jamb mathematics- lesson notes on fractions, decimals, approximations, and percentages

Fractions, decimals, approximations and percentage | Jamb Mathematics

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Dear seeker of knowledge,
As the examination draws near, let it not be a mere test of memory but a reflective journey through the vast expanse of understanding you have cultivated. Approach your preparation not with haste, but with the deliberate curiosity of one seeking wisdom, for every concept mastered is a step toward greater enlightenment. Remember, the measure of success is not solely in answers given, but in the depth of thought and insight you bring to the endeavor.

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Are you preparing for your JAMB Mathematics exam and feeling a bit uncertain about how to approach the topic of Fractions, decimals, approximations and percentage? Don’t worry—you’re in the right place! This lesson is here to break it down in a simple, clear, and engaging way, helping you build the strong foundation you need to succeed. Whether you're struggling with complex questions or just seeking a quick refresher, this guide will boost your understanding and confidence. Let’s tackle Fractions, decimals, approximations and percentage together and move one step closer to achieving your exam success! Blissful learning.

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Calculation problems on decimals

Convert $\frac{3}{4}$ $\frac{3}{4}$ to a decimal.
- Step 1: Divide numerator by denominator: $3 \div 4$ .
- Step 2: Perform division: $3 \div 4 = 0.75$ .
- Final result: $\frac{3}{4} = 0.75$ .

Convert $0.125$ $0.125$ to a fraction
- Step 1: Write as $\frac{125}{1000}$ .
- Step 2: Simplify by dividing numerator and denominator by their GCD (125):
  $\frac{125}{1000} = \frac{1}{8}.$
- Final result: $0.125 = \frac{1}{8}$ .

Add $\frac{1}{2}$ $\frac{1}{2}$ and $\frac{2}{3}$ $\frac{2}{3}$ .
- Step 1: Find the LCM of denominators $2$ and $3$ : $6$ .
- Step 2: Convert fractions to equivalent fractions with denominator 6: $\frac{1}{2} = \frac{3}{6}, \quad \frac{2}{3} = \frac{4}{6}.$
- Step 3: Add numerators: $3 + 4 = 7$ .
- Final result: $\frac{7}{6}$ or $1 \frac{1}{6}$ .

Subtract $0.75$ $0.75$ from $1.5$ $1.5$ .
- Step 1: Convert $0.75$ to a fraction: $\frac{3}{4}$ .
- Step 2: Convert $1.5$ to a fraction: $\frac{3}{2}$ .
- Step 3: Subtract: $\frac{3}{2} - \frac{3}{4} = \frac{6}{4} - \frac{3}{4} = \frac{3}{4}.$
- Final result: $0.75$ .

Multiply $\frac{5}{6}$ $\frac{5}{6}$ by $\frac{3}{4}$ $\frac{3}{4}$ .
- Step 1: Multiply numerators: $5 \times 3 = 15$ .
- Step 2: Multiply denominators: $6 \times 4 = 24$ .
- Step 3: Simplify: $\frac{15}{24} = \frac{5}{8}$ .
- Final result: $\frac{5}{6} \times \frac{3}{4} = \frac{5}{8}$ .

Divide $\frac{7}{8}$ $\frac{7}{8}$ by $\frac{2}{3}$ $\frac{2}{3}$ .
- Step 1: Multiply by the reciprocal of $\frac{2}{3}$ :
  $\frac{7}{8} \div \frac{2}{3} = \frac{7}{8} \times \frac{3}{2}.$
- Step 2: Multiply numerators and denominators: $\frac{7 \times 3}{8 \times 2} = \frac{21}{16}.$
- Final result: $\frac{21}{16}$ or $1 \frac{5}{16}$ .

Simplify $\frac{18}{24}$ $\frac{18}{24}$ .
- Step 1: Find the GCD of $18$ and $24$ : $6$ .
- Step 2: Divide numerator and denominator by 6: $\frac{18}{24} = \frac{3}{4}.$
- Final result: $\frac{3}{4}$ .

Write $2.25$ $2.25$ as a fraction
- Step 1: Write as $\frac{225}{100}$ .
- Step 2: Simplify by dividing numerator and denominator by their GCD (25):
  $\frac{225}{100} = \frac{9}{4}.$
- Final result: $2.25 = \frac{9}{4}$ .

Convert $\frac{5}{8}$ $\frac{5}{8}$ to a decimal.
- Step 1: Divide $5 \div 8 = 0.625$ .
- Final result: $\frac{5}{8} = 0.625$ .

Add $1.5$ and $2.75$ .

Step 1: Add as decimals: $1.5 + 2.75 = 4.25$ .
Final result: $4.25$ .

Subtract $\frac{4}{5}$ from $1$ .

Step 1: Convert $1$ to $\frac{5}{5}$ .
Step 2: Subtract: $\frac{5}{5} - \frac{4}{5} = \frac{1}{5}.$
Final result: $\frac{1}{5}$ .

Simplify $\frac{45}{60}$

Step 1: Find GCD of $45$ and $60$ 15 $.
Step 2: Divide numerator and denominator by $15$ : $\frac{45}{60} = \frac{3}{4}.$
Final result: $\frac{3}{4}$ .

Convert $0.333...$ to a fraction.

Step 1: Let $x = 0.333...$ .
Step 2: Multiply by 10: $10x = 3.333...$ .
Step 3: Subtract: $10x - x = 9x = 3$ .
Step 4: Solve: $x = \frac{3}{9} = \frac{1}{3}$ .
Final result: $0.333... = \frac{1}{3}$ .

Convert $0.875$ to a fraction.

Step 1: Write as $\frac{875}{1000}$ .
Step 2: Simplify by dividing numerator and denominator by their GCD (125):
$\frac{875}{1000} = \frac{7}{8}.$
Final result: $0.875 = \frac{7}{8}$ .

Multiply $2.5$ by $ \frac45 .

Step 1: Convert $2.5$ to $\frac{5}{2}$ .
Step 2: Multiply: $\frac{5}{2} \times \frac{4}{5} = \frac{20}{10} = 2.$
Final result: $2$ .

Divide $\frac{3}{4}$ by $\frac{5}{6}$ .

Step 1: Multiply by the reciprocal: $\frac{3}{4} \div \frac{5}{6} = \frac{3}{4} \times \frac{6}{5}.$
Step 2: Multiply numerators and denominators: $\frac{3 \cdot 6}{4 \cdot 5} = \frac{18}{20} = \frac{9}{10}.$
Final result: $\frac{9}{10}$ .

Add $0.5 + \frac{3}{4}$ .

Step 1: Convert $0.5$ to $\frac{1}{2}$ .
Step 2: Find LCM of denominators $2$ and $4$ : $4$ .
Step 3: Convert: $\frac{1}{2} = \frac{2}{4}, \quad \frac{3}{4} = \frac{3}{4}.$
Step 4: Add numerators: $\frac{2}{4} + \frac{3}{4} = \frac{5}{4} = 1 \frac{1}{4}.$
Final result: $1 \frac{1}{4}$ .

Subtract $0.2$ from $0.8$ .

Step 1: Subtract decimals: $0.8 - 0.2 = 0.6.$
Final result: $0.6$ .

Write $\frac{7}{20}$ as a decimal.

Step 1: Divide $7 \div 20 = 0.35$ .
Final result: $0.35$ .

Add $\frac{2}{3}$ and $0.5$ .

Step 1: Convert $0.5$ to $\frac{1}{2}$ .
Step 2: Find LCM of denominators $3$ and $2$ : $6$ .
Step 3: Convert: $\frac{2}{3} = \frac{4}{6}, \quad \frac{1}{2} = \frac{3}{6}.$
Step 4: Add numerators: $\frac{4}{6} + \frac{3}{6} = \frac{7}{6}.$
Final result: $1 \frac{1}{6}$ .
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Significant figures problems

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Question 1:

How many significant figures are in the number 0.00540?

Answer:

Ignore leading zeros.
- The zeros before the 5 are not significant.
The digits 5 and 4 are significant.
The trailing zero is significant because it follows a decimal point.

Significant Figures: 3

Question 2:

What is the number of significant figures in 7000?

Answer:

Trailing zeros in a number without a decimal point are not significant.
Only the digit 7 is significant.

Significant Figures: 1

Question 3:

Round 12.3467 to 3 significant figures.

Answer:

Identify the first 3 significant digits: 12.3.
The 4th digit is 4, so no rounding up is needed.

Rounded Value: 12.3

Question 4:

Express 0.000456 in scientific notation with 3 significant figures.

Answer:

Convert to scientific notation: $4.56 \times 10^{-4}$ .
Retain 3 significant figures in $4.56$ .

Answer:

4.56 \times 10^{-4}

Question 5:

How many significant figures are in 500.0?

Answer:

The decimal point makes all zeros significant.
Include all digits (5, 0, 0, and 0).

Significant Figures: 4

Question 6:

Perform the calculation

6.38 \times 2.1

and report the result with the correct number of significant figures.

Answer:

Multiply: $6.38 \times 2.1 = 13.398$ .
The least number of significant figures is 2 (in 2.1).
Round to 2 significant figures: 13.

Answer: 13

Question 7

Add

5.34 + 2.1 + 0.007

and report the result to the correct number of decimal places.

Answer:

Align decimal places:
$5.34$
$2.10$ $0.007$
Sum: $7.447$ .
The least number of decimal places is 1 (in 2.1).
Round to 1 decimal place: 7.4.

Answer: 7.4

Question 8:

How many significant figures are in 0.03040?

Answer:

Ignore leading zeros.
Digits 3, 0, 4, and the trailing 0 after the decimal are significant.

Significant Figures: 4

Question 9:

Divide

8.45 \div 2.1

and report the result with the correct number of significant figures.

Answer:

Perform division: $8.45 \div 2.1 = 4.0238$ .
The least number of significant figures is 2 (in 2.1).
Round to 2 significant figures: 4.0.

Answer: 4.0

Question 10:

Write

0.004560

with 3 significant figures in scientific notation.

Answer:

Convert to scientific notation: $4.560 \times 10^{-3}$ .
Retain 3 significant figures: $4.56 \times 10^{-3}$ .

Answer:

4.56 \times 10^{-3}

Question 11:

How many significant figures are in

1.000 \times 10^{3}

Answer:

All digits in $1.000$ are significant because of the decimal point.

Significant Figures: 4

Question 12:

Subtract

7.15 - 0.006

and report the result with the correct number of decimal places.

Answer:

Align decimal places:
$7.150$ $0.006$
Subtract: $7.144$ .
The least number of decimal places is 3 (in 7.15).

Answer: 7.144

Question 13:

Multiply

0.0045 \times 200.0

and report the result with the correct number of significant figures.

Answer:

Multiply: $0.0045 \times 200.0 = 0.9$ .
The least number of significant figures is 2 (in 0.0045).
Retain 2 significant figures: 0.90.

Answer: 0.90

Question 14:

Round

0.00098765

to 4 significant figures.

Answer:

Identify the first 4 significant digits: 9876.
Write in decimal form: $0.0009877$ .

Answer:

0.0009877

Question 15:

How many significant figures are in

3.00 \times 10^{4}

Answer:

All digits in $3.00$ are significant because of the decimal point.

Significant Figures: 3

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Calculation problems on Percentage errors

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Question 1

If the measured value is 48 and the true value is 50, find the percentage error.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{50 - 48}{50} \right) \times 100$
Calculate: $\text{Percentage Error} = \left( \frac{2}{50} \right) \times 100 = 4\%$

Question 2

A measurement of 120 cm is recorded instead of the true value of 125 cm. What is the percentage error?

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{125 - 120}{125} \right) \times 100$
Calculate: $\text{Percentage Error} = \left( \frac{5}{125} \right) \times 100 = 4\%$

Question 3

The measured value of a length is 15 cm, and the percentage error is 10%. Find the true value.

Solution:

Formula: $\text{Measured Value} = \text{True Value} \times (1 - \text{Percentage Error})$
Rearrange: $\text{True Value} = \frac{\text{Measured Value}}{1 - \text{Percentage Error}}$
Substitute: $\text{True Value} = \frac{15}{1 - 0.10} = \frac{15}{0.90}$
Calculate: $\text{True Value} = 16.67 \, \text{cm}$

Question 4

A measurement of 65 is recorded, with a 5% error. Find the possible range of true values.

Solution:

Formula: $\text{True Value} = \text{Measured Value} \pm (\text{Measured Value} \times \text{Percentage Error})$
Substitute: $\text{True Value} = 65 \pm (65 \times 0.05)$
Calculate: $\text{True Value} = 65 \pm 3.25$
Range: $[61.75, 68.25]$

Question 5

Find the percentage error when the measured value is 80 and the true value is 100.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{100 - 80}{100} \right) \times 100$
Calculate: $\text{Percentage Error} = 20\%$

Question 6

The true value of a mass is 250 g, and the percentage error is 8%. Find the measured value.

Solution:

Formula: $\text{Measured Value} = \text{True Value} \times (1 - \text{Percentage Error})$
Substitute: $\text{Measured Value} = 250 \times (1 - 0.08)$
Calculate: $\text{Measured Value} = 250 \times 0.92 = 230 \, \text{g}$

Question 7

If the percentage error is 12% and the measured value is 88, find the true value.

Solution:

Formula: $\text{True Value} = \frac{\text{Measured Value}}{1 - \text{Percentage Error}}$
Substitute: $\text{True Value} = \frac{88}{1 - 0.12} = \frac{88}{0.88}$
Calculate: $\text{True Value} = 100$

Question 8

The measured radius of a circle is 14 cm, and the true radius is 15 cm. Find the percentage error in the area calculation.

Solution:

Area formula: $\text{Area} = \pi r^2$
Measured area: $A_m = \pi (14)^2 = 196\pi$
True area: $A_t = \pi (15)^2 = 225\pi$
Percentage error: $\text{Percentage Error} = \left( \frac{A_t - A_m}{A_t} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{225\pi - 196\pi}{225\pi} \right) \times 100$
Simplify: $\text{Percentage Error} = \frac{29}{225} \times 100 = 12.89\%$

Question 9

A measured resistance value is 95 ohms, but the true resistance is 100 ohms. What is the percentage error?

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{100 - 95}{100} \right) \times 100$
Calculate: $\text{Percentage Error} = 5\%$

Question 10

A student calculated a volume of 500 cm³ when the true volume was 520 cm³. Find the percentage error.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{520 - 500}{520} \right) \times 100$
Calculate: $\text{Percentage Error} = 3.85\%$

Question 11

If a length is measured as 24.5 cm but the true value is 25 cm, find the percentage error.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{25 - 24.5}{25} \right) \times 100$
Calculate: $\text{Percentage Error} = 2\%$

Question 12

Find the percentage error if a value of 85 is recorded instead of 100.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{100 - 85}{100} \right) \times 100$
Calculate: $\text{Percentage Error} = 15\%$

Question 13

A true temperature is

300 \, K

, but it is measured as

310 \, K

. Find the percentage error.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{310 - 300}{300} \right) \times 100$
Calculate: $\text{Percentage Error} = 3.33\%$

Question 14

If the true value of a force is 200 N and the measured value is 210 N, calculate the percentage error.

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{Measured Value} - \text{True Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{210 - 200}{200} \right) \times 100$
Calculate: $\text{Percentage Error} = 5\%$

Question 15

A length is measured as 75 cm, but the actual value is

78 \, \text{cm}

. What is the percentage error?

Solution:

Formula: $\text{Percentage Error} = \left( \frac{\text{True Value} - \text{Measured Value}}{\text{True Value}} \right) \times 100$
Substitute: $\text{Percentage Error} = \left( \frac{78 - 75}{78} \right) \times 100$
Calculate: $\text{Percentage Error} = 3.85\%$
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Calculation problems on simple interest

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Question 1

Find the simple interest on a principal of ₹5,000 at a rate of 5% per annum for 3 years.

Solution:

Formula: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $\text{SI} = \frac{5000 \times 5 \times 3}{100}$
Calculate: $\text{SI} = \frac{75000}{100} = ₹750$

Question 2

What is the total amount after 2 years if ₹10,000 is invested at 6% per annum?

Solution:

Simple Interest: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $\text{SI} = \frac{10000 \times 6 \times 2}{100} = ₹1200$
Total Amount: $\text{A} = \text{P} + \text{SI} = 10000 + 1200 = ₹11,200$

Question 3

How many years will it take for ₹8,000 to earn ₹1,600 as interest at 5% per annum?

Solution:

Formula: $T = \frac{\text{SI} \times 100}{P \times R}$
Substitute: $T = \frac{1600 \times 100}{8000 \times 5}$
Calculate: $T = \frac{160000}{40000} = 4 \, \text{years}$

Question 4

If ₹15,000 amounts to ₹18,000 in 3 years, find the rate of interest per annum.

Solution:

Simple Interest: $\text{SI} = \text{A} - \text{P} = 18000 - 15000 = ₹3000$
Formula: $R = \frac{\text{SI} \times 100}{P \times T}$
Substitute: $R = \frac{3000 \times 100}{15000 \times 3}$
Calculate: $R = \frac{300000}{45000} = 6\%$

Question 5

A sum of ₹4,000 earns an interest of ₹480 in 2 years. What is the rate of interest per annum?

Solution:

Formula: $R = \frac{\text{SI} \times 100}{P \times T}$
Substitute: $R = \frac{480 \times 100}{4000 \times 2}$
Calculate: $R = \frac{48000}{8000} = 6\%$

Question 6

If the principal is ₹7,500 and the rate of interest is 4% per annum, find the time required to earn ₹900 as interest.

Solution:

Formula: $T = \frac{\text{SI} \times 100}{P \times R}$
Substitute: $T = \frac{900 \times 100}{7500 \times 4}$
Calculate: $T = \frac{90000}{30000} = 3 \, \text{years}$

Question 7

What is the principal if the simple interest is ₹1,200, the rate is 8%, and the time is 5 years?

Solution:

Formula: $P = \frac{\text{SI} \times 100}{R \times T}$
Substitute: $P = \frac{1200 \times 100}{8 \times 5}$
Calculate: $P = \frac{120000}{40} = ₹3,000$

Question 8

Find the simple interest on ₹9,000 at 7% per annum for 4 years.

Solution:

Formula: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $\text{SI} = \frac{9000 \times 7 \times 4}{100}$
Calculate: $\text{SI} = \frac{252000}{100} = ₹2,520$

Question 9

What is the total amount after 5 years if ₹12,000 is invested at 9% per annum?

Solution:

Simple Interest: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $\text{SI} = \frac{12000 \times 9 \times 5}{100} = ₹5,400$
Total Amount: $\text{A} = \text{P} + \text{SI} = 12000 + 5400 = ₹17,400$

Question 10

How long will it take for ₹20,000 to double itself at a rate of 10% per annum?

Solution:

Double implies $\text{SI} = \text{P}$ , so $\text{SI} = 20000$ .
Formula: $T = \frac{\text{SI} \times 100}{P \times R}$
Substitute: $T = \frac{20000 \times 100}{20000 \times 10}$
Calculate: $T = 10 \, \text{years}$

Question 11

What is the principal if the total amount after 4 years is ₹12,000 at 5% per annum?

Solution:

Total Amount: $\text{A} = \text{P} + \text{SI}$
Simple Interest: $\text{SI} = \frac{P \times R \times T}{100} = \frac{P \times 5 \times 4}{100} = \frac{20P}{100} = 0.2P$
Total Amount: $\text{A} = P + 0.2P = 1.2P$
Substitute: $12000 = 1.2P$
Calculate: $P = \frac{12000}{1.2} = ₹10,000$

Question 12

Find the rate of interest if ₹5,000 amounts to ₹6,250 in 5 years.

Solution:

Simple Interest: $\text{SI} = \text{A} - \text{P} = 6250 - 5000 = ₹1,250$
Formula: $R = \frac{\text{SI} \times 100}{P \times T}$
Substitute: $R = \frac{1250 \times 100}{5000 \times 5}$
Calculate: $R = 5\%$

Question 13

If ₹4,500 earns ₹810 as interest in 6 years, find the rate of interest.

Solution:

Formula: $R = \frac{\text{SI} \times 100}{P \times T}$
Substitute: $R = \frac{810 \times 100}{4500 \times 6}$
Calculate: $R = 3\%$

Question 14

How many years will ₹3,000 take to earn ₹450 as interest at 5% per annum?

Solution:

Formula: $T = \frac{\text{SI} \times 100}{P \times R}$
Substitute: $T = \frac{450 \times 100}{3000 \times 5}$
Calculate: $T = 3 \, \text{years}$

Question 15

What is the total amount after 10 years if ₹1,000 is invested at 2% per annum?

Solution:

Simple Interest: $\text{SI} = \frac{P \times R \times T}{100} = \frac{1000 \times 2 \times 10}{100} = ₹200$
Total Amount: $\text{A} = P + SI = 1000 + 200 = ₹1,200$

Question 16

What is the simple interest on ₹2,400 at 1.5% per annum for 5 years?

Solution:

Formula: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $\text{SI} = \frac{2400 \times 1.5 \times 5}{100}$
Calculate: $\text{SI} = ₹180$

Question 17

How much principal will amount to ₹8,500 in 5 years at 5% per annum?

Solution:

Total Amount: $\text{A} = \text{P} + \text{SI}$
Simple Interest: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $8500 = P + \frac{P \times 5 \times 5}{100} = P + 0.25P = 1.25P$
Calculate: $P = \frac{8500}{1.25} = ₹6,800$

Question 18

A sum triples itself in 20 years. Find the rate of interest.

Solution:

$\text{SI} = 2P$ since it triples.
Formula: $R = \frac{\text{SI} \times 100}{P \times T}$
Substitute: $R = \frac{2P \times 100}{P \times 20}$
Simplify: $R = 10\%$

Question 19

Find the time required for ₹2,000 to earn ₹1,000 as interest at 10% per annum.

Solution:

Formula: $T = \frac{\text{SI} \times 100}{P \times R}$
Substitute: $T = \frac{1000 \times 100}{2000 \times 10}$
Calculate: $T = 5 \, \text{years}$

Question 20

What is the simple interest on ₹10,000 at 6% per annum for 1.5 years?

Solution:

Formula: $\text{SI} = \frac{P \times R \times T}{100}$
Substitute: $\text{SI} = \frac{10000 \times 6 \times 1.5}{100}$
Calculate: $\text{SI} = ₹900$

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Calculation problems on profit and loss account

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Question 1

A shopkeeper bought an article for ₹200 and sold it for ₹250. Find the profit percent.

Solution:

Profit: $\text{Profit} = \text{Selling Price} - \text{Cost Price} = 250 - 200 = ₹50$
Profit Percent:
$\text{Profit Percent} = \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100 = \left( \frac{50}{200} \right) \times 100 = 25\%$

Question 2

A trader sells an item for ₹450 at a loss of 10%. Find the cost price.

Solution:

Formula: $\text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 - \text{Loss Percent}}$
Substitute: $\text{Cost Price} = \frac{450 \times 100}{100 - 10} = \frac{450 \times 100}{90}$
Calculate: $\text{Cost Price} = ₹500$

Question 3

An article is purchased for ₹600 and sold for ₹660. Find the profit percent.

Solution:

Profit: $\text{Profit} = \text{SP} - \text{CP} = 660 - 600 = ₹60$
Profit Percent:
$\text{Profit Percent} = \left( \frac{60}{600} \right) \times 100 = 10\%$

Question 4

If an article is sold for ₹800 at a profit of 20%, find the cost price.

Solution:

Formula: $\text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 + \text{Profit Percent}}$
Substitute: $\text{Cost Price} = \frac{800 \times 100}{100 + 20} = \frac{800 \times 100}{120}$
Calculate: $\text{Cost Price} = ₹666.67$

Question 5

A shopkeeper earns a profit of 25% by selling an item for ₹500. What is the cost price?

Solution:

Formula: $\text{Cost Price} = \frac{\text{SP} \times 100}{100 + \text{Profit Percent}}$
Substitute: $\text{Cost Price} = \frac{500 \times 100}{100 + 25} = \frac{500 \times 100}{125}$
Calculate: $\text{Cost Price} = ₹400$

Question 6

A person sold an article for ₹1200 at a 20% loss. Find the cost price.

Solution:

Formula: $\text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 - \text{Loss Percent}}$
Substitute: $\text{Cost Price} = \frac{1200 \times 100}{100 - 20} = \frac{1200 \times 100}{80}$
Calculate: $\text{Cost Price} = ₹1500$

Question 7

If an item is sold at ₹500 with a 15% profit, find the cost price.

Solution:

Formula: $\text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 + \text{Profit Percent}}$
Substitute: $\text{Cost Price} = \frac{500 \times 100}{100 + 15} = \frac{500 \times 100}{115}$
Calculate: $\text{Cost Price} = ₹434.78$

Question 8

A person buys a watch for ₹800 and sells it at a loss of 12%. What is the selling price?

Solution:

Formula: $\text{Selling Price} = \text{Cost Price} \times \left( 1 - \frac{\text{Loss Percent}}{100} \right)$
Substitute: $\text{Selling Price} = 800 \times \left( 1 - \frac{12}{100} \right) = 800 \times 0.88$
Calculate: $\text{Selling Price} = ₹704$

Question 9

Find the loss percent if an item is bought for ₹300 and sold for ₹270.

Solution:

Loss: $\text{Loss} = \text{Cost Price} - \text{Selling Price} = 300 - 270 = ₹30$
Loss Percent:
$\text{Loss Percent} = \left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100 = \left( \frac{30}{300} \right) \times 100 = 10\%$

Question 10

If the cost price of an article is ₹200 and the selling price is ₹240, find the profit percent.

Solution:

Profit: $\text{Profit} = \text{SP} - \text{CP} = 240 - 200 = ₹40$
Profit Percent:
$\text{Profit Percent} = \left( \frac{40}{200} \right) \times 100 = 20\%$

Question 11

If a loss of ₹50 is incurred on selling an item for ₹450, find the cost price.

Solution:

Formula: $\text{Cost Price} = \text{Selling Price} + \text{Loss}$
Substitute: $\text{Cost Price} = 450 + 50 = ₹500$

Question 12

Find the selling price if an article costing ₹1,200 is sold at a profit of 25%.

Solution:

Formula: $\text{Selling Price} = \text{Cost Price} \times \left( 1 + \frac{\text{Profit Percent}}{100} \right)$
Substitute: $\text{Selling Price} = 1200 \times \left( 1 + \frac{25}{100} \right) = 1200 \times 1.25$
Calculate: $\text{Selling Price} = ₹1,500$

Question 13

An article costing ₹500 is sold for ₹450. Find the loss percent.

Solution:

Loss: $\text{Loss} = \text{Cost Price} - \text{Selling Price} = 500 - 450 = ₹50$
Loss Percent:
$\text{Loss Percent} = \left( \frac{\text{Loss}}{\text{Cost Price}} \right) \times 100 = \left( \frac{50}{500} \right) \times 100 = 10\%$

Question 14

What is the selling price if the cost price is ₹2,000 and the profit percent is 15%?

Solution:

Formula: $\text{Selling Price} = \text{Cost Price} \times \left( 1 + \frac{\text{Profit Percent}}{100} \right)$
Substitute: $\text{Selling Price} = 2000 \times \left( 1 + \frac{15}{100} \right) = 2000 \times 1.15$
Calculate: $\text{Selling Price} = ₹2,300$

Question 15

An item was bought for ₹800 and sold at ₹880. Find the profit percent.

Solution:

Profit: $\text{Profit} = 880 - 800 = ₹80$
Profit Percent:
$\text{Profit Percent} = \left( \frac{80}{800} \right) \times 100 = 10\%$

Question 16

A table is sold at ₹1,800, incurring a 10% loss. What is the cost price?

Solution:

Formula: $\text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 - \text{Loss Percent}}$
Substitute: $\text{Cost Price} = \frac{1800 \times 100}{100 - 10} = \frac{1800 \times 100}{90}$
Calculate: $\text{Cost Price} = ₹2,000$

Question 17

If the cost price is ₹1,500 and the loss percent is 5%, find the selling price.

Solution:

Formula: $\text{Selling Price} = \text{Cost Price} \times \left( 1 - \frac{\text{Loss Percent}}{100} \right)$
Substitute: $\text{Selling Price} = 1500 \times \left( 1 - \frac{5}{100} \right) = 1500 \times 0.95$
Calculate: $\text{Selling Price} = ₹1,425$

Question 18

A person gains ₹200 by selling an item for ₹2,400. Find the profit percent.

Solution:

Cost Price: $\text{CP} = \text{SP} - \text{Profit} = 2400 - 200 = ₹2,200$
Profit Percent:
$\text{Profit Percent} = \left( \frac{200}{2200} \right) \times 100 = 9.09\%$

Question 19

Find the selling price if the cost price is ₹5,000 and the profit is ₹800.

Solution:

Formula: $\text{Selling Price} = \text{Cost Price} + \text{Profit}$
Substitute: $\text{Selling Price} = 5000 + 800 = ₹5,800$

Question 20

A person incurs a 12% loss by selling an article for ₹880. Find the cost price.

Solution:

Formula: $\text{Cost Price} = \frac{\text{Selling Price} \times 100}{100 - \text{Loss Percent}}$
Substitute: $\text{Cost Price} = \frac{880 \times 100}{100 - 12} = \frac{880 \times 100}{88}$
Calculate: $\text{Cost Price} = ₹1,000$

Calculation problems on ratio, proportion and rate;

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Question 1

If the ratio of apples to oranges in a basket is 3:2, and there are 15 apples, how many oranges are there?

Solution:

Let the number of oranges be $x$ Ratio: $\frac{\text{Apples}}{\text{Oranges}} = \frac{3}{2}$ .
Equation: $\frac{15}{x} = \frac{3}{2}$ .
Cross-multiply: $3x = 30$ .
Solve: $x = 10$ .

Answer: There are 10 oranges.

Question 2

The ratio of boys to girls in a class is 4:5, and there are 36 students in total. How many boys are there?

Solution:

Total parts: $4 + 5 = 9$ .
Each part: $\frac{36}{9} = 4$ .
Number of boys: $4 \times 4 = 16$ .

Answer: There are 16 boys.

Question 3

Two numbers are in the ratio 7:9. If their sum is 64, find the numbers.

Solution:

Total parts: $7 + 9 = 16$ .
Value of each part: $\frac{64}{16} = 4$ .
Numbers: $7 \times 4 = 28$ , $9 \times 4 = 36$ .

Answer: The numbers are 28 and 36.

Question 4

Divide ₹1,200 in the ratio 5:3.

Solution:

Total parts: $5 + 3 = 8$ .
Each part: $\frac{1200}{8} = 150$ .
Amounts: $5 \times 150 = ₹750$ , $3 \times 150 = ₹450$ .

Answer: ₹750 and ₹450.

Question 5

A recipe requires sugar and flour in the ratio 2:3. If 10 kg of flour is used, how much sugar is needed?

Solution:

Ratio: $\frac{\text{Sugar}}{\text{Flour}} = \frac{2}{3}$ .
Let sugar = $x$ .
Equation: $\frac{x}{10} = \frac{2}{3}$ .
Solve: $3x = 20$ , $x = 6.67 \, \text{kg}$ .

Answer: 6.67 kg of sugar.

Question 6

If 5 workers can complete a task in 8 days, how many days will it take for 10 workers to complete the same task?

Solution:

Work is inversely proportional to the number of workers.
Equation: $5 \times 8 = 10 \times x$ .
Solve: $x = \frac{40}{10} = 4 \, \text{days}$ .

Answer: 4 days.

Question 7

The ratio of speed of two cars is 3:4. If the first car travels at 60 km/h, find the speed of the second car.

Solution:

Ratio: $\frac{\text{Speed of first car}}{\text{Speed of second car}} = \frac{3}{4}$ .
Let speed of the second car = $x$ .
Equation: $\frac{60}{x} = \frac{3}{4}$ .
Solve: $3x = 240$ , $x = 80 \, \text{km/h}$ .

Answer: 80 km/h.

Question 8

Three friends divide ₹6,000 in the ratio 2:3:5. How much does each get?

Solution:

Total parts: $2 + 3 + 5 = 10$ .
Each part: $\frac{6000}{10} = 600$ .
Amounts: $2 \times 600 = ₹1200$ , $3 \times 600 = ₹1800$ , $5 \times 600 = ₹3000$ .

Answer: ₹1200, ₹1800, and ₹3000.

Question 9

The ratio of water to milk in a mixture is 3:7. If the mixture contains 15 liters of water, how much milk is there?

Solution:

Ratio: $\frac{\text{Water}}{\text{Milk}} = \frac{3}{7}$ .
Let milk = $x$ .
Equation: $\frac{15}{x} = \frac{3}{7}$ .
Solve: $3x = 105$ , $x = 35 \, \text{liters}$ .

Answer: 35 liters of milk.

Question 10

If 6 pens cost ₹90, what is the cost of 10 pens?

Solution:

Cost per pen: $\frac{90}{6} = ₹15$ .
Cost of 10 pens: $10 \times 15 = ₹150$ .

Answer: ₹150.

Question 11

Two numbers are in the ratio 4:5. If their difference is 9, find the numbers.

Solution:

Difference of parts: $5 - 4 = 1$ .
Value of each part: $\frac{9}{1} = 9$ .
Numbers: $4 \times 9 = 36$ , $5 \times 9 = 45$ .

Answer: 36 and 45.

Question 12

If 8 workers complete a task in 12 days, how many workers are needed to complete it in 6 days?

Solution:

Work is inversely proportional to workers.
Equation: $8 \times 12 = x \times 6$ .
Solve: $x = \frac{96}{6} = 16$ .

Answer: 16 workers.

Question 13

A sum of ₹500 is divided among A, B, and C in the ratio 1:2:3. How much does each get?

Solution:

Total parts: $1 + 2 + 3 = 6$ .
Each part: $\frac{500}{6} = ₹83.33$ .
Amounts: $₹83.33, ₹166.67, ₹250$ .

Answer: ₹83.33, ₹166.67, and ₹250.

Question 14

If 15 men can dig a trench in 6 days, how long will it take for 10 men to dig the same trench?

Solution:

Equation: $15 \times 6 = 10 \times x$ .
Solve: $x = \frac{90}{10} = 9 \, \text{days}$ .

Answer: 9 days.

Question 15

A and B divide ₹1,800 in the ratio 5:7. How much does A get?

Solution:

Total parts: $5 + 7 = 12$ .
Each part: $\frac{1800}{12} = ₹150$ .
A’s share: $5 \times 150 = ₹750$ .

Answer: ₹750.

Question 16

A train travels 240 km in 4 hours. What is its speed?

Solution:

Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$ .
Substitute: $\text{Speed} = \frac{240}{4} = 60 \, \text{km/h}$ .

Answer: 60 km/h.

Question 17

If the ratio of the angles in a triangle is 2:3:4, find the angles.

Solution:

Total parts: $2 + 3 + 4 = 9$ .
Value of each part: $\frac{180}{9} = 20^\circ$ .
Angles: $2 \times 20 = 40^\circ, 3 \times 20 = 60^\circ, 4 \times 20 = 80^\circ$ .

Answer:

40^\circ, 60^\circ, 80^\circ

Question 18

If 5 machines produce 100 items in 2 hours, how many items will 10 machines produce in 1 hour?

Solution:

Output per machine per hour: $\frac{100}{5 \times 2} = 10$ .
Total output: $10 \times 10 \times 1 = 100 \, \text{items}$ .

Answer: 100 items.

Question 19

A sum of ₹2,400 is divided among A, B, and C in the ratio 3:4:5. How much does B get?

Solution:

Total parts: $3 + 4 + 5 = 12$ .
Each part: $\frac{2400}{12} = ₹200$ .
B’s share: $4 \times 200 = ₹800$ .

Answer: ₹800.

Question 20

The ratio of red to blue balls in a bag is 5:3. If there are 40 red balls, how many blue balls are there?

Solution:

Ratio: $\frac{\text{Red}}{\text{Blue}} = \frac{5}{3}$ .
Let blue = $x$ .
Equation: $\frac{40}{x} = \frac{5}{3}$ .
Solve: $5x = 120$ , $x = 24$ .

Answer: 24 blue balls.

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Calculation problems on shares and valued added tax

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Question 1

A person invests ₹10,000 in shares of face value ₹100 at ₹120 each. How many shares does the person buy?

Solution:

Formula: $\text{Number of shares} = \frac{\text{Investment}}{\text{Market Price per Share}}$ .
Substitute: $\text{Number of shares} = \frac{10000}{120}$ .
Calculate: $\text{Number of shares} = 83.33 \approx 83$ (since shares are whole numbers).

Answer: 83 shares.

Question 2

A company declares a 10% dividend on shares of face value ₹50. If a person holds 100 shares, find the dividend received.

Solution:

Formula: $\text{Dividend} = \text{Face Value} \times \text{Number of Shares} \times \frac{\text{Dividend Rate}}{100}$ .
Substitute: $\text{Dividend} = 50 \times 100 \times \frac{10}{100}$ .
Calculate: $\text{Dividend} = ₹500$ .

Answer: ₹500.

Question 3

A person buys 200 shares of ₹10 face value at a premium of ₹5. Find the total investment.

Solution:

Total Price per Share = Face Value + Premium = ₹10 + ₹5 = ₹15.
Total Investment = $200 \times 15 = ₹3,000$ .

Answer: ₹3,000.

Question 4

The market price of a share is ₹120, and its face value is ₹100. Find the premium.

Solution:

Premium = Market Price - Face Value = ₹120 - ₹100 = ₹20.

Answer: ₹20.

Question 5

A shareholder receives a dividend of ₹800 on 400 shares of face value ₹20. Find the dividend rate.

Solution:

Formula: $\text{Dividend Rate} = \frac{\text{Dividend} \times 100}{\text{Face Value} \times \text{Number of Shares}}$ .
Substitute: $\text{Dividend Rate} = \frac{800 \times 100}{20 \times 400}$ .
Calculate: $\text{Dividend Rate} = 10\%$ .

Answer: 10%.

Question 6

A person purchases 150 shares at ₹120 each. The face value of the shares is ₹100. Calculate the total premium paid.

Solution:

Premium per share = Market Price - Face Value = ₹120 - ₹100 = ₹20.
Total Premium = $150 \times 20 = ₹3,000$ .

Answer: ₹3,000.

Question 7

A person earns ₹960 as a dividend on shares of face value ₹80 with a dividend rate of 12%. Find the number of shares.

Solution:

Formula: $\text{Dividend} = \text{Face Value} \times \text{Number of Shares} \times \frac{\text{Dividend Rate}}{100}$ .
Rearrange: $\text{Number of Shares} = \frac{\text{Dividend} \times 100}{\text{Face Value} \times \text{Dividend Rate}}$ .
Substitute: $\text{Number of Shares} = \frac{960 \times 100}{80 \times 12}$ .
Calculate: $\text{Number of Shares} = 100$ .

Answer: 100 shares.

Question 8

The face value of a share is ₹50, and the dividend rate is 15%. A person owns 80 shares. Find the dividend received.

Solution:

Formula: $\text{Dividend} = \text{Face Value} \times \text{Number of Shares} \times \frac{\text{Dividend Rate}}{100}$ .
Substitute: $\text{Dividend} = 50 \times 80 \times \frac{15}{100}$ .
Calculate: $\text{Dividend} = ₹600$ .

Answer: ₹600.

Question 9

A person buys 120 shares at ₹90 each with a face value of ₹50. Calculate the total amount invested.

Solution:

Total Investment = $\text{Number of Shares} \times \text{Market Price per Share}$ .
Substitute: $120 \times 90 = ₹10,800$ .

Answer: ₹10,800.

Question 10

A company declares a 6% dividend. If the face value of the share is ₹100, how much dividend is received per share?

Solution:

Dividend per share = $\text{Face Value} \times \frac{\text{Dividend Rate}}{100}$ .
Substitute: $100 \times \frac{6}{100} = ₹6$ .

Answer: ₹6 per share.

Value Added Tax (VAT)

Question 11

A person buys an item for ₹2,000 with a VAT of 12%. Find the VAT amount.

Solution:

Formula: $\text{VAT Amount} = \text{Price} \times \frac{\text{VAT Rate}}{100}$ .
Substitute: $\text{VAT Amount} = 2000 \times \frac{12}{100}$ .
Calculate: $\text{VAT Amount} = ₹240$ .

Answer: ₹240.

Question 12

A TV is sold for ₹30,000 including 18% VAT. Find the original price before VAT.

Solution:

Formula: $\text{Price before VAT} = \frac{\text{Price including VAT}}{1 + \frac{\text{VAT Rate}}{100}}$ .
Substitute: $\text{Price before VAT} = \frac{30000}{1 + \frac{18}{100}} = \frac{30000}{1.18}$ .
Calculate: $\text{Price before VAT} = ₹25,423.73$ .

Answer: ₹25,423.73.

Question 13

An item priced at ₹5,000 has a VAT of 8%. Find the total price including VAT.

Solution:

Total Price = $\text{Original Price} + \text{VAT Amount}$ .
VAT Amount: $5000 \times \frac{8}{100} = ₹400$ .
Total Price: $5000 + 400 = ₹5,400$ .

Answer: ₹5,400.

Question 14

If an item costs ₹12,000 after 10% VAT, find the VAT amount.

Solution:

Formula: $\text{VAT Amount} = \frac{\text{Price Including VAT} \times \text{VAT Rate}}{100 + \text{VAT Rate}}$ .
Substitute: $\text{VAT Amount} = \frac{12000 \times 10}{100 + 10} = \frac{12000 \times 10}{110}$ .
Calculate: $\text{VAT Amount} = ₹1,090.91$ .

Answer: ₹1,090.91.

Question 15

A person purchases a washing machine for ₹20,000 with VAT of 15%. What is the total amount paid?

Solution:

VAT Amount: ( 20000 \times \frac15100 = ₹3,000 ).
Total Price: ( 20000 + 3000 = ₹23,000 ).

Answer: ₹23,000.

Question 16

The price of a book is ₹500 without VAT. If VAT is charged at 5%, find the total price including VAT.

Solution:

VAT Amount: $500 \times \frac{5}{100} = ₹25$ .
Total Price: $500 + 25 = ₹525$ .

Answer: ₹525.

Question 17

A person pays ₹16,000 for an item including 20% VAT. Find the price before VAT.

Solution:

Formula: $\text{Price before VAT} = \frac{\text{Price including VAT}}{1 + \frac{\text{VAT Rate}}{100}}$ .
Substitute: $\text{Price before VAT} = \frac{16000}{1 + \frac{20}{100}} = \frac{16000}{1.2}$ .
Calculate: $\text{Price before VAT} = ₹13,333.33$ .

Answer: ₹13,333.33.

Question 18

An item costs ₹800 after adding 10% VAT. What is the VAT amount?

Solution:

VAT Amount: $\text{Price including VAT} \times \frac{\text{VAT Rate}}{100 + \text{VAT Rate}}$ .
Substitute: $800 \times \frac{10}{100 + 10} = 800 \times \frac{10}{110}$ .
Calculate: $\text{VAT Amount} = ₹72.73$ .

Answer: ₹72.73.

Question 19

An item costs ₹1,500 before VAT. If 18% VAT is added, what is the total price?

Solution:

VAT Amount: $1500 \times \frac{18}{100} = ₹270$ .
Total Price: $1500 + 270 = ₹1,770$ .

Answer: ₹1,770.

Question 20

If the total cost of an item is ₹12,500 including VAT at 25%, find the price excluding VAT.

Solution:

Formula: $\text{Price excluding VAT} = \frac{\text{Price including VAT}}{1 + \frac{\text{VAT Rate}}{100}}$ .
Substitute: $\text{Price excluding VAT} = \frac{12500}{1.25}$ .
Calculate: $\text{Price excluding VAT} = ₹10,000$ .

Answer: ₹10,000.

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