Chapter 2 — Fractions And Decimals

Definition: A fraction represents a part of a whole or a collection. It is written as a/b where a (numerator) is the number of equal parts considered and b (denominator) is the total equal parts in one whole (b ≠ 0).

Parts of a fraction:

• Numerator — top number (parts taken).
• Denominator — bottom number (total equal parts).

Types of fractions:

• Proper fraction: numerator < denominator (e.g., 3/5).
• Improper fraction: numerator ≥ denominator (e.g., 7/4).
• Mixed number: a whole number and a proper fraction together (e.g., 2 3/4).
• Equivalent fractions: different fractions that represent the same value (e.g., 1/2 = 2/4 = 3/6).
• Like fractions: same denominators. Unlike fractions: different denominators.

Simplest form (lowest terms): A fraction is in simplest form when numerator and denominator have no common factor other than 1. To simplify, divide both by their greatest common divisor (GCD).

Representation: Fractions can be shown by:

• Area models (pie or rectangle divided into equal parts, shaded parts show the fraction).
• Number line (mark the part between 0 and 1 or beyond for improper fractions/mixed numbers).
• Fraction strips or tiles (rectangles of equal length cut into equal pieces).

Comparing fractions: If denominators are equal, compare numerators. If not, use cross-multiplication: a/b ? c/d → compare a×d and c×b. Larger product means larger fraction.

Basic operations (overview):

• Addition/Subtraction:
  • Like denominators: add/subtract numerators, keep denominator.
  • Unlike denominators: convert to like denominators (use LCM), then add/subtract numerators.
• Multiplication: Multiply numerators and multiply denominators, then simplify: (a/b)×(c/d) = (a×c)/(b×d). Cancel common factors before multiplying when possible.
• Division: Divide by a fraction by multiplying by its reciprocal: (a/b) ÷ (c/d) = (a/b)×(d/c), c ≠ 0.

Mixed numbers and improper fractions: Convert mixed to improper: for m n/p, value = (m×p + n)/p. Convert improper to mixed by dividing numerator by denominator: quotient = whole part, remainder/denominator = fractional part.

Connection with decimals and percentages: To get a decimal, divide numerator by denominator (a ÷ b). To get a percentage, multiply the fraction by 100%: (a/b)×100%.

Tips: Always simplify fractions when possible; use LCM for addition/subtraction; use reciprocal for division; represent fractions on number line to build intuition.

Equivalent fractions are different fractions that represent the same part of a whole. Two fractions a/b and c/d are equivalent if they have the same value, i.e. a/b = c/d. You can create equivalent fractions by multiplying or dividing the numerator and denominator of a fraction by the same non‑zero integer.

Example: 2/3 = (2×2)/(3×2) = 4/6 = (2×3)/(3×3) = 6/9.

Useful test: To check if a/b and c/d are equivalent use cross multiplication: a×d = b×c.

Simplest form (or lowest terms) of a fraction means the numerator and denominator have no common factor other than 1. To write a fraction in simplest form, divide both numerator and denominator by their greatest common divisor (GCD).

Example: 8/12. GCD(8,12)=4, so 8/12 = (8÷4)/(12÷4) = 2/3. This 2/3 is the simplest form.

How to simplify (stepwise):

• Find common factor(s) of numerator and denominator (or compute GCD).
• Divide numerator and denominator by the common factor (or by the GCD) until no common factor > 1 remains.
• Ensure denominator is positive; move any negative sign to the numerator.

Methods to find GCD: prime factorization (list prime factors and take common ones) or the Euclidean algorithm for larger numbers.

Notes: Every nonzero fraction has infinitely many equivalent forms, but exactly one simplest form (up to sign). Simplification applies to proper, improper fractions and mixed numbers (convert to an improper fraction, simplify, then convert back if needed).

Introduction: Fractions represent parts of a whole. Two main types for comparison are like fractions (same denominator) and unlike fractions (different denominators). To compare means to decide which is greater, smaller or if they are equal; to order means to arrange several fractions from smallest to largest (ascending) or largest to smallest (descending).

Like fractions (same denominator): If denominators are equal, the fraction with the larger numerator is larger because each fraction has the same sized parts.

• Example rule: For a/b and c/b, if a > c then a/b > c/b; if a = c then a/b = c/b; if a < c then a/b < c/b.

Unlike fractions (different denominators): To compare, make denominators the same (a common denominator) or use cross-multiplication or convert to decimals.

• Common denominator method: Convert fractions to equivalent fractions with the same denominator (often use LCM of denominators) then compare numerators.
• Cross-multiplication method: For a/b and c/d, compute a×d and c×b; compare those products. If a×d > c×b then a/b > c/d, etc.
• Decimal method: Divide numerator by denominator to get decimal values and compare (useful when decimal forms terminate or for calculators).

Ordering several fractions: Convert all to like fractions (common denominator) or compare pairwise using cross-multiplication or decimals, then arrange in ascending/descending order. For mixed numbers, first convert to improper fractions or compare whole parts first, then fractional parts.

Equivalent fractions: Multiplying numerator and denominator by the same nonzero number gives an equivalent fraction. Use this to create common denominators.

Special notes: Always simplify fractions when possible. When comparing negative fractions, remember the more negative value is smaller (e.g., -3/4 < -1/2).

What are we doing? Addition and subtraction of fractions means combining or removing parts of a whole. The basic idea: make the fractions have the same type of parts (same denominator), then add or subtract the numerators. Finally, simplify the result.

Cases:

• Like denominators (denominators equal): add/subtract numerators directly: a/b ± c/b = (a ± c)/b. Reduce if possible.
• Unlike denominators (denominators different): find a common denominator (preferably the LCM), convert each fraction to an equivalent fraction with that denominator, then add/subtract numerators and simplify.
• Mixed numbers: convert mixed numbers to improper fractions before operating, then convert the final improper result back to a mixed number if needed.

Step-by-step procedure for unlike denominators

1. Find the least common multiple (LCM) of the denominators — this is the least common denominator (LCD).
2. Convert each fraction to an equivalent fraction with the LCD: multiply numerator and denominator by the same number.
3. Add or subtract the numerators; keep the common denominator.
4. Simplify the result (reduce to lowest terms). If improper, convert to a mixed number if required.

Important notes: Always simplify answers. When subtracting larger minus smaller fraction, you may need to borrow (for mixed numbers) or get a negative result if appropriate.

What it means: Multiplication of fractions finds a fraction of another quantity. For example, (2/3) × (3/4) means "two-thirds of three-quarters."

Steps / Rule:

• If any factor is a mixed number, convert it to an improper fraction. Example: 1 1/2 = 3/2.
• Write whole numbers as fractions by giving them denominator 1 (e.g., 5 = 5/1).
• 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).
• Before multiplying, cancel (cross-simplify) any common factors between a numerator and a denominator to keep numbers small.
• After multiplying, simplify the result. If required, convert an improper fraction back to a mixed number.
• Sign rule: If exactly one factor is negative, the product is negative; if both (or none) are negative, the product is positive.

Why it works (intuitions):

• Area model: If one side of a rectangle is a/b of a unit and the other side is c/d of a unit, the area is (a/b)×(c/d).
• Scaling / number-line: Multiplying by a fraction scales a length. For example, multiplying by 1/2 gives half the length.

Tips:

• Always look to cancel common factors before multiplying to simplify calculations.
• When multiplying several fractions, multiply all numerators and all denominators (with prior cancellation when possible).

What it means: Division of fractions answers the question "How many groups of one fraction are in another?" or "If we share some amount into groups of a given fractional size, how many groups do we get?"

Rule (reciprocal method): To divide by a fraction, multiply by its reciprocal. For two nonzero fractions a/b and c/d,

(a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d) / (b × c).

Why it works (brief): Division by a number x is the same as multiplying by 1/x. The reciprocal (d/c) is 1 ÷ (c/d), so multiplying by (d/c) gives the correct number of groups.

Step-by-step method:

• Convert mixed numbers to improper fractions (if any).
• Change the division sign to multiplication and swap numerator and denominator of the divisor (take reciprocal).
• Multiply numerators together and denominators together.
• Cancel common factors before multiplying if possible (to simplify calculations).
• Simplify the result; convert to a mixed number if required.
• Remember: you cannot divide by 0.

Special cases:

• Dividing by a whole number n: a/b ÷ n = a/b × 1/n = a/(b×n).
• Dividing a whole number m by a fraction c/d: m ÷ (c/d) = m × (d/c).
• When the divisor (the fraction you divide by) is less than 1, the result is larger than the dividend.

What is simplification? Simplification of a fraction means writing it in its simplest (lowest-term) form so that numerator and denominator have no common factor except 1. To simplify, divide numerator and denominator by their greatest common divisor (GCD).

Key steps and techniques

• Reduce by GCD: If fraction is a/b, divide a and b by gcd(a,b).
• Cancel common factors: In a product of fractions cancel common factors between any numerator and any denominator before multiplying.
• Convert mixed <-> improper: To add or multiply reliably, convert mixed numbers to improper fractions: if you have m n/d then improper = (m*d + n)/d. Convert back for final answers if needed.
• Common denominator for addition/subtraction: For a/b + c/d, use LCM of b and d (often bd if denominators coprime) and then add numerators.

Order of operations with fractions

Follow the usual BODMAS/BIDMAS rules: Brackets first, then Orders (powers and roots), then Division and Multiplication (from left to right), and finally Addition and Subtraction (from left to right). These rules apply equally when terms are fractions. Important special points:

• Division by a fraction: a/b ÷ c/d = a/b × d/c (multiply by reciprocal). Do this after handling brackets and orders.
• Left-to-right for × and ÷: If both appear, compute in the order they occur left to right, simplifying by cancellation when possible.
• Left-to-right for + and −: Do addition and subtraction in sequence from left to right after × and ÷ are done.

Practical tips

• Always simplify at each step when possible to keep numbers small.
• When adding/subtracting, convert to lowest common denominator (use LCM) before combining numerators.
• Use cancellation before multiplying to avoid large numbers.

What is a decimal? A decimal is a way of writing numbers that are not whole by using a decimal point. The decimal point separates the whole-number part (on the left) from the fractional part (on the right). Decimals are another form of fractions and represent parts of a whole using powers of ten.

Place value in decimals — each place to the right of the decimal point has a value one-tenth of the place before it. For example:

In 4.370, the digit 3 is in the tenths place (3 × 10⁻¹ = 0.3), 7 is in the hundredths place (7 × 10⁻² = 0.07) and 0 is in the thousandths place (0 × 10⁻³ = 0).

Reading and writing decimals — read the whole-number part, then say 'point', then read each digit of the fractional part separately: 5.204 is read as 'five point two zero four'. You may also say the fractional value: 5.2 = five and two tenths.

Converting between fractions and decimals — fractions with denominators 10, 100, 1000, ... convert easily by placing the numerator in the corresponding decimal place. For other fractions, divide the numerator by the denominator to get a decimal. Some decimals terminate (end after finite digits) and some repeat (recurring decimals).

Comparing and ordering decimals — compare digits from left to right, aligning decimal points. If needed, add zeros to the right to make the same number of digits after the decimal point before comparing.

Basic operations and rules — for addition and subtraction, align the decimal points. For multiplication and division, use place-value understanding or move the decimal point according to multiplication/division by powers of ten; otherwise perform the operation and place the decimal point so that the result has the correct total number of decimal places (multiplication) or by shifting (division).

Visual models — decimals can be shown on a number line, in place-value charts, or with area models such as a 10×10 grid where each small square represents one hundredth.

What it means: Converting between fractions and decimals means writing the same number in two different forms. A fraction a/b shows parts of a whole; a decimal shows the same quantity using place value (tenths, hundredths, etc.).

Fraction → Decimal (how)

1. Divide the numerator by the denominator (use long division): numerator ÷ denominator = decimal.
2. If the division ends (remainder becomes 0), you get a terminating decimal.
3. If remainders repeat in a cycle, you get a repeating (recurring) decimal — show the repeating block with a vinculum or parentheses, e.g. 0.333... = 0.(3).

Decimal → Fraction (how)

1. If decimal terminates: multiply by 10^n (n = number of decimal places) and put over 10^n, then simplify. Example: 0.75 = 75/100 = 3/4.
2. If decimal repeats: use an algebraic method. Put x = repeating decimal, multiply by a power of 10 to shift a full repeat, subtract to remove the repeating part, solve for x. Example: x = 0.(6) → 10x = 6.(6) → 9x = 6 → x = 6/9 = 2/3.
3. For mixed (non-repeating + repeating) decimals, shift so the non-repeating part aligns, subtract, and simplify. There is a general formula using 10^m and 10^{m+n} (explained below).

Types to expect: terminating decimals (finite digits), repeating decimals (one or more digits repeat), and mixed numbers (whole part + fractional part).

Tips: Always simplify the fraction to its lowest terms. When converting fraction → decimal, watch for long repeating cycles (common with denominators having prime factors other than 2 or 5).

What they are
Terminating decimals are decimals that stop after a finite number of digits (for example, 0.25, 3.125). Recurring (repeating) decimals are decimals in which one or more digits repeat forever (for example, 0.333... = 0.3, 0.272727... = 0.27).

How to recognise them
Take a fraction in simplest form a/b. If the denominator b has only the prime factors 2 and/or 5, its decimal form terminates. If b has any other prime factor (3, 7, 11, etc.), the decimal repeats.

Converting a fraction to a decimal
Use long division: divide the numerator by the denominator. If the remainder becomes 0 at some step, the decimal terminates. If remainders start repeating, the decimal repeats; the repeating pattern of digits begins where that remainder first occurred.

Converting a recurring decimal to a fraction (methods)
1) Pure recurring decimal (only repeating part), e.g. x = 0.3. Let x = 0.3. Multiply by a power of 10 to shift one full repeat:

x = 0.3
10x = 3.3
10x - x = 3 => 9x = 3 => x = 3/9 = 1/3

Let A = all digits up to end of first repeat block (as integer)
Let B = digits of the non-repeating part (as integer)
x = (A - B) / (10^m (10^n - 1))

Decimal notation for repeating block
We usually show repeating digits with a bar above them: 0.666... = 0.6, 2.583333... = 2.583.

Why this matters (intuition)
Every fraction (rational number) either ends or repeats in decimal form because there are only finitely many possible remainders in long division. When a remainder repeats, the subsequent quotient digits repeat forever.

What are decimals? Decimals are numbers that represent parts of a whole using a decimal point. Each digit after the decimal point has a place value: tenths (0.1), hundredths (0.01), thousandths (0.001), and so on.

Place-value idea (key to comparing): To compare two decimals, look at digits from left to right starting with the whole-number part, then tenths, then hundredths, etc. A larger digit at the first place where they differ means a larger decimal.

Rules and method:

1. If whole-number parts are different, the number with the larger whole part is larger.
2. If whole-number parts are equal, compare tenths. If tenths are equal, compare hundredths, then thousandths, etc.
3. If one decimal has fewer digits after the point, we may add trailing zeros to make the same number of decimal places (e.g., 0.4 = 0.40 = 0.400) and then compare digit by digit.
4. Alternatively, convert both decimals to fractions with denominator 10, 100, 1000... by writing 0.abcd = abcd / 10^n, simplify if needed, and compare fractions.

Ordering decimals: To order a list of decimals from smallest to largest (ascending) or largest to smallest (descending), use the same comparison method pairwise, or add trailing zeros so all have the same number of decimal places and then sort by digits from left to right.

Special notes: Equal decimals can look different (0.5 and 0.50). Always remember to add zeros where needed. For practical situations (money, measurements), rounding to a given place may be required; use the usual rounding rule (5 or more round up, less than 5 round down).

What are decimals? Decimals are another way to show fractions with denominators that are powers of 10 (tenths, hundredths, thousandths, ...). Each place to the right of the decimal point has a value: tenths (0.1), hundredths (0.01), thousandths (0.001), etc.

Basic idea for adding and subtracting: Always line up the decimal points so that digits of the same place value (units with units, tenths with tenths, hundredths with hundredths) are in the same column. If necessary, add zeros at the end of a number to give both numbers the same number of decimal places. Then add or subtract column by column from right to left, carrying or borrowing as needed. The decimal point in the answer goes directly below the other decimal points.

1. Step 1 — Line up decimal points: Write the numbers one above the other with decimal points aligned.
2. Step 2 — Equalize decimal places: Add zeros to the end of a number so both have the same number of digits after the decimal point.
3. Step 3 — Operate right to left: For addition, add digits and carry when a column sum is 10 or more. For subtraction, subtract digits and borrow from the next left column when needed.
4. Step 4 — Place the decimal point: Put the decimal point in the result directly under the aligned decimal points.
5. Step 5 — Check with estimation: Round numbers to 1 decimal place (or whole numbers) to estimate the result and verify the answer is reasonable.

Important points: Addition of decimals is commutative (a + b = b + a) and associative ((a + b) + c = a + (b + c)). Subtraction is not commutative (a - b ≠ b - a). You can convert decimals to fractions with a common denominator for understanding, but column method is faster for calculation.

What it means
Multiplication of decimals is the same process as multiplying whole numbers, but you must also take care of the decimal places. When you multiply two (or more) decimal numbers, the product is found by multiplying their digit forms as integers and then placing the decimal point so that the total number of digits to the right of the decimal point in the product equals the sum of the digits to the right of the decimal points in the factors.

Step-by-step method

1. Ignore the decimal points and multiply the numbers as whole numbers.
2. Count the number of digits to the right of the decimal point in each factor. Add these counts to get the total number of decimal places.
3. Place the decimal point in the product so that there are exactly that many digits to the right of the decimal point. If needed, add leading zeros to the left of the product or trailing zeros to the right to get the required number of decimal places.
4. Check your result roughly by estimation (round factors to 1 significant figure and multiply) and by the sign or magnitude.

Why this works (short explanation)
Each decimal can be written as an integer divided by a power of 10. For example, 2.34 = 234/100. If a = A/10^m and b = B/10^n (A and B integers), then a × b = (A×B)/10^(m+n). That is why the number of decimal places in the product equals m + n.

Special tips

• Multiplying by 10, 100, 1000 etc. shifts the decimal point to the right by 1, 2, 3 places respectively.
• Multiplying by 0.1, 0.01, 0.001 etc. shifts the decimal left by 1, 2, 3 places respectively.
• If the product has fewer digits than the required decimal places, add leading zeros before the integer part (e.g., 6 × 3 = 18 but with 3 decimal places → 0.018).

Worked example (detailed)

Find 2.3 × 4.5.

1. Ignore decimals: 23 × 45 = 1035.
2. Count decimal places: 2.3 has 1, 4.5 has 1 → total = 2.
3. Place decimal so 2 digits are to the right: 10.35.

Checking with fractions
2.3 = 23/10 and 4.5 = 45/10, so product = (23×45)/100 = 1035/100 = 10.35.

Common mistakes to avoid

• Forgetting to count all decimal places from every factor.
• Placing the decimal point by copying a position from one factor — use the sum of the places instead.
• Not adding leading or trailing zeros when needed to reach the required number of decimal places.

What it means: Division of decimals is the operation of sharing or grouping numbers that are written with decimal points. You can treat decimal division the same way as whole-number division by first removing the decimal from the divisor.

Main idea / rule: To divide by a decimal, multiply both the divisor and the dividend by the same power of 10 (10, 100, 1000, ...) so that the divisor becomes a whole number. Then divide as usual and place the decimal point in the quotient directly above the decimal point in the adjusted dividend.

1. Step 1: Count how many decimal places are in the divisor (call this n).
2. Step 2: Multiply both divisor and dividend by 10^n to remove decimals from the divisor.
3. Step 3: Perform long division with the new numbers (the divisor is now an integer).
4. Step 4: Place the decimal point in the quotient directly above the decimal point in the (adjusted) dividend. If needed, add zeros to the dividend to continue division.

Example procedure (short): To compute 6.75 ÷ 0.25: divisor 0.25 has two decimal places, so multiply both numbers by 100 → 675 ÷ 25 = 27. So 6.75 ÷ 0.25 = 27.

Tips:

• If the divisor is 1 or 10, 100, etc., dividing is just moving the decimal point left.
• If the dividend is smaller than the divisor after adjustment, your quotient will be less than 1 (put 0 before the decimal point).
• You can also turn the division into a fraction and simplify before converting to a decimal if that makes calculation easier.
• Watch for repeating decimals in the quotient; round only when instructed.

What is estimation? Estimation means finding an approximate value that is close enough to the exact value for practical purposes. In Class 7, estimation is used to make calculations easier and faster by replacing complicated numbers (fractions or decimals) with simpler ones.

Why we estimate: To check answers, to make quick mental calculations, to plan (money, time, materials) and to simplify measurements in everyday life.

Two main ideas:

• Rounding: Replace a number by the nearest value at a chosen place value (nearest whole number, nearest tenth, nearest hundredth, or nearest simple fraction such as 1/2 or 1/4).
• Using compatible numbers or benchmarks: Replace numbers by close numbers that are easy to compute with (for example 0.49 ≈ 0.5, 7/13 ≈ 1/2, or 48 ≈ 50).

Rules for rounding decimals:

• Decide the place to round to (units, tenths, hundredths, etc.).
• Look at the digit immediately to the right of that place:
• If it is 5 or more, increase the rounding place by 1 (round up).
• If it is 0–4, keep the rounding place the same (round down).
• Replace all digits to the right by zeros (or drop them if writing in decimal form).

Rounding fractions: Two common methods:

• Convert the fraction to a decimal, then round using the decimal rules.
• Compare the fraction with common benchmarks (0, 1/4, 1/2, 3/4, 1) to choose the nearest simple fraction. Example: 5/8 is near 3/4 because 5/8 = 0.625 and 3/4 = 0.75, but 5/8 is closer to 1/2 = 0.5? (0.625 is 0.125 from 0.5 and 0.125 from 0.75 — tie: choose a convention or convert to decimal.)

Estimation techniques:

• Front-end estimation: Keep the leading (largest) digits and change the rest to zeros or easy values. Useful for quick sums.
• Compatible numbers: Adjust numbers to nearby values that make arithmetic easy (e.g., use 0.75 instead of 0.74 when multiplying by 4).
• Benchmark fractions: Use 0, 1/4, 1/2, 3/4, 1 to judge which simple value a fraction is nearest.

Error bounds: When you round, the maximum possible error is half of the unit you rounded to. Example: when rounding to the nearest tenth, the error is at most 0.05.

How to choose rounding level: Depends on required accuracy. For money you often round to two decimal places (hundredths). For quick mental sums you might round to the nearest whole number or tenth.

Practical steps for solving problems:

• Decide what degree of accuracy is needed.
• Choose rounding place or compatible numbers.
• Apply rounding rules or convert fractions to decimals when helpful.
• Estimate result and check whether the estimate is reasonable compared to exact calculation (if available).

What this topic covers

Applications and word problems in the Fractions and Decimals chapter teach how to translate everyday situations into mathematical expressions involving fractions and decimals, and then solve them using arithmetic rules. Problems include sharing, measuring, mixing, money, time, speed/rate, area/length with fractional measures, and conversion between fractions, decimals and percentages.

Steps to solve word problems

1. Read carefully: Identify what is given and what is asked.
2. Translate to math: Convert words into fractions/decimals and write equations.
3. Simplify units: Make sure all quantities use the same unit (litres, metres, rupees, etc.).
4. Choose method: Use addition/subtraction/multiplication/division of fractions or decimals as needed. For division by a fraction, multiply by its reciprocal.
5. Compute stepwise: Use LCM to add/subtract fractions; align decimal points for decimals; convert mixed numbers to improper fractions for multiplication/division.
6. Answer and check: Put result in required form (fraction, mixed number or decimal), check with estimation and units.

Common problem types

• Parts of a whole: fraction of quantity (eg, 3/8 of a kg)
• Sharing and dividing: split quantities among people using fractions/decimals
• Mixing/ratio problems: combine liquids/ingredients in fractional parts
• Measurement problems: length, area, time given in fractions/decimals
• Money problems: payments, discounts, conversions involving decimals

Tips

• Convert mixed numbers to improper fractions before multiplying or dividing.
• For addition/subtraction of fractions, find LCM of denominators (common denominator).
• To add/subtract decimals, align decimal points; for multiplication, ignore decimals then place decimal in product (sum of decimal places).
• When dividing by decimals, multiply dividend and divisor by a power of 10 to make the divisor an integer.