction because the pieces are the same size. For example, 5/9 > 3/9 because 5 parts of size 1/9 are more than 3 parts of size 1/9. Use number lines or fraction strips to visualise this idea: equal parts, count which has more.

Comparing with same numerators: If two fractions have the same numerator, the one with the smaller denominator is larger because the parts are bigger. For example, 3/4 > 3/5 because one fourth is larger than one fifth, so three fourths is larger than three fifths. Visual models such as bars help show that dividing into fewer parts makes each part larger.

Comparing with different numerators and denominators: Use either the LCM/common denominator method or cross-multiplication. The LCM method rewrites both fractions with the least common denominator and then compares numerators. Cross-multiplication compares products: for a/b and c/d, compute a×d and c×b. If a×d > c×b then a/b > c/d. Cross-multiplication is quick and avoids finding full equivalent fractions.

Why these methods work: Both methods put fractions on the same scale. Making denominators equal gives parts of equal size. Cross-multiplication compares the same sized counts indirectly by multiplying across.

Practical checks and orderings: Use estimation and approximate decimals to check results if needed. For ordering several fractions, convert to a common denominator or convert all to decimals. Encourage students to draw number lines, compare in pairs, and write explanations for choices since explaining the step helps understanding.

Rule and reason: When two or more fractions have the same denominator, you add the numerators and keep the denominator unchanged. This works because the denominators tell us the size of each part; if parts are equal, adding is just counting more parts. Example: 2/7 + 3/7 = (2+3)/7 = 5/7. Using fraction strips or shaded shapes shows this clearly — combine shaded parts and count how many of the equal pieces you have.

Steps to follow: 1) Ensure denominators are identical. 2) Add numerators. 3) Simplify the resulting fraction if possible. 4) If numerator is greater than or equal to denominator, convert to a mixed number. For example, 7/9 + 5/9 = 12/9 = 1 3/9 which simplifies to 1 1/3.

Working with mixed numbers: Add whole parts and fractions separately, then adjust. For example, 2 2/5 + 1 3/5: add whole numbers 2+1=3, add fractions 2/5+3/5=5/5=1. Combine to get 3+1 = 4. Or convert to improper fractions, add, and convert back; both methods work but choose the clearer one for the problem.

Common mistakes to avoid: Do not add denominators. Always check if the sum can be simplified. Use visual checks and estimation: if adding two numbers less than 1 gives a number greater than 1, confirm by converting to mixed number or decimal.

Practice: Use class exercises with objects and fraction bars so students perform physical addition before computing. This builds intuition and reduces errors in symbolic work.

Why a common denominator is needed: Fractions with different denominators have parts of different sizes. To add them, we must express them in terms of equal-sized parts. This means finding a common denominator that both fractions can share. The least common multiple (LCM) of the denominators gives the smallest such common denominator and keeps numbers small.

Step-by-step method: 1) Find the LCM of the denominators. 2) Convert each fraction into an equivalent fraction with the LCM as denominator by multiplying numerator and denominator by the appropriate factor. 3) Add the new numerators and keep the common denominator. 4) Simplify the result and convert to a mixed number if the numerator is larger than the denominator.

Example and checks: To add 2/3 + 1/4, LCM of 3 and 4 is 12. Convert: 2/3 = 8/12 and 1/4 = 3/12. Add: 8/12 + 3/12 = 11/12. Check by using decimal approximations: 2/3≈0.666 and 1/4=0.25, sum≈0.916 which matches 11/12≈0.916.

Shortcut and caution: A shortcut is to use the product of denominators: a/b + c/d = (a×d + b×c)/(b×d). This always works but may give larger numbers which need simplification. Prefer LCM for neat answers. Always simplify the final result.

Practical tips: Teach using fraction strips and pie models converting to a common number of parts. Practice problems that require both LCM and product methods so students learn when to choose which method. Emphasise showing steps and simplifying answers.

Like denominators: When denominators are equal, subtract numerators and keep the denominator. For example, 5/8 − 2/8 = 3/8. As with addition, visual models such as fraction strips help: remove shaded parts and count remaining equal parts. Simplify the result if possible and convert to mixed numbers if numerator ≥ denominator.

Unlike denominators: First get a common denominator using LCM or product, convert both fractions to equivalent ones with that denominator, then subtract numerators. Example: 5/6 − 1/4: LCM is 12, so 10/12 − 3/12 = 7/12. Always simplify the final answer.

Borrowing when subtracting mixed numbers: If the fractional part of the minuend is smaller than fractional part of the subtrahend, borrow 1 from the whole part. For example, 2 1/5 − 1 3/5: borrow 1 from 2 to make 1 6/5, then subtract to get 1 6/5 − 1 3/5 = 3/5. Alternatively convert mixed numbers to improper fractions, subtract, then convert back to mixed if needed.

Checks and estimation: After subtraction, add the result to the subtracted amount to see if you return to the original. Estimation by converting to decimals gives a quick reasonableness check. Use diagrams and number lines to show the subtraction visually and to avoid mistakes.

Practice problems: Include straightforward like-denominator exercises, unlike-denominator tasks needing LCM, and mixed-number borrowing problems to build confidence and accuracy.

Rule and reasoning: To multiply two fractions, multiply the numerators to get the new numerator and multiply the denominators to get the new denominator: (a/b)×(c/d) = (a×c)/(b×d). This gives a part of a part: if you take 2/3 of a ribbon and then 4/5 of that piece, you have 2/3×4/5 of the whole.

Simplifying before multiplying: Always look for common factors between any numerator and any denominator and cancel them before multiplying. This reduces calculation and keeps numbers small. Example: (2/3)×(9/4) cancel 9 and 3 (divide both by 3): becomes (2/1)×(3/4)=6/4=3/2.

Working with mixed numbers: Convert mixed numbers to improper fractions first, multiply, simplify, and then convert back to mixed numbers for final answers when needed. For instance, 1 1/2 × 2/3 → convert 3/2 × 2/3 → simplifies to 1.

Area model and understanding: Use a rectangle split into rows and columns to model multiplication of fractions. For 2/3×4/5 draw a rectangle split into 3 rows and 5 columns; shade 2 rows then 4 columns; the overlapping shaded small rectangles show the product 8 out of 15 equal pieces, so 8/15.

Common mistakes: Do not add denominators or numerators; do not forget to simplify. Practice with a variety of numbers and mixed forms to build fluency. Teach cancellation as an early step so students avoid big multiplications and make fewer errors.

Reciprocal method: To divide by a fraction, multiply by its reciprocal. For non-zero fractions c/d, (a/b) ÷ (c/d) = (a/b) × (d/c). For example, (3/4) ÷ (2/5) = (3/4) × (5/2) = 15/8. The reciprocal swaps numerator and denominator and turns the division into multiplication, which is easier to perform.

Steps to follow: 1) Convert any mixed numbers to improper fractions. 2) Take the reciprocal of the divisor (swap its numerator and denominator). 3) Multiply the dividend by this reciprocal. 4) Simplify the product and convert to a mixed number if required. Always check the divisor is not zero — division by zero is not allowed.

Why reciprocal works: Division asks how many times the divisor fits into the dividend. Multiplying by the reciprocal gives the number of such groups. For example, how many 1/3-litre bottles can be filled from 2 litres? Compute 2 ÷ 1/3 = 2 × 3 = 6 bottles.

Use cancellation: Cancel common factors between numerators and denominators before multiplying to simplify work. For example, 5/6 ÷ 2/3 = 5/6 × 3/2. Cancel 3 with 6 to get 5/2 = 2 1/2.

Practical problems and checks: Use word problems for context: splitting ropes, measuring liquids, or packaging items. Check answers by reversing the operation: multiply the result by the divisor to see if you get the original dividend. Practise with mixed numbers and improper fractions to be comfortable with all forms.

Where fractions appear: Fractions are used in cooking (recipes), measuring (lengths and weights), sharing (dividing items among people), money (parts of a rupee or rupee fractions), and many other everyday tasks. Translating a real situation into fractions and operations builds practical mathematical skill.

How to approach a word problem: 1) Read the question carefully and underline key numbers and words. 2) Identify what each fraction represents and decide the needed operation — addition for combining parts, subtraction for remaining parts, multiplication for part of a part, and division for sharing or finding how many parts fit into a whole. 3) Convert mixed numbers to improper fractions if calculation will be easier. 4) Carry out calculations, simplify the answer, and give final result with units.

Common types and examples: Sharing problems: split food or money among people using division of fractions. Scaling recipes: multiply or divide fraction quantities when changing number of servings. Measurement problems: add and subtract lengths given in fractional metres or kilograms. Conversion problems: convert mixed numbers to improper fractions and vice versa when necessary.

Model and check: Draw diagrams such as bar models, pies or number lines to represent the given data visually. After solving, check the result by estimation or reversing the operation. For instance, if you divided by a fraction, multiply your answer by that fraction to confirm you get the original amount.

Practice and explanation: Encourage students to write each step clearly, show conversions, and explain reasoning. Word problems help students see why fractions matter and prepare them for higher work in decimals, ratios and algebra.