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ICSE • Class 7 • Mathematics • Ch 7
Estimated Time: 45 Mins
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Percentage

In ICSE Class 7 Mathematics, "Percentage" provides an authoritative, commercial arithmetic master study guide analyzing the mathematical concept of parts per hundred ($x\% = \frac{x}{100}$). This comprehensive chapter explores Meaning and Conversion (Converting percentages to fractions and decimals; Converting fractions, decimals, and ratios to percentages), Expressing One Quantity as a Percentage of Another, Percentage Increase and Decrease (Formulas: $\frac{\text{Increase}}{\text{Original}} \times 100\%$ and $\frac{\text{Decrease}}{\text{Original}} \times 100\%$), Finding the Original Value given the increased or decreased value, Commercial Applications: Profit and Loss (Cost Price [CP], Selling Price [SP], Profit $= SP - CP$, Loss $= CP - SP$, Profit $\% = \frac{\text{Profit}}{CP} \times 100$, Loss $\% = \frac{\text{Loss}}{CP} \times 100$), and Simple Interest (Principal $[P]$, Rate percent per annum $[R]$, Time in years $[T]$, Simple Interest $I = \frac{P \times R \times T}{100}$, Total Amount $A = P + I$) aligned with the 2026–27 CISCE curriculum.

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If a Store Slashes Prices by 20% Today and Then Increases Them by 20% Tomorrow, Why Do You NOT End Up with the Original Price?

Imagine walking into your favorite electronics shop. A smartphone has a price tag of $\text{Rs. } 10,000$. During a flash Diwali festival sale, the shopkeeper slashes the price by 20%! The phone now costs $\text{Rs. } 8,000$. The next day, the sale ends, and the store announces: "Prices increased by 20% back to normal!" You pull out $\text{Rs. } 10,000$ to buy it, but the cashier shakes his head: 20% of $\text{Rs. } 8,000$ is only $\text{Rs. } 1,600$, so the new price is $\text{Rs. } 9,600$! You just saved $\text{Rs. } 400$! How can a 20% drop followed by a 20% rise result in a lower price? Because in the language of Percentage, the base quantity upon which the percentage is calculated changes every time! From Latin per centum (out of every hundred), percentages rule global commerce, banking interest rates, tax calculations, and sports statistics. What is the fundamental difference between Profit Percentage calculated on Cost Price versus Selling Price? How does the Simple Interest formula ($I = \frac{PRT}{100}$) govern banking loans? Let's master percentage.

Why This Chapter Matters

Percentage is the universal language of commerce, personal banking, salary increments, income tax brackets, scientific yields, and data analytics. Mastery of profit-and-loss margins and simple interest calculations is a core, high-weightage topic in ICSE board examinations and lifelong financial literacy.

Before You Begin (Prerequisites)

  • Fractions and decimals: Conversion and basic operations.
  • Ratio and proportion from Chapter 6.
  • Solving simple linear algebraic equations.

What You Will Learn (Core Objectives)

  • Convert between percentages, fractions, decimals, and ratios seamlessly.
  • Calculate a given percentage of a quantity and find what percent one quantity is of another.
  • Calculate percentage increase and percentage decrease using standard formulas.
  • Determine the original value when the increased/decreased value is provided.
  • Compute Cost Price, Selling Price, Profit, Loss, and Profit/Loss percentages.
  • Calculate Simple Interest ($I = \frac{PRT}{100}$) and Total Maturity Amount ($A = P + I$).

Chapter Roadmap & Progression

1 1. Concept of Percentage & Inter-Co...
2 2. Percentage Change & Finding Orig...
3 3. Commercial Applications: Profit...
4 4. Simple Interest & Total Maturity...

Complete Concept Guide (100% Curriculum Coverage)

1. Concept of Percentage & Inter-Conversions

Understand
A. Meaning of Percentage:

The word Percent is derived from the Latin phrase per centum, meaning "out of one hundred". It is denoted by the symbol %:

$$x\% = \frac{x}{100}$$

A percentage is simply a fraction whose denominator is always $100$.

B. Inter-conversion Rules:
  • Percentage to Fraction / Decimal: Drop the % sign and divide by 100: $$45\% = \frac{45}{100} = \frac{9}{20} = 0.45$$
  • Fraction / Decimal / Ratio to Percentage: Multiply by 100 and annex the % sign: $$\frac{3}{8} = \left( \frac{3}{8} \times 100 \right)\% = \frac{75}{2}\% = 37.5\%$$ $$0.625 = (0.625 \times 100)\% = 62.5\%$$ $$4 : 5 = \left( \frac{4}{5} \times 100 \right)\% = 80\%$$

2. Percentage Change & Finding Original Values

Percentage Change
A. Percentage Increase and Decrease:
$$\text{Percentage Increase} = \frac{\text{Absolute Increase}}{\text{Original Initial Value}} \times 100\%$$ $$\text{Percentage Decrease} = \frac{\text{Absolute Decrease}}{\text{Original Initial Value}} \times 100\%$$

Golden Rule: The denominator is always the original initial quantity, never the new quantity!

B. Finding the Original Value:
  • If a quantity is increased by $x\%$, the new value is $(100 + x)\%$ of the original. $$\text{Original Value} = \frac{\text{New Increased Value}}{100 + x} \times 100$$
  • If a quantity is decreased by $x\%$, the new value is $(100 - x)\%$ of the original. $$\text{Original Value} = \frac{\text{New Decreased Value}}{100 - x} \times 100$$

3. Commercial Applications: Profit and Loss

Profit & Loss
A. Fundamental Terms:
  • Cost Price (CP): The purchase price of an article plus overhead expenses (transport, repairs).
  • Selling Price (SP): The price at which an article is sold to a customer.
B. Profit vs Loss Formulas:
ConditionAbsolute AmountPercentage Formula (Always on CP!)
Profit (Gain): $SP > CP$$\text{Profit} = SP - CP$$\text{Profit } \% = \frac{\text{Profit}}{CP} \times 100$
Loss: $CP > SP$$\text{Loss} = CP - SP$$\text{Loss } \% = \frac{\text{Loss}}{CP} \times 100$

Calculating SP from CP: $SP = CP \times \left( \frac{100 + \text{Profit } \%}{100} \right)$ or $SP = CP \times \left( \frac{100 - \text{Loss } \%}{100} \right)$.

4. Simple Interest & Total Maturity Amount

Simple Interest
A. Terms of Simple Interest:
  • Principal ($P$): The sum of money borrowed or deposited.
  • Rate of Interest ($R$): The interest charged per $\text{Rs. } 100$ per annum (p.a.).
  • Time ($T$): The duration for which money is borrowed, strictly in years (if given in months, divide by 12; if in days, divide by 365).
  • Simple Interest ($I$): The additional money paid for using the lender's money: $$I = \frac{P \times R \times T}{100}$$
  • Total Amount ($A$): The combined maturity sum returned at the end of the period: $$A = P + I = P + \left( \frac{P \times R \times T}{100} \right) = P \left( 1 + \frac{RT}{100} \right)$$

Key Formulas, Identities & Theorems

Simple Interest Formula
$$I = \frac{P \cdot R \cdot T}{100}$$
Time T must strictly be expressed in years.
Profit and Loss Percentages
$$\text{Profit}\% = \frac{SP - CP}{CP} \times 100, \quad \text{Loss}\% = \frac{CP - SP}{CP} \times 100$$
Always calculated on the Cost Price (CP).

Percentage Foundations, Profit/Loss & Simple Interest Matrix

Percentage: Conversions, Profit/Loss & Simple Interest CONVERSIONS • Percent: per centum (/100) • % → Fraction: Divide by 100   45% = 45/100 = 9/20 • Fraction → %: Multiply by 100   3/8 × 100% = 37.5% • % Change Formula:   (Change / Original) × 100%   *Denominator ALWAYS original! • Original = New × 100 / (100±x) PROFIT & LOSS • CP = Cost Price • SP = Selling Price • Profit (SP > CP):   Profit = SP - CP   Profit % = (Profit / CP) × 100 • Loss (CP > SP):   Loss = CP - SP   Loss % = (Loss / CP) × 100 • Crucial Rule:   Both % ALWAYS computed on CP! SIMPLE INTEREST I = (P × R × T) / 100 • P = Principal sum borrowed • R = Rate % per annum (p.a.) • T = Time strictly in YEARS   (Months ÷ 12 • Days ÷ 365) Amount: A = P + I • P = (100 × I) / (R × T) • Linear interest without compounding PROFIT% & LOSS% ARE ALWAYS CALCULATED ON CP • SIMPLE INTEREST: I = (PRT)/100

Chapter Summary & 10 Key Takeaways

Takeaway 1
Percentage means per hundred ($x\% = \frac{x}{100}$); convert to percentage by multiplying by 100.
Takeaway 2
Percentage increase or decrease is always calculated on the initial original value.
Takeaway 3
If an amount is increased by $x\%$, the original value is $\frac{\text{New Value}}{100 + x} \times 100$.
Takeaway 4
Profit occurs when Selling Price exceeds Cost Price ($SP > CP$); $\text{Profit} = SP - CP$.
Takeaway 5
Loss occurs when Cost Price exceeds Selling Price ($CP > SP$); $\text{Loss} = CP - SP$.
Takeaway 6
Profit% and Loss% are ALWAYS calculated on the Cost Price (CP), never on SP.
Takeaway 7
Selling Price formulas: $SP = CP \times \frac{100 + P\%}{100}$ and $SP = CP \times \frac{100 - L\%}{100}$.
Takeaway 8
Simple Interest: $I = \frac{P \times R \times T}{100}$, where time $T$ must be converted to years.
Takeaway 9
Maturity Amount equals Principal plus Simple Interest: $A = P + I$.
Takeaway 10
To find Principal from Interest: $P = \frac{100 \times I}{R \times T}$.

Check Your Understanding (Diagnostic Practice Questions)

Diagnostic questions testing core conceptual clarity. Answers are hidden initially — solve each problem first, then click to reveal the step-by-step verified solution.

1
A shopkeeper bought a cycle for $\text{Rs. } 3,600$ and spent $\text{Rs. } 400$ on repairs and transportation. He sold it for $\text{Rs. } 4,600$. Find his gain or loss percent.
Reveal Answer & Explanation
Answer: Step 1: Compute total Cost Price including overheads:
$$\text{Total CP} = 3,600 + 400 = \text{Rs. } 4,000$$
Step 2: Selling Price $SP = \text{Rs. } 4,600$.
Since $SP > CP$, there is a Gain (Profit):
$$\text{Profit} = SP - CP = 4,600 - 4,000 = \text{Rs. } 600$$
Step 3: Calculate Gain Percent (on CP):
$$\text{Gain } \% = \frac{\text{Profit}}{\text{Total CP}} \times 100 = \frac{600}{4,000} \times 100 = \frac{60}{4} = \mathbf{15\%}$$.
Add repair costs to purchase price to get total CP ($4,000$), find profit ($600$), and calculate $(600/4000) \times 100$.
2
By selling a wristwatch for $\text{Rs. } 1,440$, a trader incurs a loss of $10\%$. At what price should he sell it to gain $15\%$?
Reveal Answer & Explanation
Answer: Step 1: Find the Cost Price (CP):
$$SP = \text{Rs. } 1,440, \quad \text{Loss } \% = 10\%$$
$$CP = \frac{SP \times 100}{100 - \text{Loss } \%} = \frac{1,440 \times 100}{90} = 16 \times 100 = \text{Rs. } 1,600$$
Step 2: Calculate required SP to gain $15\%$:
$$SP_{\text{new}} = CP \times \left( \frac{100 + 15}{100} \right) = 1,600 \times \frac{115}{100} = 16 \times 115 = \mathbf{\text{Rs. } 1,840}$$.
Calculate original CP first: $1440 \times 100 / 90 = 1,600$. Then find new SP for 15% gain: $1600 \times 1.15 = 1,840$.
3
Calculate the Simple Interest and total Amount on $\text{Rs. } 8,000$ at $7\frac{1}{2}\%$ per annum for $3\text{ years and } 4\text{ months}$.
Reveal Answer & Explanation
Answer: Principal $P = \text{Rs. } 8,000$
Rate $R = 7\frac{1}{2}\% = \frac{15}{2}\% = 7.5\%$
Time $T = 3\text{ years } 4\text{ months} = 3 + \frac{4}{12} = 3 + \frac{1}{3} = \frac{10}{3}\text{ years}$
Step 1: Calculate Simple Interest ($I = \frac{PRT}{100}$):
$$I = \frac{8,000 \times \frac{15}{2} \times \frac{10}{3}}{100} = \frac{8,000 \times 15 \times 10}{100 \times 2 \times 3} = \frac{80 \times 150}{6} = \frac{12,000}{6} = \mathbf{\text{Rs. } 2,000}$$
Step 2: Calculate total Amount ($A = P + I$):
$$A = 8,000 + 2,000 = \mathbf{\text{Rs. } 10,000}$$.
Convert time to years: $3\text{ yrs } 4\text{ mos} = 10/3\text{ years}$. Plug into $I = (PRT)/100$, then add to $P$.
4
The price of sugar rises by $25\%$. By what percent must a household reduce its consumption so that the total expenditure on sugar remains unchanged?
Reveal Answer & Explanation
Answer: Let original price be $\text{Rs. } 100$ per kg and original consumption be $100\text{ kg}$ (Total expenditure $= \text{Rs. } 10,000$).
New price $= 100 + 25 = \text{Rs. } 125$ per kg.
To keep expenditure at $\text{Rs. } 10,000$, new consumption $= \frac{10,000}{125} = 80\text{ kg}$.
$$\text{Reduction in consumption} = 100 - 80 = 20\text{ kg}$$
$$\text{Percentage Reduction} = \frac{20}{100} \times 100 = \mathbf{20\%}$$.
*(Formula: $\frac{R}{100 + R} \times 100\% = \frac{25}{125} \times 100 = 20\%$)*.
Use the formula $\frac{R}{100 + R} \times 100\% = \frac{25}{125} \times 100 = 20\%$.
5
A sum of money doubles itself in $8\text{ years}$ under Simple Interest. What is the rate percent per annum?
Reveal Answer & Explanation
Answer: Let the principal sum be $P$. If the money doubles, Amount $A = 2P$.
$$\text{Simple Interest } I = A - P = 2P - P = P$$
Given Time $T = 8\text{ years}$. Apply $I = \frac{PRT}{100}$:
$$P = \frac{P \times R \times 8}{100}$$
Divide both sides by $P$ ($P \neq 0$):
$$1 = \frac{8R}{100} \implies 8R = 100 \implies R = \frac{100}{8} = \mathbf{12.5\% \text{ p.a.}}$$.
If sum doubles, $I = P$. Then $P = (P \times R \times 8) / 100 \implies R = 100 / 8 = 12.5\%$.
6
In an examination, $35\%$ students failed in Hindi, $45\%$ failed in English, and $20\%$ failed in both subjects. Find the percentage of students who passed in both subjects.
Reveal Answer & Explanation
Answer: Let $H = \text{Failed in Hindi}$ ($35\%$) and $E = \text{Failed in English}$ ($45\%$).
Failed in both $H \cap E = 20\%$.
Step 1: Total percentage of students who failed in at least one subject:
$$\text{Failed in } H \cup E = 35\% + 45\% - 20\% = 80\% - 20\% = 60\%$$
Step 2: Percentage of students who passed in both subjects:
$$\text{Passed in Both} = 100\% - 60\% = \mathbf{40\%}$$.
Calculate total failed using $35 + 45 - 20 = 60\%$. Those who passed both $= 100 - 60 = 40\%$.
7
A man sold two chairs for $\text{Rs. } 990$ each. On one he gained $10\%$, and on the other he lost $10\%$. Find his overall gain or loss percentage on the whole transaction.
Reveal Answer & Explanation
Answer:

• Chair 1 ($SP = 990, \text{Gain } 10\%$):

$$CP_1 = \frac{990 \times 100}{110} = \text{Rs. } 900$$


• Chair 2 ($SP = 990, \text{Loss } 10\%$):

$$CP_2 = \frac{990 \times 100}{90} = \text{Rs. } 1,100$$


• Total Transaction:

$$\text{Total CP} = 900 + 1,100 = \text{Rs. } 2,000$$


$$\text{Total SP} = 990 + 990 = \text{Rs. } 1,980$$


Since $\text{Total CP} > \text{Total SP}$, there is an overall Loss:

$$\text{Loss} = 2,000 - 1,980 = \text{Rs. } 20$$


$$\text{Loss } \% = \frac{20}{2,000} \times 100 = \mathbf{1\% \text{ Loss}}$$

.
(Shortcut rule: When two articles are sold at the same SP with equal gain and loss percent $x$, there is always an overall loss of $\frac{x^2}{100}\% = \frac{10^2}{100} = 1\%$).


Total CP $= 900 + 1100 = 2000$. Total SP $= 1980$. Loss is Rs 20. $(20/2000) \times 100 = 1\%$.
8
After spending $80\%$ of his monthly salary and giving $10\%$ of the remainder to charity, a man has $\text{Rs. } 4,500$ left. Find his monthly salary.
Reveal Answer & Explanation
Answer:

Let the total monthly salary be $S$.
After spending $80\%$, remainder left $= (100 - 80)\% = 20\%$ of $S = 0.20 S$.
He gives $10\%$ of this remainder to charity, so he retains $90\%$ of the remainder:

$$\text{Amount left} = 90\% \text{ of } (0.20 S) = 0.90 \times 0.20 S = 0.18 S$$


We are given that $0.18 S = 4,500$:

$$S = \frac{4,500}{0.18} = \frac{450,000}{18} = \mathbf{\text{Rs. } 25,000}$$

.
His monthly salary is $\text{Rs. } 25,000$.


Remainder is $20\%$ of salary. Retaining $90\%$ of remainder gives $0.90 \times 0.20 = 18\%$ of salary. $18\% = 4,500$.
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