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Class 7 Mathematics Chapter 10 Ratios, Proportions and Percentages — Formulas & Key Points

Class 7 Mathematics Chapter 10 introduces students to three interconnected concepts that form the backbone of commercial mathematics and data interpretation. Ratios simplify comparisons, proportions solve scaling problems, and percentages provide a standard way to express parts of a whole. Mastering these formulas is essential not only for CBSE Class 7 exams but also for everyday applications like calculating discounts, comparing offers, and understanding interest rates.

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Key takeaways

✓A ratio compares two quantities in simplest form; proportion states that two ratios are equal (a:b = c:d means ad = bc)
✓Convert fractions to percentages by multiplying by 100; convert percentages to fractions by dividing by 100
✓Profit = SP - CP; Loss = CP - SP; always compare selling price against cost price for商业 calculations
✓Simple Interest = (P × R × T)/100 where P is principal, R is rate per annum, T is time in years
✓Unitary method solves proportion problems: find value of one unit, then multiply for required quantity
✓Percentage increase or decrease = (Change/Original) × 100 is the universal formula for all percentage change problems
✓In proportion a:b = c:d, the product of extremes (a×d) always equals product of means (b×c)

Core Formulas: Ratios and Their Properties

A ratio is a comparison between two quantities of the same kind, expressed in simplest form after cancelling common factors. Ratios are fundamental to understanding proportions and scaling in mathematics. When working with ratios, always ensure both quantities are in the same unit before comparison. The order matters critically — the ratio of boys to girls is different from the ratio of girls to boys. NCERT Class 7 Mathematics emphasizes simplification: divide both terms by their HCF to get the simplest form. Compound ratios (formed by multiplying corresponding terms of two or more ratios) appear in advanced problems, while inverse ratios (swapping antecedent and consequent) help solve reverse proportion problems.

Ratio of a to b = a:b = a/b (where b ≠ 0)
Equivalent ratios: a:b = (a×k):(b×k) for any non-zero k
Simplest form: divide both terms by their HCF
Compound ratio of a:b and c:d is (a×c):(b×d)
Inverse ratio of a:b is b:a

Proportion Formulas and Properties

When two ratios are equal, they form a proportion. The statement 'a is to b as c is to d' is written as a:b::c:d or a:b = c:d. This is the foundation of the unitary method taught in CBSE 7 Mathematics. In any proportion, the product of extremes (first and last terms) equals the product of means (middle two terms). This cross-multiplication property is the most powerful tool for solving proportion problems. If three quantities are in continued proportion (a:b = b:c), then b is called the mean proportional between a and c, and b² = ac. Understanding these properties helps students solve complex word problems involving scaling, map distances, recipe adjustments, and speed-distance relationships efficiently.

Proportion: a:b = c:d means a/b = c/d
Product of extremes = Product of means: a×d = b×c
If a:b = c:d, then b:a = d:c (invertendo)
If a:b = c:d, then a:c = b:d (alternendo)
Mean proportional between a and c: b = √(ac) when a:b = b:c

Percentage: Conversion Formulas Table

Percentage means 'per hundred' and provides a standard way to express fractions and decimals. The symbol % represents division by 100. Class 7 Mathematics solutions emphasize that percentage is just another form of representing fractions — the denominator is always 100. Converting between fractions, decimals, and percentages is a core skill tested in every CBSE exam. When converting a fraction to a percentage, multiply by 100 and simplify. When converting a percentage to a fraction, write it over 100 and reduce to lowest terms. For decimals, remember that the decimal point moves two places when converting to or from percentages. These conversions form the basis for all practical percentage applications in profit-loss, discount, tax, and interest calculations that students will encounter throughout their academic journey.

Fraction to percentage: (Fraction) × 100 %
Percentage to fraction: (Percentage/100) in simplest form
Decimal to percentage: (Decimal) × 100 %
Percentage to decimal: Percentage ÷ 100
To find x% of quantity Q: (x/100) × Q

Percentage Change and Comparison Formulas

Percentage increase and decrease problems appear frequently in CBSE Class 7 Mathematics Chapter 10 examinations. The universal formula for percentage change compares the absolute change to the original value. Always identify which value is the 'original' or 'base' — this is critical for correct calculation. When a quantity increases from A to B, percentage increase = [(B-A)/A]×100. When it decreases from A to B, percentage decrease = [(A-B)/A]×100. Notice that the denominator is always the original value. A common application is comparing two quantities: to find by what percent A is more or less than B, use B as the base. Students often confuse which value should be in the denominator — a mistake that costs marks. Remember: percentage change is always calculated on the original amount, not the new amount.

Percentage increase = [(New Value - Original Value)/Original Value] × 100
Percentage decrease = [(Original Value - New Value)/Original Value] × 100
If a value increases by x%, new value = Original × (1 + x/100)
If a value decreases by x%, new value = Original × (1 - x/100)
A is what percent of B? = (A/B) × 100%

Complete Profit and Loss Formula Table

Commercial mathematics begins in Class 7 with profit and loss calculations. Cost Price (CP) is what a shopkeeper pays to acquire goods; Selling Price (SP) is what the customer pays. When SP exceeds CP, there is profit; when CP exceeds SP, there is loss. All profit and loss percentages are calculated on Cost Price unless stated otherwise — this is a crucial convention in NCERT Class 7 Mathematics. Mark Price (MP) and Discount are introduced in advanced problems: MP is the printed price, discount is the reduction offered, and SP = MP - Discount. Students must distinguish clearly between absolute profit (in rupees) and profit percentage (relative to CP). Many word problems involve successive discounts or finding CP when SP and profit percentage are given, requiring formula manipulation. Understanding these relationships prepares students for compound interest and banking concepts in higher classes.

Profit = Selling Price (SP) - Cost Price (CP) when SP > CP
Loss = Cost Price (CP) - Selling Price (SP) when CP > SP
Profit % = (Profit/CP) × 100 = [(SP-CP)/CP] × 100
Loss % = (Loss/CP) × 100 = [(CP-SP)/CP] × 100
SP = CP × (1 + Profit%/100) or CP × (100 + Profit%)/100
SP = CP × (1 - Loss%/100) or CP × (100 - Loss%)/100
CP = [SP × 100]/(100 + Profit%) when profit is known
CP = [SP × 100]/(100 - Loss%) when loss is known

Simple Interest: Complete Formula Set

Simple Interest is the extra amount paid for borrowing money or earned for lending money, calculated uniformly over the entire time period. The three pillars of simple interest are Principal (P) — the original sum borrowed or invested; Rate (R) — the interest percent per annum; and Time (T) — the duration in years. The fundamental formula SI = (P×R×T)/100 must be memorized perfectly as it forms the basis for all banking and finance problems. Amount is always the sum of Principal and Interest. When time is given in months, convert to years by dividing by 12; when given in days, divide by 365. Class 7 Mathematics notes emphasize that Rate is always annual unless specified otherwise. Problems often require finding P, R, or T when the other values and SI are given — this needs algebraic manipulation of the base formula. Students preparing for CBSE exams should practice reverse calculations where Amount and one other parameter are given, requiring them to first find SI by subtracting P from Amount, then using the formula to find the unknown quantity.

Simple Interest (SI) = (Principal × Rate × Time)/100 = (P × R × T)/100
Amount (A) = Principal + Simple Interest = P + SI
Principal (P) = (SI × 100)/(R × T) when SI, R, T are known
Rate (R) = (SI × 100)/(P × T) when SI, P, T are known
Time (T) = (SI × 100)/(P × R) when SI, P, R are known
Time conversion: months to years = months/12; days to years = days/365

Key Definitions and Mathematical Terms

Understanding precise terminology is essential for solving word problems and interpreting questions correctly in CBSE 7 Mathematics examinations. The antecedent is the first term in a ratio, the consequent is the second term. In a proportion, extremes are the first and fourth terms, means are the second and third terms. Base value is the reference quantity to which we compare — in percentage problems, identifying the base correctly is half the solution. Discount is always calculated on Marked Price, while profit and loss are always on Cost Price unless the question explicitly states otherwise. Overhead expenses (transport, labour, rent) are added to purchase price to get the total Cost Price. When items are bought in bulk at one price and sold individually at another, unit conversion becomes critical. These concepts interconnect: a 20% discount on MP of ₹500 gives SP of ₹400; if CP was ₹320, profit is ₹80 which is 25% of CP. Clarity on these terms prevents conceptual errors that appear in Class 7 Mathematics solutions discussions.

Antecedent: First term of a ratio a:b (here a)
Consequent: Second term of a ratio a:b (here b)
Extremes in proportion a:b::c:d are a and d; Means are b and c
Base: The reference value (denominator) for percentage calculation
Discount: Reduction on Marked Price; Discount = MP - SP
Overhead Expenses: Additional costs added to purchase price to get CP
Unitary Method: Finding value of one unit, then scaling to required quantity

Memory Tricks and Mnemonics for Quick Recall

Smart memory techniques help students recall formulas under exam pressure. For proportion, remember 'EXTREME MEAN MULTIPLICATION' — product of extremes equals product of means, or simply PEPM (Product of Extremes = Product of Means). When calculating percentage change, the mnemonic 'CHANGE OVER ORIGINAL' reminds you that the difference always goes in the numerator and the original value in the denominator. For profit-loss, think 'CP is KING' — all percentages bow to Cost Price (are calculated on CP). In simple interest, 'PaRTy divided by 100' helps recall SI = (P×R×T)/100, where you are multiplying Principal, Rate, Time and having a party dividing by 100. To remember whether to add or subtract in percentage increase/decrease: UP means multiply by (1 + %), DOWN means multiply by (1 - %). For ratios, 'Same Units First' — always convert to same units before forming a ratio. These mnemonics, used by toppers and recommended on CBSETUTOR.ai doubt-solving sessions, significantly reduce formula recall time during examinations, allowing more time for problem-solving and verification.

Proportion check: PEPM — Product of Extremes = Product of Means
Percentage change: COO — Change Over Original (then ×100)
Profit/Loss: 'CP is KING' — all % calculated on Cost Price
Simple Interest: 'PaRTy/100' — (Principal × Rate × Time)/100
Increase/Decrease: UP adds (+%), DOWN subtracts (-%) from 1
Ratio rule: 'SUF' — Same Units First before comparison

Common Mistakes: Signs, Units and Notation Errors

Class 7 students frequently lose marks due to avoidable mistakes rather than conceptual weakness. The most common error in ratios is comparing quantities in different units — always convert kilometres to metres, rupees to paise, hours to minutes before forming ratios. In percentage problems, students often use the wrong base: when finding what percent A is greater than B, B must be the denominator, not A. Order matters in ratios: 3:5 is not the same as 5:3; swapping changes meaning entirely. In profit-loss, calculating profit percent on SP instead of CP is a frequent error — remember 'CP is KING'. Writing 25% as 25 in calculations instead of 25/100 or 0.25 is another common slip. In simple interest, forgetting to convert months to years (divide by 12) or days to years (divide by 365) leads to grossly incorrect answers. When solving proportions, some students incorrectly multiply extremes with means instead of equating their products. Always write the % symbol after percentage values and the ratio symbol (:) between ratio terms — notation clarity prevents misinterpretation. During revision, CBSE 7 Mathematics students should maintain an error log of their personal common mistakes and review it before exams.

NEVER compare quantities in different units without conversion
In percentage calculations, identify the BASE correctly (denominator)
Order in ratios matters: 2:3 ≠ 3:2; do not swap without reason
Profit% and Loss% are ALWAYS on CP, not on SP (unless stated)
Convert time: months÷12 and days÷365 to get years for SI formula
Write 25% as 0.25 or 25/100 in calculations, not as 25
In proportion cross-multiplication: a×d = b×c (equals, not multiply)
Always simplify ratios to lowest terms by dividing by HCF

Three Solved Mini-Examples with Step-by-Step Working

Worked examples solidify understanding and demonstrate systematic problem-solving approaches expected in CBSE Class 7 Mathematics Chapter 10 answer sheets. Each problem here represents a common exam pattern: ratio comparison with unit conversion, percentage decrease in a practical context, and simple interest with Amount given. Following the structured format shown here — writing Given, To Find, Formula, Solution steps, and Final Answer — ensures full marks even if the final answer has a minor calculation slip, because method marks are awarded. These three examples cover the integration of concepts: the first combines ratios with practical measurement; the second links percentages to real-world shopping scenarios familiar to Class 7 students; the third demonstrates reverse calculation in simple interest where students must first extract SI from Amount, then find the rate. Practicing such diverse problem types available in Class 7 Mathematics solutions helps build confidence and speed, essential for scoring well in the 80-mark CBSE term exam where Chapter 10 typically carries 10-12 marks across multiple question types from VSA to Long Answer.

One-Glance Last-Minute Revision Box

This rapid-fire checklist is designed for the final 15 minutes before your CBSE Class 7 Mathematics exam when you need to refresh all key formulas and concepts from Chapter 10 without reading lengthy explanations. Each line is a self-contained formula or fact that appears frequently in question papers. Pin this section or screenshot it for quick mobile reference during revision breaks. The formulas are presented in the exact format you should write in your answer sheet — clean, standard mathematical notation that examiners expect. Remember that formula recall is just the first step; applying them correctly to word problems requires identifying which formula fits the given situation. CBSETUTOR.ai users get additional practice with photo-upload solving where you can snap any problem from your textbook or worksheet and receive step-by-step solutions that show exactly which formula to use and when, for just ₹999/month covering all subjects from Class 6 to 12 with a 3-day free trial to experience AI-powered doubt clearing available 24×7, especially helpful when preparing for chapter tests or revising for term exams late at night when no teacher or tutor is available.

Ratio a:b = a/b, always simplify by dividing both by HCF
Proportion: a:b::c:d means a×d = b×c (product of extremes = product of means)
Fraction to % = (Fraction)×100%; % to fraction = %/100 in simplest form
Percentage of Q: x% of Q = (x/100)×Q
% Increase = [(New-Old)/Old]×100; % Decrease = [(Old-New)/Old]×100
Profit = SP - CP (when SP>CP); Loss = CP - SP (when CP>SP)
Profit% = (Profit/CP)×100; Loss% = (Loss/CP)×100
SP = CP(100+Profit%)/100; SP = CP(100-Loss%)/100
Simple Interest = (P×R×T)/100; Amount = P + SI
To find P,R,T from SI: Transpose the SI formula algebraically
Time conversions: months÷12 → years; days÷365 → years
Unitary method: Find value of 1 unit, multiply by required units

Frequently asked questions

What is the difference between ratio and proportion in Class 7 Mathematics Chapter 10?+

A ratio is a comparison of two quantities (like 3:5), while a proportion is an equation stating that two ratios are equal (like 3:5 = 6:10). Ratio is a single comparison; proportion involves four terms where product of extremes equals product of means.

How do I convert a percentage to a fraction in simplest form?+

Write the percentage as a fraction with denominator 100, then simplify by dividing both numerator and denominator by their HCF. For example, 45% = 45/100 = 9/20 after dividing both by 5.

Why are profit and loss percentages always calculated on Cost Price?+

CP is the base investment by the seller, and business profitability is measured relative to what was originally invested. This is a standard commercial convention taught in NCERT Class 7 Mathematics and used universally unless a question specifically states otherwise.

What should I do if time is given in months or days in a simple interest problem?+

Always convert time to years before using the SI formula. Divide months by 12 (for example, 9 months = 9/12 = 0.75 years) and divide days by 365 (for example, 73 days = 73/365 = 0.2 years).

How can I remember which value to use as the base when calculating percentage increase or decrease?+

The base is always the ORIGINAL or INITIAL value before the change occurred. Use the mnemonic COO: Change Over Original. The change (difference) goes in the numerator, the original value in the denominator, then multiply by 100.

What is the fastest way to check if two ratios form a proportion?+

Use cross-multiplication: For a:b and c:d, calculate a×d and b×c. If these two products are equal, the ratios form a proportion (a:b::c:d). This is called the product of extremes equals product of means test.

If a shopkeeper marks up goods by 25% but gives 10% discount, what is the actual profit percent?+

Let CP = 100. MP = 100 + 25% of 100 = 125. Discount = 10% of 125 = 12.50. SP = 125 - 12.50 = 112.50. Profit = 112.50 - 100 = 12.50. Profit% = 12.50%. This successive percentage problem requires step-by-step calculation.

Do I always need to convert quantities to the same unit before finding their ratio?+

Yes, absolutely. Ratio compares like quantities. If comparing 2 km to 500 m, convert both to metres (2000 m and 500 m) giving ratio 2000:500 = 4:1. Comparing different units directly gives meaningless results and loses marks in CBSE exams.

How can CBSETUTOR.ai help with Class 7 Mathematics Chapter 10 practice?+

CBSETUTOR.ai offers 24×7 AI-powered doubt solving where you photograph any problem from your NCERT textbook or worksheets and receive step-by-step solutions showing exactly which formula to apply. At ₹999/month for Classes 6-12 all subjects with a 3-day free trial, it is like having a personal tutor available anytime, especially helpful for percentage and profit-loss word problems.

What is unitary method and how is it related to proportion?+

Unitary method solves proportion problems by first finding the value of one unit, then multiplying to get the required quantity. If 5 pens cost ₹50, find cost of 8 pens: cost of 1 pen = 50/5 = ₹10; cost of 8 pens = 10×8 = ₹80. It is a practical application of proportion where 5:50::8:x gives x=80.

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