Chapter 7 — Fractions

Definition: A fraction shows a part of a whole or a part of a collection. It is written as a/b where a (numerator) is the number of equal parts taken and b (denominator) is the total number of equal parts the whole is divided into. The denominator cannot be zero.

What the two parts tell us

• Denominator (b): how many equal pieces make the whole.
• Numerator (a): how many of those pieces we have or are considering.

Understanding with value: A fraction a/b means the same as the division a ÷ b. For example, 3/4 = 3 ÷ 4 = 0.75.

Types of fractions:

• Proper fraction: numerator < denominator (e.g., 3/5).
• Improper fraction: numerator ≥ denominator (e.g., 7/4).
• Mixed number: combination of a whole number and a proper fraction (e.g., 1 3/4).

Other important ideas: Equivalent fractions name the same part of a whole (for example, 1/2 = 2/4 = 4/8). A fraction is in simplest form when numerator and denominator have no common factor other than 1.

Use on a number line and area model: Fractions can be shown by shading part of a shape (a circle, rectangle) or by marking equal divisions on a number line between 0 and 1 (or beyond for improper fractions).

What is a fraction on the number line?
The number line is a straight line with points that represent numbers. A fraction a/b (with b > 0) represents the point at distance a times the unit part 1/b from 0, measured to the right if a >= 0. To plot a/b between 0 and 1, divide the segment from 0 to 1 into b equal parts and count a parts from 0.

How to represent fractions — step by step

• Decide the unit interval: usually between 0 and 1.
• Divide the interval 0–1 into b equal equal parts for denominator b (each part is a unit fraction 1/b).
• Count a parts from 0; that point is a/b. If a >= b (improper fraction), first mark whole numbers 1, 2, ... then mark the fractional parts after the last whole number.

Equivalent fractions on the number line
Equivalent fractions (for example 1/2 and 2/4) fall at the same point. To show this, divide 0–1 into the first denominator parts and then again into a multiple of those parts (refiner subdivision); the points coincide.

Comparing fractions using the number line
A fraction that lies to the right on the number line is larger. If two fractions share the same denominator, compare numerators. If they have different denominators, either convert to a common denominator or use cross-multiplication (see formulas) to decide which point is to the right.

Improper fractions and mixed numbers
An improper fraction a/b where a >= b is plotted by moving to the whole-number part q = floor(a/b) and then plotting the remaining r/b where r = a - qb. This corresponds to the mixed number q r/b.

Why this matters
Using the number line builds a visual understanding of size, equivalence and ordering of fractions and connects fractions with whole numbers and integers.

A fraction represents a part of a whole and is written as numerator/denominator where denominator ≠ 0. In Class 6 you learn several types of fractions and how to convert and compare them.

• Proper fraction: numerator < denominator. Value < 1. Example: 3/8.
• Improper fraction: numerator ≥ denominator. Value ≥ 1. Example: 7/4.
• Mixed fraction (mixed number): a whole number and a proper fraction together, e.g. 1 3/4. It is another way to write an improper fraction.
• Unit fraction: numerator = 1, e.g. 1/5. These are basic building blocks of other fractions.
• Like fractions: fractions with the same denominator, e.g. 2/7 and 3/7. They are easy to compare and add/subtract.
• Unlike fractions: fractions with different denominators, e.g. 2/5 and 3/8. To operate on them, convert to like denominators first.
• Equivalent fractions: different numerators and denominators but same value, e.g. 1/2 = 2/4 = 3/6. Formed by multiplying/dividing numerator and denominator by the same non-zero number.
• Simplest (lowest terms) or irreducible fraction: numerator and denominator have no common factor other than 1. Example: 3/8 is in simplest form; 6/8 reduces to 3/4.

Short procedures:

• To convert a mixed number to an improper fraction use multiplication and addition (see formulas below).
• To convert an improper fraction to a mixed number use division: quotient is whole part, remainder gives the proper fraction.
• To get equivalent fractions multiply or divide numerator and denominator by the same non-zero integer.
• To simplify a fraction divide numerator and denominator by their greatest common divisor (GCD).

Visual models (area, bar, number line) help to understand and compare types of fractions and to see equivalence, conversion and simplification clearly.

Equivalent fractions are different fractions that represent the same part of a whole. For example, 1/2, 2/4 and 3/6 are equivalent because each denotes the same amount.

How to make equivalent fractions:

• Multiply both numerator and denominator by the same nonzero number. Example: (1/2)×(2/2)=2/4.
• Divide numerator and denominator by their common factor to simplify. Example: 8/12 divided by 4/4 = 2/3.

How to check if two fractions are equivalent: use cross multiplication. Fractions a/b and c/d are equivalent if and only if a×d = b×c. Example: check 3/4 and 6/8: 3×8 = 24 and 4×6 = 24, so they are equivalent.

Lowest terms (or simplest form) means the numerator and denominator have no common factor other than 1. Every fraction has one unique equivalent fraction in lowest terms.

Visual models such as pie charts, fraction bars and number lines help students see why fractions with different numerators and denominators can be equal in value.

Like and Unlike Fractions — Definitions

Fractions have two parts: numerator (top) and denominator (bottom). Like fractions are fractions with the same denominator, for example 3/8 and 5/8. Unlike fractions have different denominators, for example 2/5 and 3/7.

Equivalent fractions are different-looking fractions that represent the same value, e.g. 1/2 = 2/4 = 3/6. You get equivalent fractions by multiplying or dividing numerator and denominator by the same non-zero number.

Why the distinction matters: Operations (addition and subtraction) are simple for like fractions — you just add/subtract numerators and keep the common denominator. For unlike fractions you must first convert them to like fractions (common denominator) before adding or subtracting.

How to convert unlike fractions to like fractions:

• Find the least common denominator (LCD), which is the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the LCD by multiplying numerator and denominator by the same number.
• Now add or subtract the numerators and keep the LCD as denominator.

Simplify the result by dividing numerator and denominator by their greatest common divisor (GCD) if possible.

Example process (brief): To add 2/3 and 1/4: LCD of 3 and 4 is 12. Convert: 2/3 = 8/12, 1/4 = 3/12. Add: 8/12 + 3/12 = 11/12. Already simplified.

Comparing fractions means deciding which of two (or more) fractions is greater, or whether they are equal. Several simple methods help compare fractions reliably:

• Same denominator: If denominators are equal, the fraction with the larger numerator is greater. Example: 5/9 > 3/9 because 5 > 3.
• Same numerator: If numerators are equal (and positive), the fraction with the smaller denominator is greater. Example: 3/4 < 3/5 is false; 3/4 > 3/5 because 4 < 5.
• Make denominators equal (like denominators): Change both fractions to equivalent fractions with the same denominator (usually the LCM of the two denominators), then compare numerators.
• Cross-multiplication: For a/b and c/d, compute a×d and c×b. If a×d > c×b then a/b > c/d; if a×d < c×b then a/b < c/d; if equal, the fractions are equal. This avoids finding LCMs.
• Convert to decimals: Divide numerator by denominator to get decimals and compare (useful with calculators).
• Use benchmarks: Compare each fraction to common benchmarks like 0, 1/2 and 1. For example, if one fraction > 1/2 and another < 1/2, you can tell which is larger immediately.

Choose the method that is quickest for the given fractions. For small denominators cross-multiplication is fast and error-free. Converting to like denominators is instructive and useful for adding/subtracting later.

Definition: A fraction is in its simplest form (or lowest terms) when the numerator and denominator have no common factor other than 1. In other words, gcd(numerator, denominator) = 1.

Why reduce? Reducing makes fractions easier to compare, add, subtract and use in real life (recipes, measurements, sharing).

How to reduce a fraction — methods:

1. Divide by common factors: Find a number >1 that divides both numerator and denominator and divide both by it. Repeat until no common factor remains. Example: 8/12 → divide by 2 → 4/6 → divide by 2 → 2/3.
2. Use the highest common factor (HCF or gcd): Find d = gcd(a,b). Then (a/b) = (a/d)/(b/d). Example: gcd(18,24)=6, so 18/24 = (18/6)/(24/6) = 3/4.
3. Prime factorization: Write numerator and denominator as products of primes, cancel matching prime factors, multiply remaining primes to get simplest numerator and denominator.

Notes: Improper fractions and the fractional part of mixed numbers are reduced in the same way (e.g., 10/8 → divide by 2 → 5/4). If the numerator is 0, the fraction 0/b is already in simplest form (provided b≠0).

Tip: Use small divisibility tests (by 2,3,5) to spot common factors quickly.

Definitions

An improper fraction is a fraction where the numerator (top number) is greater than or equal to the denominator (bottom number), for example 7/4 or 8/3. A mixed number is a number made of a whole number and a proper fraction (where the numerator is less than the denominator), for example 1 3/4 or 2 2/3.

Why convert? Converting makes it easier to understand quantities in everyday contexts (like measuring, cooking, sharing) and to perform certain operations (addition, subtraction) more easily.

Convert an improper fraction to a mixed number — Steps

1. Divide the numerator by the denominator.
2. The quotient (whole number part) is the whole number of the mixed number.
3. The remainder becomes the numerator of the fractional part; keep the same denominator.
4. If the remainder is 0, the mixed number is just the whole number (no fraction part).

Example: 7/4 → divide 7 by 4: quotient 1, remainder 3 → 1 3/4.

Convert a mixed number to an improper fraction — Steps

1. Multiply the whole number by the denominator.
2. Add the numerator of the fractional part to that product. This sum is the new numerator.
3. Use the same denominator.

Formula form: whole \times denominator + numerator all over denominator. Example: 1 3/4 → (1×4 + 3)/4 = 7/4.

Negative numbers

If the fraction is negative, the negative sign applies to the whole mixed number. Example: -7/4 = -1 3/4. When converting a negative mixed number to improper form, keep the negative sign: -(1×4 + 3)/4 = -7/4.

Tips

• Always simplify the fractional part of the mixed number if possible.
• Check your work by converting back: converting the result should return the original value.

What is addition of fractions? Addition of fractions means combining two or more fractional parts to get a larger fraction or a whole. We add fractions by making their denominators the same, then adding their numerators and simplifying the result.

Step-by-step methods

1. When denominators are the same (like fractions):

   To add a/b + c/b, keep the denominator b and add the numerators: (a + c)/b. Then simplify if possible.

2. When denominators are different (unlike fractions):

   Find a common denominator (preferably the LCM of the denominators). Convert each fraction to an equivalent fraction with that common denominator, add the numerators, and simplify.

3. When mixed numbers are involved:

   Either add whole-number parts and fractional parts separately (after making fractions like denominators) or convert mixed numbers to improper fractions, add, then convert back and simplify.

4. Simplifying:

   Always reduce the resulting fraction to its simplest form and, if it is an improper fraction (numerator >= denominator), convert it to a mixed number if required.

Important notes: Use the least common multiple (LCM) to keep work simple. Check if result can be simplified by a common factor. For visual understanding, use number lines, area (pie) models, or bar models.

What is subtraction of fractions?
Subtraction of fractions means finding how much is left when one fractional part is taken away from another or from a whole. The result should be written in simplest form.

Basic rules (stepwise)

1. Same denominator: If the denominators are equal, subtract the numerators and keep the denominator. Example:  3/7 − 1/7 = (3−1)/7 = 2/7. Simplify if possible.
2. Different denominators: Change the fractions to equivalent fractions with a common denominator (preferably the LCM of the denominators). Then subtract the numerators and simplify.
   Steps: (a) Find LCM of denominators. (b) Convert each fraction to an equivalent fraction with that denominator. (c) Subtract the numerators. (d) Simplify the result.
3. Mixed numbers: Either (A) convert each mixed number to an improper fraction, perform subtraction, then simplify; or (B) subtract whole parts and fractional parts separately using borrowing if needed. Converting to improper fractions is the simplest method for Class 6.
4. Check sign: If the subtrahend is larger than the minuend, the result will be negative.

Example of different denominators (brief): 3/4 − 1/6. LCM of 4 and 6 is 12. Convert: 3/4 = 9/12, 1/6 = 2/12. Subtract: 9/12 − 2/12 = 7/12.

Real-life contexts: subtracting fractions appears when sharing or using parts of items—e.g., eating slices of a pizza, using some length of ribbon from a roll, measuring ingredients while cooking, or removing time (hours and half-hours) from a schedule. In each case, convert to common units (same denominator) and subtract.

Simplify the result: Always reduce the fraction to lowest terms (divide numerator and denominator by their GCD). If the result is an improper fraction and a mixed number is preferred, convert back to a mixed number.

Idea: Multiplying a fraction by a whole number means taking that fraction repeatedly the given number of times. For example, 3 × (2/5) means add 2/5 three times: 2/5 + 2/5 + 2/5.

Rule / Method:

1. Write the fraction as a/b and the whole number as n.
2. Multiply the numerator by the whole number: n × (a/b) = (n × a) / b.
3. Then simplify the resulting fraction if possible. If the numerator is larger than the denominator, convert to a mixed number.

Why it works: A fraction a/b means a parts out of b equal parts. Taking n such fractions gives n·a parts of size 1/b each, so total = (n·a)/b.

Worked example:

Find 3 × (2/5):

1. Repeated addition: 2/5 + 2/5 + 2/5 = (2+2+2)/5 = 6/5.
2. Using rule: 3 × (2/5) = (3×2)/5 = 6/5 = 1 1/5 (mixed number).

Tip: You may also convert the whole number to a fraction (n = n/1) and multiply: (a/b) × (n/1) = (a×n)/(b×1) = (n×a)/b.

What this topic covers
Word Problems and Applications with fractions teach how to translate real-life situations into fraction expressions and solve them using operations on fractions (addition, subtraction, multiplication, division) and mixed numbers. Problems include finding a fraction of a quantity, sharing, scaling (recipes, maps), comparing, and converting units.

Step-by-step approach to solve word problems

1. Read the problem carefully and underline what is asked.
2. Identify quantities and express them as fractions or mixed numbers.
3. Decide which operation(s) are needed: add/subtract (combine or find remainder), multiply (fraction of a quantity or scaling), divide (sharing or how many groups), or a sequence of operations.
4. If needed, convert mixed numbers to improper fractions, and use a common denominator for addition/subtraction.
5. Perform the operation, simplify the result (use HCF), and if needed convert an improper fraction to a mixed number.
6. Check the answer in the context of the problem (units, reasonableness).

Key methods

• Add/Subtract: make denominators same, add/subtract numerators, simplify.
• Multiply: multiply numerators and denominators; simplify before or after multiplying.
• Divide: multiply by the reciprocal of the divisor (a/b ÷ c/d = a/b × d/c).
• Mixed numbers: convert to improper fraction to compute, then convert back if required.
• Fraction of a quantity: multiply the fraction by the total quantity (e.g., 3/5 of 250 = (3/5)×250).

Tips for word problems

• Use diagrams (fraction strips, bar models, pie charts) to visualize the situation.
• Label units clearly (kg, m, cups, slices, etc.).
• Break multi-step problems into parts and solve stepwise.

Overview: Practice and problem‑solving techniques for fractions help students understand, compute and check answers quickly and accurately. Focus on visualizing fractions, converting between forms, using equivalent fractions, and choosing the right method for addition, subtraction, multiplication and division.

Step-by-step problem-solving approach

1. Read and understand: Identify the whole, the parts, and what the problem asks (add, subtract, multiply, divide, compare, or convert).
2. Draw or model: Use a number line, pie diagram, fraction strip or bar model to picture the fractions. This reveals relationships like which fraction is larger or how pieces combine.
3. Make denominators compatible: For addition/subtraction, change fractions to equivalent fractions with the Least Common Denominator (LCD) using LCM. For multiplication, work directly with numerators and denominators.
4. Perform the operation: Use the appropriate rule (see formulas). Simplify the result using GCD. Convert improper fractions to mixed numbers when needed.
5. Estimate and check: Round or convert to decimal to see if the answer is reasonable. Re-model the result with a diagram or do the inverse operation to verify.
6. Write the answer clearly: In simplest form; give mixed number or improper as required by the question.

Useful techniques and strategies

• Equivalent fractions: Multiply or divide numerator and denominator by the same number to make denominators same or simplify a fraction.
• Number line: Place fractions on a number line to compare sizes and to perform addition/subtraction by moving right/left.
• Estimation: Round fractions (close to 0, 1/2, or 1) to check answers quickly.
• Cross‑multiplication: Compare two fractions a/b and c/d by comparing a·d and b·c (no need to find LCM).
• Simplify early: When multiplying or dividing, cancel common factors before multiplying to keep numbers small.
• Break complex problems: Split mixed-number operations into whole and fractional parts, or use fraction strips to add piecewise.

Small worked example (addition with unlike denominators):

Problem: 1/3 + 2/5

1. Find LCM of 3 and 5 = 15.
2. Convert: 1/3 = 5/15, 2/5 = 6/15.
3. Add: 5/15 + 6/15 = 11/15 (already in simplest form).
4. Estimate check: 1/3 (~0.33) + 2/5 (0.4) ≈ 0.73; 11/15 ≈ 0.733 — reasonable.

Practice tips: Start with visual models, do mixed sets of problems (compare, simplify, convert), time short drills for basic skills (finding equivalents, simplifying), and explain solutions to a peer or teacher to strengthen understanding.