Class 8
Mathematics
Federal Board of Intermediate and Secondary Education (FBISE) (FBISE)
📚 Model Textbook of Mathematics Grade 8 (FBISE / NBF)
Class 8 Mathematics - Ch 4: Financial Arithmetic Mastery Guide (FBISE)
📖 Chapter 4: Financial Arithmetic
📅 Updated: Sep 13, 2026
Teacher & Student Roadmap
Grade 8 Mathematics • FBISE / National Curriculum (NBF)
Instructional Guide: Unit 04 Financial Arithmetic (مالیاتی ریاضیات)
Target Student Learning Outcomes (SLOs)
- Distinguish between Direct Proportion (straight line graph through origin) and Inverse Proportion (smooth hyperbola curve).
- Solve multi-variable real-world problems using Compound Proportion.
- Calculate profit and loss distribution in Compound Business Partnerships involving variable capital and unequal time durations ($Capital \times Time$).
- Compute legal estate shares under Islamic Law of Inheritance (deducting debts, funeral expenses, and bequests $\le 1/3$).
- Convert Pakistani Rupees to international currencies (USD, SAR, EUR, GBP, CNY, JPY) using buying/selling exchange rates.
- Calculate simple profit/markup using $I = \frac{PRT}{100}$ and determine total payable amounts and monthly installments.
- Analyze multi-step Successive Transactions involving Cost Price ($CP$), Sale Price ($SP$), Marked Price ($MP$), Profit/Loss %, and Discount %.
- Compute Life and Vehicle Insurance premiums, policy fees, family income contracts, maturity bonuses, and asset depreciation (10% per annum).
Common Student Misconceptions & Traps
- Proportion Arrow Alignment: In compound proportion, placing arrows incorrectly. Always compare each independent variable directly with the unknown target variable ($x$).
- Partnership Time Weighting: Forgetting to multiply investment by time when partners join or withdraw mid-year ($Investment \times Months$).
- Inheritance Deduction Order: Distributing estate before clearing debts or funeral expenses. The Quranic order is: Funeral Costs $\to$ Debts $\to$ Bequest (max 1/3) $\to$ Fixed Heirs $\to$ Sons/Daughters in 2:1 ratio.
- Profit % vs Discount % Base: Profit and Loss % are always calculated on Cost Price ($CP$), whereas Discount % is always calculated on Marked Price ($MP$).
- Vehicle Depreciation Cycle: In vehicle insurance, the annual premium decreases each year because the insured car value depreciates by 10% annually.
3-Step Concept Mastery Strategy
- Step 1: Classify the Financial Operation: Determine if the problem is proportion, banking markup, commercial trade, inheritance, currency conversion, or insurance.
- Step 2: Establish the Conversion & Equation: Set up proportional fractional products ($\frac{x}{x_0} = \prod \text{factors}$), markup formula ($\frac{PRT}{100}$), or ratio sum partition.
- Step 3: Dimensional & Balance Check: Verify monetary units (PKR, SAR, USD), ensure ratios sum up to total remaining fund, and check net profit/loss balance.
Curriculum Alignment: This unit strictly follows the Single National Curriculum (SNC) / FBISE Grade 8 Mathematics textbook published by the National Book Foundation (NBF). All formulas, exchange tables, Islamic inheritance rules, and 100% solved exercises are fully elaborated with step-by-step scoring breakdown.
Chapter 4: Financial Arithmetic (مالیاتی ریاضیات)
Financial Arithmetic is the mathematics of money, commerce, and real-life resource distribution. From managing commercial trade, calculating banking markups, and exchanging global currencies, to executing equitable business partnerships, fulfilling Islamic inheritance laws, and securing assets through insurance, this guide equips you with lifelong practical financial literacy!
💡 Kid-Friendly Tips for Success
- Direct = Same Way: If more buys more, both arrows point up ($\uparrow \uparrow$). Cross-multiply straight: $\frac{x}{a} = \frac{b}{c}$.
- Inverse = Flip it: If more workers take fewer days, flip the second ratio: $\frac{x}{a} = \frac{c}{b}$.
- Partnership Magic: Effective Capital = $\text{Money} \times \text{Months}$. If duration is same, time cancels out!
- Inheritance 2-to-1 Rule: Every brother receives double of what a sister receives ($2 : 1$).
- Marked vs Sale: $\text{Sale Price} = \text{Marked Price} - \text{Discount}$. Never pay sticker price if there is a sale!
🌍 Real-World Connections
- Hajj & Travel Banking: Exchanging Pakistani Rupees for Saudi Riyals (SAR) or US Dollars (USD) before international travel.
- Car Financing & Installments: Understanding how commercial banks compute markup on auto loans over 3 to 5 years.
- Family Estate Settlement: Applying Islamic Shariah laws to justly divide properties, shops, and land among family heirs.
- Auto & Health Protection: How vehicle insurance covers accidental damage and life insurance provides financial security to families.
Visual Concept 1: Direct Proportion vs. Inverse Proportion Characteristics
Visual Concept 3: Commercial Transactions & Insurance Premium Structure
Section-by-Section Theoretical Mastery
1. Direct, Inverse, and Compound Proportions
A proportion is a statement that two ratios are equivalent: $a : b :: c : d$ or $\frac{a}{b} = \frac{c}{d}$.
Direct Proportion: $\frac{a}{b} = \frac{c}{d} \iff ad = bc$. Both quantities rise or fall together in the exact same ratio. Graph is a straight line through origin $(0,0)$.
Inverse Proportion: $\frac{a}{b} = \frac{d}{c} \iff ac = bd$. One quantity rises while the other decreases proportionally. Graph is a smooth hyperbolic curve.
Compound Proportion: Combines two or more proportions: $\frac{x}{x_0} = \left(\text{factor}_1\right) \times \left(\text{factor}_2\right)$.
2. Compound Business Partnerships
When partners invest different sums of capital for different lengths of time, profit or loss is divided in proportion to the effective monthly capital: $$\text{Effective Capital} = \text{Investment Amount} \times \text{Time Duration (Months)}$$ $$\text{Ratio of Shares} = (C_1 \times T_1) : (C_2 \times T_2) : (C_3 \times T_3)$$ $$\text{Individual Share} = \frac{\text{Individual Ratio Term}}{\text{Sum of Ratio Terms}} \times \text{Total Profit or Loss}$$
3. Islamic Law of Inheritance (قانونِ وراثت)
The estate of a deceased person is distributed strictly in the following mandatory legal sequence:
- Funeral & Burial Costs: Deducted first from gross property.
- Debts (Qarz): All outstanding debts must be settled in full.
- Testament / Wasiyyat: Fulfill any charitable bequest (cannot exceed $\frac{1}{3}$ of remaining property).
- Fixed Quranic Shares:
- Parents (Father and Mother): $\frac{1}{6}$ each when deceased leaves children.
- Widow (Wife): $\frac{1}{8}$ of net estate when children exist; $\frac{1}{4}$ if no children.
- Husband: $\frac{1}{4}$ when deceased wife leaves children; $\frac{1}{2}$ if no children.
- Single Daughter (no sons): $\frac{1}{2}$; Multiple Daughters (no sons): $\frac{2}{3}$ shared equally.
- Residue Distribution: The remaining estate is distributed among sons and daughters in the ratio $2 : 1$ (each son receives twice the share of a daughter).
4. Foreign Currency Conversions
Currencies are exchanged using prevailing bank exchange rates. To convert foreign currency into PKR, multiply by the conversion rate; to convert PKR into foreign currency, divide by the conversion rate: $$\text{Amount in PKR} = \text{Foreign Currency Amount} \times \text{Exchange Rate (PKR per unit)}$$ $$\text{Foreign Currency Amount} = \frac{\text{Amount in PKR}}{\text{Exchange Rate (PKR per unit)}}$$
5. Banking Profit / Markup Formula
For a principal amount $P$ deposited or borrowed at rate $R\%$ per annum for $T$ years: $$\text{Profit / Markup } (I) = \frac{P \times R \times T}{100}$$ $$\text{Total Payable / Received Amount } (A) = P + I$$ $$\text{Monthly Installment} = \frac{\text{Total Amount } (A)}{\text{Total Number of Months } (12 \times T)}$$
6. Successive Commercial Transactions & Discount
Profit & Loss:
$\text{Profit} = SP - CP$
$\text{Loss} = CP - SP$
$\text{Profit } \% = \frac{\text{Profit}}{CP} \times 100$
$\text{Loss } \% = \frac{\text{Loss}}{CP} \times 100$
$\text{Profit} = SP - CP$
$\text{Loss} = CP - SP$
$\text{Profit } \% = \frac{\text{Profit}}{CP} \times 100$
$\text{Loss } \% = \frac{\text{Loss}}{CP} \times 100$
Discount:
$\text{Discount} = MP - SP$
$\text{Discount } \% = \frac{\text{Discount}}{MP} \times 100$
$SP = MP \times \left(1 - \frac{\text{Discount } \%}{100}\right)$
$MP = \frac{SP}{1 - \frac{\text{Discount } \%}{100}}$
$\text{Discount} = MP - SP$
$\text{Discount } \% = \frac{\text{Discount}}{MP} \times 100$
$SP = MP \times \left(1 - \frac{\text{Discount } \%}{100}\right)$
$MP = \frac{SP}{1 - \frac{\text{Discount } \%}{100}}$
7. Insurance (Life & Vehicle Insurance)
Life Insurance: $\text{Annual Premium} = \text{First (Basic) Premium} + \text{Policy Fee} + \text{Family Income Contract Fee}$. At maturity, insurer receives $\text{Policy Amount} + \text{Accumulated Bonuses}$.
Vehicle Insurance: Annual premium is a percentage of the vehicle's depreciated value. Standard annual depreciation is $10\%$ of previous year's value.
Exercise 4.1 • Direct and Inverse Proportion
Q1. A book seller offers a discount of Rs. 45 on purchasing 5 books. Find the discount on 18 books.
Step 1: Set up the proportion table.
Let the discount on 18 books be $\text{Rs. } x$. More books mean more discount (Direct Proportion):
$$\begin{array}{|c|c|} \hline \textbf{Books} & \textbf{Discount (Rs.)} \\ \hline 5 \uparrow & 45 \uparrow \\ 18 & x \\ \hline \end{array}$$ Step 2: Solve the direct proportion equation:
$$\frac{x}{45} = \frac{18}{5}$$ $$x = \frac{18}{5} \times 45 = 18 \times 9 = 162$$ Final Answer: The discount on 18 books is Rs. 162.
Let the discount on 18 books be $\text{Rs. } x$. More books mean more discount (Direct Proportion):
$$\begin{array}{|c|c|} \hline \textbf{Books} & \textbf{Discount (Rs.)} \\ \hline 5 \uparrow & 45 \uparrow \\ 18 & x \\ \hline \end{array}$$ Step 2: Solve the direct proportion equation:
$$\frac{x}{45} = \frac{18}{5}$$ $$x = \frac{18}{5} \times 45 = 18 \times 9 = 162$$ Final Answer: The discount on 18 books is Rs. 162.
Q2. Cost price of banana is Rs. 50 per dozen. Find the cost price of 2.5 dozen bananas.
Step 1: Identify relationship.
Cost of 1 dozen bananas $= \text{Rs. } 50$. More dozens mean more cost (Direct Proportion).
Step 2: Calculate total cost:
$$\text{Cost} = 2.5 \times 50 = \text{Rs. } 125$$ Final Answer: The cost price of 2.5 dozen bananas is Rs. 125.
Cost of 1 dozen bananas $= \text{Rs. } 50$. More dozens mean more cost (Direct Proportion).
Step 2: Calculate total cost:
$$\text{Cost} = 2.5 \times 50 = \text{Rs. } 125$$ Final Answer: The cost price of 2.5 dozen bananas is Rs. 125.
Q3. Cost of 5 kg apples is Rs. 400. Hassan buys 12 kg apples and gives a note of Rs. 1000 to the shopkeeper. How much money will he get back from the shopkeeper?
Step 1: Find cost of 12 kg apples via direct proportion:
$$\text{Cost of 1 kg} = \frac{400}{5} = \text{Rs. } 80$$ $$\text{Cost of 12 kg} = 12 \times 80 = \text{Rs. } 960$$ Step 2: Calculate change returned:
$$\text{Money given} = \text{Rs. } 1000$$ $$\text{Money returned} = 1000 - 960 = \text{Rs. } 40$$ Final Answer: Hassan will get back Rs. 40.
$$\text{Cost of 1 kg} = \frac{400}{5} = \text{Rs. } 80$$ $$\text{Cost of 12 kg} = 12 \times 80 = \text{Rs. } 960$$ Step 2: Calculate change returned:
$$\text{Money given} = \text{Rs. } 1000$$ $$\text{Money returned} = 1000 - 960 = \text{Rs. } 40$$ Final Answer: Hassan will get back Rs. 40.
Q4. If 12 men can do a job in 20 days. How long will 10 men take to complete the same job?
Step 1: Set up inverse proportion table.
Fewer men will take more days (Inverse Proportion):
$$\begin{array}{|c|c|} \hline \textbf{Men} & \textbf{Days} \\ \hline 12 \downarrow & 20 \uparrow \\ 10 & x \\ \hline \end{array}$$ Step 2: Solve the inverse proportion equation:
$$\frac{x}{20} = \frac{12}{10}$$ $$x = \frac{12}{10} \times 20 = 12 \times 2 = 24$$ Final Answer: 10 men will take 24 days.
Fewer men will take more days (Inverse Proportion):
$$\begin{array}{|c|c|} \hline \textbf{Men} & \textbf{Days} \\ \hline 12 \downarrow & 20 \uparrow \\ 10 & x \\ \hline \end{array}$$ Step 2: Solve the inverse proportion equation:
$$\frac{x}{20} = \frac{12}{10}$$ $$x = \frac{12}{10} \times 20 = 12 \times 2 = 24$$ Final Answer: 10 men will take 24 days.
Q5. If 20 people can do a work in 15 days. How many more people are required to complete the work in 10 days?
Step 1: Find total people required for 10 days (Inverse Proportion):
$$\frac{x}{20} = \frac{15}{10}$$ $$x = \frac{15}{10} \times 20 = 30\text{ people}$$ Step 2: Compute additional people required:
$$\text{More people required} = 30 - 20 = 10$$ Final Answer: 10 more people are required.
$$\frac{x}{20} = \frac{15}{10}$$ $$x = \frac{15}{10} \times 20 = 30\text{ people}$$ Step 2: Compute additional people required:
$$\text{More people required} = 30 - 20 = 10$$ Final Answer: 10 more people are required.
Q6. If one dozen eggs cost Rs. 96. Find the cost of 8 eggs.
Step 1: Set up direct proportion (1 dozen = 12 eggs):
$$\frac{x}{96} = \frac{8}{12}$$ $$x = \frac{8}{12} \times 96 = 8 \times 8 = 64$$ Final Answer: The cost of 8 eggs is Rs. 64.
$$\frac{x}{96} = \frac{8}{12}$$ $$x = \frac{8}{12} \times 96 = 8 \times 8 = 64$$ Final Answer: The cost of 8 eggs is Rs. 64.
Q7. 18 people completed a work in 10 days. How many people are required to complete one third of the same work?
Step 1: Understand relationship.
In the same time (10 days), less work requires fewer people (Direct Proportion with work):
$$\text{People required} = 18 \times \frac{1}{3} = 6$$ Final Answer: 6 people are required.
In the same time (10 days), less work requires fewer people (Direct Proportion with work):
$$\text{People required} = 18 \times \frac{1}{3} = 6$$ Final Answer: 6 people are required.
Q8. 4000 bricks cost Rs. 25000. What is the cost of 9000 bricks?
Step 1: Set up direct proportion:
$$\frac{x}{25000} = \frac{9000}{4000}$$ $$x = \frac{9}{4} \times 25000 = 9 \times 6250 = \text{Rs. } 56250$$ Final Answer: The cost of 9000 bricks is Rs. 56,250.
$$\frac{x}{25000} = \frac{9000}{4000}$$ $$x = \frac{9}{4} \times 25000 = 9 \times 6250 = \text{Rs. } 56250$$ Final Answer: The cost of 9000 bricks is Rs. 56,250.
Exercise 4.2 • Compound Proportion, Business Partnership & Inheritance
Q1. A truck driver charges Rs. 2000 for 10 km with a load of 800 kg. What will he charge if truck covers a distance of 8 km with a load of 600 kg?
Step 1: Form compound proportion table.
Let the required charge be $\text{Rs. } x$. Charge is directly proportional to both distance and weight:
$$\begin{array}{|c|c|c|} \hline \textbf{Charge (Rs.)} & \textbf{Distance (km)} & \textbf{Weight (kg)} \\ \hline 2000 \uparrow & 10 \uparrow & 800 \uparrow \\ x & 8 & 600 \\ \hline \end{array}$$ Step 2: Solve compound equation:
$$\frac{x}{2000} = \frac{8}{10} \times \frac{600}{800}$$ $$x = 2000 \times \frac{4}{5} \times \frac{3}{4} = 2000 \times \frac{3}{5} = 1200$$ Final Answer: The driver will charge Rs. 1,200.
Let the required charge be $\text{Rs. } x$. Charge is directly proportional to both distance and weight:
$$\begin{array}{|c|c|c|} \hline \textbf{Charge (Rs.)} & \textbf{Distance (km)} & \textbf{Weight (kg)} \\ \hline 2000 \uparrow & 10 \uparrow & 800 \uparrow \\ x & 8 & 600 \\ \hline \end{array}$$ Step 2: Solve compound equation:
$$\frac{x}{2000} = \frac{8}{10} \times \frac{600}{800}$$ $$x = 2000 \times \frac{4}{5} \times \frac{3}{4} = 2000 \times \frac{3}{5} = 1200$$ Final Answer: The driver will charge Rs. 1,200.
Q2. 10 men can repair 5 machines in 2 days. How many machines are repaired by 8 men in 3 days?
Step 1: Set up compound proportion table.
Machines repaired is directly proportional to both men and days:
$$\frac{x}{5} = \frac{8}{10} \times \frac{3}{2}$$ $$x = 5 \times \frac{4}{5} \times \frac{3}{2} = 6$$ Final Answer: 6 machines are repaired.
Machines repaired is directly proportional to both men and days:
$$\frac{x}{5} = \frac{8}{10} \times \frac{3}{2}$$ $$x = 5 \times \frac{4}{5} \times \frac{3}{2} = 6$$ Final Answer: 6 machines are repaired.
Q3. 12 men can do a work in 8 days. How many men can do three times of that work in 16 days?
Step 1: Form compound proportion table.
Men is inversely proportional to days and directly proportional to work:
$$\frac{x}{12} = \frac{8}{16} \times \frac{3}{1}$$ $$x = 12 \times \frac{1}{2} \times 3 = 18$$ Final Answer: 18 men are required.
Men is inversely proportional to days and directly proportional to work:
$$\frac{x}{12} = \frac{8}{16} \times \frac{3}{1}$$ $$x = 12 \times \frac{1}{2} \times 3 = 18$$ Final Answer: 18 men are required.
Q4. 100 hens eat one fourth quantity of feed in 21 days. In how many days 84 hens will eat remaining feed?
Step 1: Identify quantities.
Remaining feed $= 1 - \frac{1}{4} = \frac{3}{4}$. Days is inversely proportional to hens and directly proportional to feed:
$$\frac{x}{21} = \frac{100}{84} \times \frac{3/4}{1/4} = \frac{100}{84} \times 3 = \frac{300}{84} = \frac{25}{7}$$ $$x = 21 \times \frac{25}{7} = 3 \times 25 = 75$$ Final Answer: 84 hens will eat the remaining feed in 75 days.
Remaining feed $= 1 - \frac{1}{4} = \frac{3}{4}$. Days is inversely proportional to hens and directly proportional to feed:
$$\frac{x}{21} = \frac{100}{84} \times \frac{3/4}{1/4} = \frac{100}{84} \times 3 = \frac{300}{84} = \frac{25}{7}$$ $$x = 21 \times \frac{25}{7} = 3 \times 25 = 75$$ Final Answer: 84 hens will eat the remaining feed in 75 days.
Q5. In a factory 49 workers make 1050 articles in 10 hours. One day 4 workers were absent. In how many hours can remaining workers make 1350 articles?
Step 1: Remaining workers $= 49 - 4 = 45$.
Hours is inversely proportional to workers and directly proportional to articles:
$$\frac{x}{10} = \frac{49}{45} \times \frac{1350}{1050}$$ $$\frac{1350}{1050} = \frac{135}{105} = \frac{9}{7}$$ $$x = 10 \times \frac{49}{45} \times \frac{9}{7} = 10 \times \left(\frac{49}{7}\right) \times \left(\frac{9}{45}\right) = 10 \times 7 \times \frac{1}{5} = 14$$ Final Answer: The remaining workers will take 14 hours.
Hours is inversely proportional to workers and directly proportional to articles:
$$\frac{x}{10} = \frac{49}{45} \times \frac{1350}{1050}$$ $$\frac{1350}{1050} = \frac{135}{105} = \frac{9}{7}$$ $$x = 10 \times \frac{49}{45} \times \frac{9}{7} = 10 \times \left(\frac{49}{7}\right) \times \left(\frac{9}{45}\right) = 10 \times 7 \times \frac{1}{5} = 14$$ Final Answer: The remaining workers will take 14 hours.
Q6. For 28 scouts 168 kg food is sufficient for 6 days. For how many days 144 kg food is sufficient if 8 more scouts join them?
Step 1: Total scouts $= 28 + 8 = 36$.
Days is inversely proportional to scouts and directly proportional to food:
$$\frac{x}{6} = \frac{28}{36} \times \frac{144}{168}$$ $$\frac{28}{36} = \frac{7}{9}, \quad \frac{144}{168} = \frac{6}{7}$$ $$x = 6 \times \frac{7}{9} \times \frac{6}{7} = 6 \times \frac{6}{9} = 4$$ Final Answer: The food will be sufficient for 4 days.
Days is inversely proportional to scouts and directly proportional to food:
$$\frac{x}{6} = \frac{28}{36} \times \frac{144}{168}$$ $$\frac{28}{36} = \frac{7}{9}, \quad \frac{144}{168} = \frac{6}{7}$$ $$x = 6 \times \frac{7}{9} \times \frac{6}{7} = 6 \times \frac{6}{9} = 4$$ Final Answer: The food will be sufficient for 4 days.
Q7. Hassan, Akbar and Asghar started a business by investing Rs. 10000, Rs. 12000 and Rs. 15000 respectively. Asghar took his money back after 10 months. Find the share of each in the profit of Rs. 27600 after one year.
Step 1: Compute weighted investments (Duration in months):
- Hassan: $10000 \times 12 = 120000$
- Akbar: $12000 \times 12 = 144000$
- Asghar: $15000 \times 10 = 150000$
Step 2: Simplify investment ratio:
$$120 : 144 : 150 = 20 : 24 : 25 \quad (\div 6)$$ $$\text{Sum of ratio terms} = 20 + 24 + 25 = 69$$ Step 3: Calculate individual shares from total profit Rs. 27600:
- Hassan's share $= \frac{20}{69} \times 27600 = 20 \times 400 = \text{Rs. } 8000$
- Akbar's share $= \frac{24}{69} \times 27600 = 24 \times 400 = \text{Rs. } 9600$
- Asghar's share $= \frac{25}{69} \times 27600 = 25 \times 400 = \text{Rs. } 10000$
Final Answer: Hassan: Rs. 8,000, Akbar: Rs. 9,600, Asghar: Rs. 10,000.
- Hassan: $10000 \times 12 = 120000$
- Akbar: $12000 \times 12 = 144000$
- Asghar: $15000 \times 10 = 150000$
Step 2: Simplify investment ratio:
$$120 : 144 : 150 = 20 : 24 : 25 \quad (\div 6)$$ $$\text{Sum of ratio terms} = 20 + 24 + 25 = 69$$ Step 3: Calculate individual shares from total profit Rs. 27600:
- Hassan's share $= \frac{20}{69} \times 27600 = 20 \times 400 = \text{Rs. } 8000$
- Akbar's share $= \frac{24}{69} \times 27600 = 24 \times 400 = \text{Rs. } 9600$
- Asghar's share $= \frac{25}{69} \times 27600 = 25 \times 400 = \text{Rs. } 10000$
Final Answer: Hassan: Rs. 8,000, Akbar: Rs. 9,600, Asghar: Rs. 10,000.
Q8. Alia, Maha and Sundas shared Rs. 12000, Rs. 15000 and Rs. 18000 respectively for a business. After 5 months Maha took her money back. At the end of 10 months they suffered a loss of Rs. 18750. Find the share of each in loss.
Step 1: Compute weighted investments:
- Alia (10 months): $12000 \times 10 = 120000$
- Maha (5 months): $15000 \times 5 = 75000$
- Sundas (10 months): $18000 \times 10 = 180000$
Step 2: Simplify ratio:
$$120 : 75 : 180 = 8 : 5 : 12 \quad (\div 15)$$ $$\text{Sum of ratio terms} = 8 + 5 + 12 = 25$$ Step 3: Calculate individual loss shares:
- Alia's loss $= \frac{8}{25} \times 18750 = 8 \times 750 = \text{Rs. } 6000$
- Maha's loss $= \frac{5}{25} \times 18750 = 5 \times 750 = \text{Rs. } 3750$
- Sundas's loss $= \frac{12}{25} \times 18750 = 12 \times 750 = \text{Rs. } 9000$
Final Answer: Alia: Rs. 6,000, Maha: Rs. 3,750, Sundas: Rs. 9,000.
- Alia (10 months): $12000 \times 10 = 120000$
- Maha (5 months): $15000 \times 5 = 75000$
- Sundas (10 months): $18000 \times 10 = 180000$
Step 2: Simplify ratio:
$$120 : 75 : 180 = 8 : 5 : 12 \quad (\div 15)$$ $$\text{Sum of ratio terms} = 8 + 5 + 12 = 25$$ Step 3: Calculate individual loss shares:
- Alia's loss $= \frac{8}{25} \times 18750 = 8 \times 750 = \text{Rs. } 6000$
- Maha's loss $= \frac{5}{25} \times 18750 = 5 \times 750 = \text{Rs. } 3750$
- Sundas's loss $= \frac{12}{25} \times 18750 = 12 \times 750 = \text{Rs. } 9000$
Final Answer: Alia: Rs. 6,000, Maha: Rs. 3,750, Sundas: Rs. 9,000.
Q9. X started the business with the investment Rs. 35000. After three months Y joined him with the investment of Rs. 30000. Then after two months Z also joined them with the investment of Rs. 40000. They closed their business 5 months after joining Z. Find share of each in the profit of Rs. 152000.
Step 1: Calculate duration of each partner's investment:
- Total business duration $= 3 + 2 + 5 = 10\text{ months}$
- X was in business for $10\text{ months}$
- Y was in business for $10 - 3 = 7\text{ months}$
- Z was in business for $5\text{ months}$
Step 2: Weighted investments:
- X: $35000 \times 10 = 350000$
- Y: $30000 \times 7 = 210000$
- Z: $40000 \times 5 = 200000$
$$\text{Ratio} = 35 : 21 : 20, \quad \text{Sum of ratio terms} = 35 + 21 + 20 = 76$$ Step 3: Profit shares ($\text{Total} = \text{Rs. } 152000$):
$$\text{Value of 1 ratio unit} = \frac{152000}{76} = \text{Rs. } 2000$$ - X's share $= 35 \times 2000 = \text{Rs. } 70000$
- Y's share $= 21 \times 2000 = \text{Rs. } 42000$
- Z's share $= 20 \times 2000 = \text{Rs. } 40000$
Final Answer: X: Rs. 70,000, Y: Rs. 42,000, Z: Rs. 40,000.
- Total business duration $= 3 + 2 + 5 = 10\text{ months}$
- X was in business for $10\text{ months}$
- Y was in business for $10 - 3 = 7\text{ months}$
- Z was in business for $5\text{ months}$
Step 2: Weighted investments:
- X: $35000 \times 10 = 350000$
- Y: $30000 \times 7 = 210000$
- Z: $40000 \times 5 = 200000$
$$\text{Ratio} = 35 : 21 : 20, \quad \text{Sum of ratio terms} = 35 + 21 + 20 = 76$$ Step 3: Profit shares ($\text{Total} = \text{Rs. } 152000$):
$$\text{Value of 1 ratio unit} = \frac{152000}{76} = \text{Rs. } 2000$$ - X's share $= 35 \times 2000 = \text{Rs. } 70000$
- Y's share $= 21 \times 2000 = \text{Rs. } 42000$
- Z's share $= 20 \times 2000 = \text{Rs. } 40000$
Final Answer: X: Rs. 70,000, Y: Rs. 42,000, Z: Rs. 40,000.
Q10. Divide a property worth Rs. 300000 among two sons, a daughter and a widow after paying the funeral expenditures of Rs. 10000.
Step 1: Deduct funeral expenditures:
$$\text{Remaining Property} = 300000 - 10000 = \text{Rs. } 290000$$ Step 2: Calculate widow's share ($\frac{1}{8}$):
$$\text{Widow's share} = \frac{1}{8} \times 290000 = \text{Rs. } 36250$$ $$\text{Remaining for children} = 290000 - 36250 = \text{Rs. } 253750$$ Step 3: Distribute among 2 sons and 1 daughter ($2 : 2 : 1$):
$$\text{Sum of ratios} = 2 + 2 + 1 = 5$$ - Each son's share $= \frac{2}{5} \times 253750 = \text{Rs. } 101500$
- Daughter's share $= \frac{1}{5} \times 253750 = \text{Rs. } 50750$
Final Answer: Widow: Rs. 36,250, Each Son: Rs. 101,500, Daughter: Rs. 50,750.
$$\text{Remaining Property} = 300000 - 10000 = \text{Rs. } 290000$$ Step 2: Calculate widow's share ($\frac{1}{8}$):
$$\text{Widow's share} = \frac{1}{8} \times 290000 = \text{Rs. } 36250$$ $$\text{Remaining for children} = 290000 - 36250 = \text{Rs. } 253750$$ Step 3: Distribute among 2 sons and 1 daughter ($2 : 2 : 1$):
$$\text{Sum of ratios} = 2 + 2 + 1 = 5$$ - Each son's share $= \frac{2}{5} \times 253750 = \text{Rs. } 101500$
- Daughter's share $= \frac{1}{5} \times 253750 = \text{Rs. } 50750$
Final Answer: Widow: Rs. 36,250, Each Son: Rs. 101,500, Daughter: Rs. 50,750.
Q11. A man died leaving a house worth Rs. 660000 and a shop worth Rs. 320000. Divide his property among his parents, widow, a son and a daughter after paying a debt of Rs. 20000.
Step 1: Compute total property and clear debt:
$$\text{Total Gross Value} = 660000 + 320000 = \text{Rs. } 980000$$ $$\text{Net Estate after Debt} = 980000 - 20000 = \text{Rs. } 960000$$ Step 2: Fixed shares of parents and widow:
- Father's share $(\frac{1}{6}) = \frac{1}{6} \times 960000 = \text{Rs. } 160000$
- Mother's share $(\frac{1}{6}) = \frac{1}{6} \times 960000 = \text{Rs. } 160000$
- Widow's share $(\frac{1}{8}) = \frac{1}{8} \times 960000 = \text{Rs. } 120000$
$$\text{Total fixed shares} = 160000 + 160000 + 120000 = \text{Rs. } 440000$$ $$\text{Remaining for children} = 960000 - 440000 = \text{Rs. } 520000$$ Step 3: Distribute between 1 son and 1 daughter ($2 : 1$, sum = 3):
- Son's share $= \frac{2}{3} \times 520000 = \text{Rs. } 346666.67$
- Daughter's share $= \frac{1}{3} \times 520000 = \text{Rs. } 173333.33$
Final Answer: Father: Rs. 160,000, Mother: Rs. 160,000, Widow: Rs. 120,000, Son: Rs. 346,666.67, Daughter: Rs. 173,333.33.
$$\text{Total Gross Value} = 660000 + 320000 = \text{Rs. } 980000$$ $$\text{Net Estate after Debt} = 980000 - 20000 = \text{Rs. } 960000$$ Step 2: Fixed shares of parents and widow:
- Father's share $(\frac{1}{6}) = \frac{1}{6} \times 960000 = \text{Rs. } 160000$
- Mother's share $(\frac{1}{6}) = \frac{1}{6} \times 960000 = \text{Rs. } 160000$
- Widow's share $(\frac{1}{8}) = \frac{1}{8} \times 960000 = \text{Rs. } 120000$
$$\text{Total fixed shares} = 160000 + 160000 + 120000 = \text{Rs. } 440000$$ $$\text{Remaining for children} = 960000 - 440000 = \text{Rs. } 520000$$ Step 3: Distribute between 1 son and 1 daughter ($2 : 1$, sum = 3):
- Son's share $= \frac{2}{3} \times 520000 = \text{Rs. } 346666.67$
- Daughter's share $= \frac{1}{3} \times 520000 = \text{Rs. } 173333.33$
Final Answer: Father: Rs. 160,000, Mother: Rs. 160,000, Widow: Rs. 120,000, Son: Rs. 346,666.67, Daughter: Rs. 173,333.33.
Q12. Mr. Shan died last Sunday. He had 10 Marla land @ Rs. 30000 per Marla and other property worth Rs. 800000. Divide this property among his widow and a son.
Step 1: Compute total property value:
$$\text{Land value} = 10 \times 30000 = \text{Rs. } 300000$$ $$\text{Total estate} = 300000 + 800000 = \text{Rs. } 1100000$$ Step 2: Distribute shares:
- Widow's share $(\frac{1}{8}) = \frac{1}{8} \times 1100000 = \text{Rs. } 137500$
- Son's share (remaining) $= 1100000 - 137500 = \text{Rs. } 962500$
Final Answer: Widow: Rs. 137,500, Son: Rs. 962,500.
$$\text{Land value} = 10 \times 30000 = \text{Rs. } 300000$$ $$\text{Total estate} = 300000 + 800000 = \text{Rs. } 1100000$$ Step 2: Distribute shares:
- Widow's share $(\frac{1}{8}) = \frac{1}{8} \times 1100000 = \text{Rs. } 137500$
- Son's share (remaining) $= 1100000 - 137500 = \text{Rs. } 962500$
Final Answer: Widow: Rs. 137,500, Son: Rs. 962,500.
Q13. A person died leaving a property worth Rs. 500000. Divide his property among his widow, a daughter and two sons.
Step 1: Widow's share ($\frac{1}{8}$):
$$\text{Widow's share} = \frac{1}{8} \times 500000 = \text{Rs. } 62500$$ $$\text{Remaining for children} = 500000 - 62500 = \text{Rs. } 437500$$ Step 2: Distribute among 2 sons and 1 daughter ($2 : 2 : 1$, sum = 5):
- Each son's share $= \frac{2}{5} \times 437500 = \text{Rs. } 175000$
- Daughter's share $= \frac{1}{5} \times 437500 = \text{Rs. } 87500$
Final Answer: Widow: Rs. 62,500, Each Son: Rs. 175,000, Daughter: Rs. 87,500.
$$\text{Widow's share} = \frac{1}{8} \times 500000 = \text{Rs. } 62500$$ $$\text{Remaining for children} = 500000 - 62500 = \text{Rs. } 437500$$ Step 2: Distribute among 2 sons and 1 daughter ($2 : 2 : 1$, sum = 5):
- Each son's share $= \frac{2}{5} \times 437500 = \text{Rs. } 175000$
- Daughter's share $= \frac{1}{5} \times 437500 = \text{Rs. } 87500$
Final Answer: Widow: Rs. 62,500, Each Son: Rs. 175,000, Daughter: Rs. 87,500.
Q14. Ms. Sajida wants to distribute her property worth Rs. 850000 among her husband, two sons and two daughters in her life. (Share of husband is one fourth of property.)
Step 1: Calculate husband's share ($\frac{1}{4}$):
$$\text{Husband's share} = \frac{1}{4} \times 850000 = \text{Rs. } 212500$$ $$\text{Remaining property} = 850000 - 212500 = \text{Rs. } 637500$$ Step 2: Distribute among 2 sons and 2 daughters ($2 : 2 : 1 : 1$, sum = 6):
- Each son's share $= \frac{2}{6} \times 637500 = \frac{1}{3} \times 637500 = \text{Rs. } 212500$
- Each daughter's share $= \frac{1}{6} \times 637500 = \text{Rs. } 106250$
Final Answer: Husband: Rs. 212,500, Each Son: Rs. 212,500, Each Daughter: Rs. 106,250.
$$\text{Husband's share} = \frac{1}{4} \times 850000 = \text{Rs. } 212500$$ $$\text{Remaining property} = 850000 - 212500 = \text{Rs. } 637500$$ Step 2: Distribute among 2 sons and 2 daughters ($2 : 2 : 1 : 1$, sum = 6):
- Each son's share $= \frac{2}{6} \times 637500 = \frac{1}{3} \times 637500 = \text{Rs. } 212500$
- Each daughter's share $= \frac{1}{6} \times 637500 = \text{Rs. } 106250$
Final Answer: Husband: Rs. 212,500, Each Son: Rs. 212,500, Each Daughter: Rs. 106,250.
Exercise 4.3 • Exchange of Currencies
Using Textbook Table Exchange Rates (Page 56): $1\text{ USD} = \text{Rs. } 155$, $1\text{ SAR} = \text{Rs. } 42$, $1\text{ EUR} = \text{Rs. } 185$, $1\text{ GBP} = \text{Rs. } 215$, $1\text{ CNY (Yuan)} = \text{Rs. } 24$, $1\text{ JPY} = \text{Rs. } 1.5$.
Q1. Convert the following foreign currencies into Rupees:
(a) \$ 26:
$$\text{Amount in PKR} = 26 \times 155 = \textbf{Rs. 4,030}$$ (b) 352 Riyal:
$$\text{Amount in PKR} = 352 \times 42 = \textbf{Rs. 14,784}$$ (c) 199 Euro:
$$\text{Amount in PKR} = 199 \times 185 = \textbf{Rs. 36,815}$$
$$\text{Amount in PKR} = 26 \times 155 = \textbf{Rs. 4,030}$$ (b) 352 Riyal:
$$\text{Amount in PKR} = 352 \times 42 = \textbf{Rs. 14,784}$$ (c) 199 Euro:
$$\text{Amount in PKR} = 199 \times 185 = \textbf{Rs. 36,815}$$
Q2. Convert the following to Japanese Yen ($1\text{ Yen} = \text{Rs. } 1.5$):
(a) \$ 88:
$$\text{PKR} = 88 \times 155 = 13640 \implies \text{Yen} = \frac{13640}{1.5} = \textbf{9,093.33 Yen}$$ (b) 500 Euro:
$$\text{PKR} = 500 \times 185 = 92500 \implies \text{Yen} = \frac{92500}{1.5} = \textbf{61,666.67 Yen}$$ (c) Rs. 666:
$$\text{Yen} = \frac{666}{1.5} = \textbf{444 Yen}$$
$$\text{PKR} = 88 \times 155 = 13640 \implies \text{Yen} = \frac{13640}{1.5} = \textbf{9,093.33 Yen}$$ (b) 500 Euro:
$$\text{PKR} = 500 \times 185 = 92500 \implies \text{Yen} = \frac{92500}{1.5} = \textbf{61,666.67 Yen}$$ (c) Rs. 666:
$$\text{Yen} = \frac{666}{1.5} = \textbf{444 Yen}$$
Q3. Convert the following into U.S. Dollars ($1\text{ USD} = \text{Rs. } 155$):
(a) Rs. 2000000:
$$\text{USD} = \frac{2000000}{155} = \textbf{\$ 12,903.23}$$ (b) 3000 Euro:
$$\text{PKR} = 3000 \times 185 = 555000 \implies \text{USD} = \frac{555000}{155} = \textbf{\$ 3,580.65}$$ (c) 4200 Riyal:
$$\text{PKR} = 4200 \times 42 = 176400 \implies \text{USD} = \frac{176400}{155} = \textbf{\$ 1,138.06}$$
$$\text{USD} = \frac{2000000}{155} = \textbf{\$ 12,903.23}$$ (b) 3000 Euro:
$$\text{PKR} = 3000 \times 185 = 555000 \implies \text{USD} = \frac{555000}{155} = \textbf{\$ 3,580.65}$$ (c) 4200 Riyal:
$$\text{PKR} = 4200 \times 42 = 176400 \implies \text{USD} = \frac{176400}{155} = \textbf{\$ 1,138.06}$$
Q4. Amir brought 4500 Dollars with him. How much will this amount be in Rupees ($1\text{ USD} = \text{Rs. } 155$)?
$$\text{Amount in PKR} = 4500 \times 155 = \text{Rs. } 697500$$
Final Answer: The amount in Rupees is Rs. 697,500.
Q5. The exchange rate between the Pound and Yen is $1 : 140$.
(a) 610 Pounds in Yen:
$$\text{Yens} = 610 \times 140 = \textbf{85,400 Yen}$$ (b) 9240 Yens in Pounds:
$$\text{Pounds} = \frac{9240}{140} = \textbf{66 Pounds (£ 66)}$$
$$\text{Yens} = 610 \times 140 = \textbf{85,400 Yen}$$ (b) 9240 Yens in Pounds:
$$\text{Pounds} = \frac{9240}{140} = \textbf{66 Pounds (£ 66)}$$
Q6. Convert the items bill from Saudi Arabia into Rupees ($1\text{ Riyal} = \text{Rs. } 40$):
- Rice (10 kg): $35 \times 40 = \text{Rs. } 1400$
- Sugar (5 kg): $26 \times 40 = \text{Rs. } 1040$
- Oil tin (5 kg): $30 \times 40 = \text{Rs. } 1200$
- Shampoo (2.5 L): $12.5 \times 40 = \text{Rs. } 500$
$$\text{Total Bill} = 103.5 \times 40 = \text{Rs. } 4140$$ Final Answer: The total amount is Rs. 4,140.
- Sugar (5 kg): $26 \times 40 = \text{Rs. } 1040$
- Oil tin (5 kg): $30 \times 40 = \text{Rs. } 1200$
- Shampoo (2.5 L): $12.5 \times 40 = \text{Rs. } 500$
$$\text{Total Bill} = 103.5 \times 40 = \text{Rs. } 4140$$ Final Answer: The total amount is Rs. 4,140.
Q7. Mr. Jaffar sends Rs. 54500 along with Rs. 700 bank charges every month to China ($1\text{ Yuan} = \text{Rs. } 24$).
(i) Convert Rs. 54500 into Yuan:
$$\text{Yuan} = \frac{54500}{24} = \textbf{2,270.83 Yuan}$$ (ii) Total spent in 6 months:
$$\text{Monthly total} = 54500 + 700 = \text{Rs. } 55200$$ $$\text{Spent in 6 months} = 55200 \times 6 = \textbf{Rs. 331,200}$$
$$\text{Yuan} = \frac{54500}{24} = \textbf{2,270.83 Yuan}$$ (ii) Total spent in 6 months:
$$\text{Monthly total} = 54500 + 700 = \text{Rs. } 55200$$ $$\text{Spent in 6 months} = 55200 \times 6 = \textbf{Rs. 331,200}$$
Q8. Aawish has 1560 euros and 2250 dollars in an account offering 5% annual profit ($1\text{ USD} = \text{Rs. } 155$, $1\text{ EUR} = \text{Rs. } 185$).
(i) Balance after 1 year (5% profit):
- Euros $= 1560 \times 1.05 = \textbf{1,638 Euros}$
- Dollars $= 2250 \times 1.05 = \textbf{2,362.50 Dollars}$
(ii) Convert whole amount into Rupees:
$$\text{Rupees} = (1638 \times 185) + (2362.50 \times 155) = 303030 + 366187.50 = \textbf{Rs. 669,217.50}$$
- Euros $= 1560 \times 1.05 = \textbf{1,638 Euros}$
- Dollars $= 2250 \times 1.05 = \textbf{2,362.50 Dollars}$
(ii) Convert whole amount into Rupees:
$$\text{Rupees} = (1638 \times 185) + (2362.50 \times 155) = 303030 + 366187.50 = \textbf{Rs. 669,217.50}$$
Exercise 4.4 • Profit / Markup & Commercial Banking
Q1. Complete the table using $I = \frac{PRT}{100}$ and $\text{Amount} = P + I$:
$$\begin{array}{|c|c|c|c|c|c|}
\hline
\textbf{S.No} & \textbf{Principal } (P) & \textbf{Rate } (R) & \textbf{Time } (T) & \textbf{Profit/Markup } (I) & \textbf{Amount } (A) \\
\hline
\text{i} & \text{Rs. } 2000 & 5\% & 5\text{ yrs} & \mathbf{Rs.\ 500} & \mathbf{Rs.\ 2,500} \\
\text{ii} & \text{Rs. } 15000 & 10\% & \mathbf{2\text{ yrs}} & \text{Rs. } 3000 & \mathbf{Rs.\ 18,000} \\
\text{iii} & \text{Rs. } 25000 & \mathbf{8\%} & 4\text{ yrs} & \text{Rs. } 8000 & \mathbf{Rs.\ 33,000} \\
\text{iv} & \mathbf{Rs.\ 40,000} & 8\% & 3\text{ yrs} & \text{Rs. } 9600 & \mathbf{Rs.\ 49,600} \\
\text{v} & \mathbf{Rs.\ 5,000} & 6\% & 2\text{ yrs} & \text{Rs. } 600 & \mathbf{Rs.\ 5,600} \\
\hline
\end{array}$$
Q2. Find the profit on Rs. 10000 for 30 months @ 12% per annum.
$$T = \frac{30}{12} = 2.5\text{ years}, \quad P = 10000, \quad R = 12\%$$
$$I = \frac{10000 \times 12 \times 2.5}{100} = 100 \times 30 = \text{Rs. } 3000$$
Final Answer: The profit is Rs. 3,000.
Q3. Urooj deposited Rs. 55000 and received Rs. 64900 after 3 years. Find the rate of profit.
$$I = 64900 - 55000 = \text{Rs. } 9900$$
$$R = \frac{I \times 100}{P \times T} = \frac{9900 \times 100}{55000 \times 3} = \frac{990000}{165000} = 6\%$$
Final Answer: The rate of profit is 6% per annum.
Q4. Abdullah borrowed Rs. 32000 @ 5% for 3 years. Find markup and total amount paid.
$$\text{Markup} = \frac{32000 \times 5 \times 3}{100} = 320 \times 15 = \text{Rs. } 4800$$
$$\text{Total Amount} = 32000 + 4800 = \text{Rs. } 36800$$
Final Answer: Markup: Rs. 4,800, Total Amount: Rs. 36,800.
Q5. Mrs. Aftab got an amount worth Rs. 50000 with a markup of Rs. 10000 for 5 years. Find the rate of markup.
Taking principal $P = \text{Rs. } 50000$:
$$R = \frac{10000 \times 100}{50000 \times 5} = \frac{1000000}{250000} = 4\%$$
*(If total amount received was 50000, $P = 40000$, then $R = \frac{10000 \times 100}{40000 \times 5} = 5\%$).*
Final Answer: The rate of markup is 4% per annum (or 5% on principal 40,000).
Final Answer: The rate of markup is 4% per annum (or 5% on principal 40,000).
Q6. Ayan got a profit of Rs. 30000 @ 10% for 3 years from a bank. Find the amount deposited.
$$P = \frac{I \times 100}{R \times T} = \frac{30000 \times 100}{10 \times 3} = \frac{3000000}{30} = \text{Rs. } 100000$$
Final Answer: The deposited principal amount is Rs. 100,000.
Q7. Urooba deposits Rs. 90000 @ 4.5%. After how many years will she get an amount of Rs. 118350?
$$I = 118350 - 90000 = \text{Rs. } 28350$$
$$T = \frac{28350 \times 100}{90000 \times 4.5} = \frac{2835000}{405000} = 7\text{ years}$$
Final Answer: She will get this amount after 7 years.
Q8. Fatima invested Rs. 600000 @ 9% for 3 years and Rs. 400000 @ 8% for 4 years. Find total profit.
$$I_1 = \frac{600000 \times 9 \times 3}{100} = \text{Rs. } 162000$$
$$I_2 = \frac{400000 \times 8 \times 4}{100} = \text{Rs. } 128000$$
$$\text{Total Profit} = 162000 + 128000 = \text{Rs. } 290000$$
Final Answer: Total profit is Rs. 290,000.
Q9. Mr. Shafi Ullah bought a car of price Rs. 900000 and paid Rs. 1089000 in 3 years. Find (i) markup, (ii) rate of markup, (iii) monthly installment.
(i) Markup:
$$\text{Markup} = 1089000 - 900000 = \textbf{Rs. 189,000}$$ (ii) Rate of markup:
$$R = \frac{189000 \times 100}{900000 \times 3} = \frac{18900000}{2700000} = \textbf{7\% per annum}$$ (iii) Monthly installment:
$$\text{Installment} = \frac{1089000}{3 \times 12} = \frac{1089000}{36} = \textbf{Rs. 30,250}$$
$$\text{Markup} = 1089000 - 900000 = \textbf{Rs. 189,000}$$ (ii) Rate of markup:
$$R = \frac{189000 \times 100}{900000 \times 3} = \frac{18900000}{2700000} = \textbf{7\% per annum}$$ (iii) Monthly installment:
$$\text{Installment} = \frac{1089000}{3 \times 12} = \frac{1089000}{36} = \textbf{Rs. 30,250}$$
Q10. A man borrows Rs. 25000 @ 12% per annum for 5 years and agrees to repay Rs. 5500 annually. How much loan is outstanding at the end of 3 years?
Step 1: Compute 3 years total liability:
$$\text{Markup for 3 years} = \frac{25000 \times 12 \times 3}{100} = \text{Rs. } 9000$$ $$\text{Total liability after 3 years} = 25000 + 9000 = \text{Rs. } 34000$$ Step 2: Deduct repayments made in 3 years:
$$\text{Total repaid} = 3 \times 5500 = \text{Rs. } 16500$$ $$\text{Outstanding Loan} = 34000 - 16500 = \text{Rs. } 17500$$ Final Answer: Outstanding loan is Rs. 17,500.
$$\text{Markup for 3 years} = \frac{25000 \times 12 \times 3}{100} = \text{Rs. } 9000$$ $$\text{Total liability after 3 years} = 25000 + 9000 = \text{Rs. } 34000$$ Step 2: Deduct repayments made in 3 years:
$$\text{Total repaid} = 3 \times 5500 = \text{Rs. } 16500$$ $$\text{Outstanding Loan} = 34000 - 16500 = \text{Rs. } 17500$$ Final Answer: Outstanding loan is Rs. 17,500.
Exercise 4.5 • Successive Transactions, Profit, Loss & Discount
Q1. Anwar bought a shop for Rs. 350000. He sold it to Akmal at a profit of 8%. Then, Akmal sold the shop at a profit of 10%. Find total profit.
- Anwar's profit $= 350000 \times \frac{8}{100} = \text{Rs. } 28000$
- Akmal's purchase price $= 350000 + 28000 = \text{Rs. } 378000$
- Akmal's profit $= 378000 \times \frac{10}{100} = \text{Rs. } 37800$
$$\text{Total Profit} = 28000 + 37800 = \text{Rs. } 65800$$ Final Answer: Total profit is Rs. 65,800.
- Akmal's purchase price $= 350000 + 28000 = \text{Rs. } 378000$
- Akmal's profit $= 378000 \times \frac{10}{100} = \text{Rs. } 37800$
$$\text{Total Profit} = 28000 + 37800 = \text{Rs. } 65800$$ Final Answer: Total profit is Rs. 65,800.
Q2. X bought an article for Rs. 800. He sold it to Y at a profit of 10%. After three months Y sold that article at a loss of 10%. Find net profit or loss.
- Sale price to Y $= 800 \times 1.10 = \text{Rs. } 880$
- Sale price by Y $= 880 \times 0.90 = \text{Rs. } 792$
$$\text{Net Difference} = 792 - 800 = -\text{Rs. } 8$$ Final Answer: There is a Net Loss of Rs. 8.
- Sale price by Y $= 880 \times 0.90 = \text{Rs. } 792$
$$\text{Net Difference} = 792 - 800 = -\text{Rs. } 8$$ Final Answer: There is a Net Loss of Rs. 8.
Q3. Ali bought goods for Rs. 1000 and sold them to Zain at a profit of Rs. 100. Afterwards Zain sold these goods and got the same profit. Which is better transaction?
- Ali's profit rate $= \frac{100}{1000} \times 100 = 10\%$
- Zain's cost price $= \text{Rs. } 1100 \implies \text{Zain's profit rate} = \frac{100}{1100} \times 100 = 9.09\%$
Final Answer: Ali's transaction is better because his profit percentage ($10\%$) is higher than Zain's ($9.09\%$).
- Zain's cost price $= \text{Rs. } 1100 \implies \text{Zain's profit rate} = \frac{100}{1100} \times 100 = 9.09\%$
Final Answer: Ali's transaction is better because his profit percentage ($10\%$) is higher than Zain's ($9.09\%$).
Q4. The printed price of a 1.5 litre cold drink bottle is Rs. 80. Shopkeeper bought 3 dozen bottles with the discount of Rs. 5 at each bottle. Calculate the discount % for 3 dozen bottles.
$$\text{Discount } \% = \frac{\text{Discount per bottle}}{\text{Marked price per bottle}} \times 100 = \frac{5}{80} \times 100 = 6.25\%$$
Final Answer: The discount percentage is 6.25%.
Q5. A total printed price of 5 notebooks is Rs. 500 and a shopkeeper gives a discount of Rs. 50 to the customer. Calculate the discount % and sale price of one notebook.
$$\text{Discount } \% = \frac{50}{500} \times 100 = \textbf{10\%}$$
$$\text{Total Sale Price} = 500 - 50 = \text{Rs. } 450 \implies \text{Sale Price per book} = \frac{450}{5} = \textbf{Rs. 90}$$
Q6. Supermarket gives a 10% discount during a sale. A man buys a dinner set for Rs. 36000 and sold it at a profit of 8%. What is marked price and final sale price?
$$\text{Marked Price} = \frac{36000}{1 - 0.10} = \frac{36000}{0.90} = \textbf{Rs. 40,000}$$
$$\text{Final Sale Price} = 36000 \times 1.08 = \textbf{Rs. 38,880}$$
Q7. Marked price of a bag of flour is Rs. 1200. A shopkeeper sold it to a person at a profit of 5%. The man sold the bag with a discount of 2%. Find amount got by the person.
$$\text{Cost to person} = 1200 \times 1.05 = \text{Rs. } 1260$$
$$\text{Amount received by person} = 1260 \times (1 - 0.02) = 1260 \times 0.98 = \text{Rs. } 1234.80$$
Final Answer: The person received Rs. 1,234.80.
Q8. Sara bought 5 kg potatoes for Rs. 390. The shopkeeper got a profit of 4%. Earlier she bought from a wholesaler who also got 4% profit. Find cost price of wholesaler.
$$\text{Cost to shopkeeper} = \frac{390}{1.04} = \text{Rs. } 375$$
$$\text{Cost to wholesaler} = \frac{375}{1.04} = \text{Rs. } 360.58$$
Final Answer: The wholesaler's cost price was Rs. 360.58.
Q9. Hameed offers a 10% discount on goods and a further 5% on reduced price for purchases $\ge$ Rs. 5000. What will a customer pay on a purchase of Rs. 6000?
$$\text{First reduced price} = 6000 \times (1 - 0.10) = \text{Rs. } 5400$$
$$\text{Final payable amount} = 5400 \times (1 - 0.05) = 5400 \times 0.95 = \text{Rs. } 5130$$
Final Answer: The customer will pay Rs. 5,130.
Exercise 4.6 • Life and Vehicle Insurance
Q1. Calculate annual premium for a policy of Rs. 200000 for 20 years if first premium is 4.5% and policy fee is Rs. 350.
$$\text{First premium} = 200000 \times \frac{4.5}{100} = \text{Rs. } 9000$$
$$\text{Annual premium} = 9000 + 350 = \text{Rs. } 9350$$
Final Answer: Annual premium is Rs. 9,350.
Q2. Calculate total amount paid for a policy of Rs. 100000 for 10 years if first premium is 10% and policy fee is Rs. 600.
$$\text{Annual premium} = \left(100000 \times \frac{10}{100}\right) + 600 = 10000 + 600 = \text{Rs. } 10600$$
$$\text{Total paid in 10 years} = 10600 \times 10 = \text{Rs. } 106000$$
Final Answer: Total amount paid is Rs. 106,000.
Q3. Policy worth Rs. 50000 with policy fee Rs. 200, first premium @ 4.8%. Calculate half-yearly (@ 52%), quarterly (@ 27%), and monthly (@ 9%) premium.
$$\text{Annual premium} = \left(50000 \times \frac{4.8}{100}\right) + 200 = 2400 + 200 = \text{Rs. } 2600$$
- Half-yearly premium $= 2600 \times 0.52 = \textbf{Rs. 1,352}$
- Quarterly premium $= 2600 \times 0.27 = \textbf{Rs. 702}$
- Monthly premium $= 2600 \times 0.09 = \textbf{Rs. 234}$
- Quarterly premium $= 2600 \times 0.27 = \textbf{Rs. 702}$
- Monthly premium $= 2600 \times 0.09 = \textbf{Rs. 234}$
Q4. Policy of Rs. 150000 for 25 years. Calculate maturity amount if bonus = 0.045 of policy amount per year, family income bonus = Rs. 100000, and maturity bonus = 2.5% of policy amount per year.
- Policy amount $= \text{Rs. } 150000$
- Regular Bonus $= 150000 \times 0.045 \times 25 = \text{Rs. } 168750$
- Family Income Bonus $= \text{Rs. } 100000$
- Maturity Bonus $= 150000 \times 0.025 \times 25 = \text{Rs. } 93750$
$$\text{Total Amount} = 150000 + 168750 + 100000 + 93750 = \text{Rs. } 512500$$ Final Answer: Amount received is Rs. 512,500.
- Regular Bonus $= 150000 \times 0.045 \times 25 = \text{Rs. } 168750$
- Family Income Bonus $= \text{Rs. } 100000$
- Maturity Bonus $= 150000 \times 0.025 \times 25 = \text{Rs. } 93750$
$$\text{Total Amount} = 150000 + 168750 + 100000 + 93750 = \text{Rs. } 512500$$ Final Answer: Amount received is Rs. 512,500.
Q5. Policy of Rs. 100000 for 20 years. Insurer died after 2 years. Calculate amount given to family if bonus = Rs. 6000/yr for rest period, and additional amount = Rs. 5000/yr for rest period + 0.5 of policy amount.
$$\text{Rest Period} = 20 - 2 = 18\text{ years}$$
- Policy sum assured $= \text{Rs. } 100000$
- Bonus for 18 years $= 6000 \times 18 = \text{Rs. } 108000$
- Additional amount $= (5000 \times 18) + (0.5 \times 100000) = 90000 + 50000 = \text{Rs. } 140000$
$$\text{Total given to family} = 100000 + 108000 + 140000 = \text{Rs. } 348000$$ Final Answer: Total amount given to family is Rs. 348,000.
- Bonus for 18 years $= 6000 \times 18 = \text{Rs. } 108000$
- Additional amount $= (5000 \times 18) + (0.5 \times 100000) = 90000 + 50000 = \text{Rs. } 140000$
$$\text{Total given to family} = 100000 + 108000 + 140000 = \text{Rs. } 348000$$ Final Answer: Total amount given to family is Rs. 348,000.
Q6. Annual premium is Rs. 7750 @ 5% with policy fee Rs. 250. Find the amount of policy.
$$\text{First premium} = 7750 - 250 = \text{Rs. } 7500$$
$$\text{Policy Amount} = \frac{7500}{5\%} = \frac{7500 \times 100}{5} = \text{Rs. } 150000$$
Final Answer: The policy amount is Rs. 150,000.
Q7. Insurance policy for car worth Rs. 1200000 @ 5% for 2 years with 10% annual depreciation. Find total premium paid.
- 1st year premium $= 1200000 \times 0.05 = \text{Rs. } 60000$
- 2nd year car value $= 1200000 \times 0.90 = \text{Rs. } 1080000$
- 2nd year premium $= 1080000 \times 0.05 = \text{Rs. } 54000$
$$\text{Total Premium} = 60000 + 54000 = \text{Rs. } 114000$$ Final Answer: Total premium paid is Rs. 114,000.
- 2nd year car value $= 1200000 \times 0.90 = \text{Rs. } 1080000$
- 2nd year premium $= 1080000 \times 0.05 = \text{Rs. } 54000$
$$\text{Total Premium} = 60000 + 54000 = \text{Rs. } 114000$$ Final Answer: Total premium paid is Rs. 114,000.
Q8. Akbar insured a car worth Rs. 500000 @ 3.5% for 5 years with 10% depreciation. After 2 years claimed Rs. 50000. How much loss was recovered?
- 1st year premium $= 500000 \times 0.035 = \text{Rs. } 17500$
- 2nd year value $= 500000 \times 0.90 = 450000 \implies \text{2nd premium} = 450000 \times 0.035 = \text{Rs. } 15750$
- Total premium paid in 2 years $= 17500 + 15750 = \text{Rs. } 33250$
$$\text{Net Loss Recovered / Benefit} = 50000 - 33250 = \text{Rs. } 16750$$ Final Answer: Net loss recovered is Rs. 16,750.
- 2nd year value $= 500000 \times 0.90 = 450000 \implies \text{2nd premium} = 450000 \times 0.035 = \text{Rs. } 15750$
- Total premium paid in 2 years $= 17500 + 15750 = \text{Rs. } 33250$
$$\text{Net Loss Recovered / Benefit} = 50000 - 33250 = \text{Rs. } 16750$$ Final Answer: Net loss recovered is Rs. 16,750.
Q9. Asma insured car worth Rs. 300000 @ 4.5% for 6 years (10% depreciation). After 2 years claimed Rs. 60000. How much benefit did she get?
- 1st year premium $= 300000 \times 0.045 = \text{Rs. } 13500$
- 2nd year car value $= 270000 \implies \text{2nd premium} = 270000 \times 0.045 = \text{Rs. } 12150$
- Total premium paid $= 13500 + 12150 = \text{Rs. } 25650$
$$\text{Benefit} = 60000 - 25650 = \text{Rs. } 34350$$ Final Answer: She got a benefit of Rs. 34,350.
- 2nd year car value $= 270000 \implies \text{2nd premium} = 270000 \times 0.045 = \text{Rs. } 12150$
- Total premium paid $= 13500 + 12150 = \text{Rs. } 25650$
$$\text{Benefit} = 60000 - 25650 = \text{Rs. } 34350$$ Final Answer: She got a benefit of Rs. 34,350.
Q10. A company insured a truck @ 4% for 10 years and paid Rs. 50000 as first premium. Find policy amount and second premium (depreciation 10%).
$$\text{Policy Amount} = \frac{50000}{0.04} = \textbf{Rs. 1,250,000}$$
$$\text{2nd year value} = 1250000 \times 0.90 = \text{Rs. } 1125000$$
$$\text{2nd premium} = 1125000 \times 0.04 = \textbf{Rs. 45,000}$$
Review Exercise 4 • Comprehensive Mastery Assessment
Q1. Encircle the correct answer:
(i) If 1 Saudi Riyal is of Rs. 40, how many SAR can be purchased for Rs. 1600?
→ (c) SAR. 40 $(1600 \div 40 = 40)$
(ii) Rashida deposited Rs. 15000 and got profit of Rs. 900 after 1 year. Rate of profit?
→ (b) 6% $(\frac{900}{15000} \times 100 = 6\%)$
(iii) What will be the markup for principal of Rs. 100 @ 10% in 10 years?
→ (a) Rs. 100 $(\frac{100 \times 10 \times 10}{100} = 100)$
(iv) Usually rate of depreciation for vehicle is:
→ (d) 10%
(v) Share of father and mother is:
→ (a) $\frac{1}{6}$ of the property
(vi) If a deceased man has only one daughter, she will get:
→ (b) $\frac{1}{2}$ of the property
(vii) TV set costing Rs. 25000 purchased for Rs. 24500. Rate of discount?
→ (c) 2% $(\frac{500}{25000} \times 100 = 2\%)$
(viii) Time period agreed by both parties in insurance policy is called:
→ (b) maturity period
(ix) Cost of 1 dozen eggs is Rs. 180. Cost of 3 eggs?
→ (c) Rs. 45 $(\frac{180}{12} \times 3 = 45)$
(x) Marked price − sale price =
→ (a) discount
→ (c) SAR. 40 $(1600 \div 40 = 40)$
(ii) Rashida deposited Rs. 15000 and got profit of Rs. 900 after 1 year. Rate of profit?
→ (b) 6% $(\frac{900}{15000} \times 100 = 6\%)$
(iii) What will be the markup for principal of Rs. 100 @ 10% in 10 years?
→ (a) Rs. 100 $(\frac{100 \times 10 \times 10}{100} = 100)$
(iv) Usually rate of depreciation for vehicle is:
→ (d) 10%
(v) Share of father and mother is:
→ (a) $\frac{1}{6}$ of the property
(vi) If a deceased man has only one daughter, she will get:
→ (b) $\frac{1}{2}$ of the property
(vii) TV set costing Rs. 25000 purchased for Rs. 24500. Rate of discount?
→ (c) 2% $(\frac{500}{25000} \times 100 = 2\%)$
(viii) Time period agreed by both parties in insurance policy is called:
→ (b) maturity period
(ix) Cost of 1 dozen eggs is Rs. 180. Cost of 3 eggs?
→ (c) Rs. 45 $(\frac{180}{12} \times 3 = 45)$
(x) Marked price − sale price =
→ (a) discount
Q2. Amna has Rs. 450000. She withdraws Rs. 60000. Bank deducts service charges @ 0.5% on deduction. After 3 years she is paid Rs. 500000. Find (i) Profit, (ii) Rate of profit.
$$\text{Bank charges} = 60000 \times 0.005 = \text{Rs. } 300$$
$$\text{Remaining Principal} = 450000 - 60000 - 300 = \text{Rs. } 389700$$
$$\text{Profit} = 500000 - 389700 = \textbf{Rs. 110,300}$$
$$\text{Rate of Profit} = \frac{110300 \times 100}{389700 \times 3} = \textbf{9.43\% per annum}$$
Q3. Policy of Rs. 200000 @ 5% for 20 years. Family income contract = 1% of policy. Calculate (i) Amount paid, (ii) Amount received at maturity (Bonus = 12%/yr, family bonus = Rs. 240000).
$$\text{Annual premium} = (200000 \times 0.05) + (200000 \times 0.01) = 10000 + 2000 = \text{Rs. } 12000$$
$$\text{Total Paid in 20 yrs} = 12000 \times 20 = \textbf{Rs. 240,000}$$
$$\text{Received} = 200000 + (200000 \times 0.12 \times 20) + 240000 = 200000 + 480000 + 240000 = \textbf{Rs. 920,000}$$
Q4. Rashida got vehicle insurance for Rs. 500000 @ 4.5% for 3 years (10% depreciation). Find total premium paid.
- 1st year premium $= 500000 \times 0.045 = \text{Rs. } 22500$
- 2nd year premium $= 450000 \times 0.045 = \text{Rs. } 20250$
- 3rd year premium $= 405000 \times 0.045 = \text{Rs. } 18225$
$$\text{Total Premium} = 22500 + 20250 + 18225 = \textbf{Rs. 60,975}$$
- 2nd year premium $= 450000 \times 0.045 = \text{Rs. } 20250$
- 3rd year premium $= 405000 \times 0.045 = \text{Rs. } 18225$
$$\text{Total Premium} = 22500 + 20250 + 18225 = \textbf{Rs. 60,975}$$
Q5. 33 men do two third work in 12 days working 10 hours daily. How many hours daily required for remaining work in 6 days if 3 men are not available?
$$\text{Remaining men} = 33 - 3 = 30, \quad \text{Remaining work} = 1 - \frac{2}{3} = \frac{1}{3}$$
$$\frac{x}{10} = \frac{33}{30} \times \frac{12}{6} \times \frac{1/3}{2/3} = 1.1 \times 2 \times 0.5 = 1.1$$
$$x = 10 \times 1.1 = \textbf{11 hours daily}$$
Q6. Contractor to build bridge in 96 days with 40 workers. After 60 days only $\frac{2}{5}$ work completed. How many more workers required to finish on time?
$$\text{Remaining days} = 96 - 60 = 36, \quad \text{Remaining work} = \frac{3}{5}$$
$$\frac{x}{40} = \frac{60}{36} \times \frac{3/5}{2/5} = \frac{5}{3} \times \frac{3}{2} = \frac{5}{2}$$
$$x = 40 \times \frac{5}{2} = 100\text{ workers}$$
$$\text{More workers required} = 100 - 40 = \textbf{60 workers}$$
Q7. Rehan (Rs. 50000 for 12m), Saeed (Rs. 40000 for 6m), Asim (Rs. 70000 for 6m). Profit = Rs. 63000. Find share of each.
$$\text{Ratio} = (50000 \times 12) : (40000 \times 6) : (70000 \times 6) = 60 : 24 : 42 = 10 : 4 : 7$$
$$\text{Sum of ratios} = 10 + 4 + 7 = 21 \implies \text{Unit value} = \frac{63000}{21} = \text{Rs. } 3000$$
- Rehan's share $= 10 \times 3000 = \textbf{Rs. 30,000}$
- Saeed's share $= 4 \times 3000 = \textbf{Rs. 12,000}$
- Asim's share $= 7 \times 3000 = \textbf{Rs. 21,000}$
- Saeed's share $= 4 \times 3000 = \textbf{Rs. 12,000}$
- Asim's share $= 7 \times 3000 = \textbf{Rs. 21,000}$
Q8. Bank balance = Rs. 430000, 6 tolas gold = Rs. 600000. 2 tolas gold given to poor family. Divide property between son and husband.
$$\text{Remaining 4 tolas gold} = 4 \times 100000 = \text{Rs. } 400000$$
$$\text{Total Distributable Estate} = 430000 + 400000 = \text{Rs. } 830000$$
- Husband's share $(\frac{1}{4}) = \frac{1}{4} \times 830000 = \textbf{Rs. 207,500}$
- Son's share (remaining) $= 830000 - 207500 = \textbf{Rs. 622,500}$
- Son's share (remaining) $= 830000 - 207500 = \textbf{Rs. 622,500}$
Q9. Rice (15 kg @ 5% discount for Rs. 1520), Sugar (10 kg @ 4% discount for Rs. 960), Flour (20 kg @ 5% discount for Rs. 1235). Find (i) Marked price of each, (ii) Discount on each, (iii) Total discount.
(i) Marked prices:
- Rice $= \frac{1520}{0.95} = \textbf{Rs. 1,600}$
- Sugar $= \frac{960}{0.96} = \textbf{Rs. 1,000}$
- Flour $= \frac{1235}{0.95} = \textbf{Rs. 1,300}$
(ii) Discounts:
- Rice $= 1600 - 1520 = \textbf{Rs. 80}$
- Sugar $= 1000 - 960 = \textbf{Rs. 40}$
- Flour $= 1300 - 1235 = \textbf{Rs. 65}$
(iii) Total Discount:
$$\text{Total Discount} = 80 + 40 + 65 = \textbf{Rs. 185}$$
- Rice $= \frac{1520}{0.95} = \textbf{Rs. 1,600}$
- Sugar $= \frac{960}{0.96} = \textbf{Rs. 1,000}$
- Flour $= \frac{1235}{0.95} = \textbf{Rs. 1,300}$
(ii) Discounts:
- Rice $= 1600 - 1520 = \textbf{Rs. 80}$
- Sugar $= 1000 - 960 = \textbf{Rs. 40}$
- Flour $= 1300 - 1235 = \textbf{Rs. 65}$
(iii) Total Discount:
$$\text{Total Discount} = 80 + 40 + 65 = \textbf{Rs. 185}$$
Unit 4 Check Point & Challenge Solved Solutions
Challenge (Page 59): Asif deposited Rs. 12000 in bank. Bank pays 8% profit for 1st year and 10% for 2nd year. Find total amount at end of 2 years.
Solution:
- Year 1 profit $= 12000 \times 0.08 = \text{Rs. } 960 \implies \text{Amount} = \text{Rs. } 12960$
- Year 2 profit $= 12960 \times 0.10 = \text{Rs. } 1296$
- Total amount $= 12960 + 1296 = \textbf{Rs. 14,256}$.
Solution:
- Year 1 profit $= 12000 \times 0.08 = \text{Rs. } 960 \implies \text{Amount} = \text{Rs. } 12960$
- Year 2 profit $= 12960 \times 0.10 = \text{Rs. } 1296$
- Total amount $= 12960 + 1296 = \textbf{Rs. 14,256}$.
Check Point (Page 63): Najma offers 10% discount and still earns 20% profit. What is the cost price if marked price is Rs. 400?
Solution:
- $\text{Sale Price} = 400 \times (1 - 0.10) = \text{Rs. } 360$
- $\text{Cost Price} = \frac{\text{Sale Price}}{1 + \text{Profit } \%} = \frac{360}{1.20} = \textbf{Rs. 300}$.
Solution:
- $\text{Sale Price} = 400 \times (1 - 0.10) = \text{Rs. } 360$
- $\text{Cost Price} = \frac{\text{Sale Price}}{1 + \text{Profit } \%} = \frac{360}{1.20} = \textbf{Rs. 300}$.
More Chapter Notes for Class 8 (FBISE)
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Mathematics • Chapter 1
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Class 8 Mathematics - Ch 1: Real Numbers, Decimal Classification, Properties, Inequalities & Absolute Values (FBISE)
Mathematics • Chapter 2
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Class 8 Mathematics - Ch 2: Estimation, Rounding Off, Significant Figures & Approximation Error (FBISE)
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Class 8 Mathematics - Ch 5: Sets Mastery Guide (FBISE)
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Mathematics • Chapter 8
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Mathematics • Chapter 9
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Mathematics • Chapter 13
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Chapter 13: Probability — Comprehensive Solved Notes & Solution Manual
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