Quick Revision Notes & Formula Sheet

Class 8 Mathematics — Chapter: Comparing Quantities (राशियों की तुलना)

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1. Ratios and Percentages (अनुपात और प्रतिशत)

• Ratio (अनुपात): A comparison of two quantities of the same kind measured in the same units.
  • Ratio of $a$ to $b$ is written as $a : b$ or $\frac{a}{b}$.
• Percentage (प्रतिशत): A fraction with denominator 100. Means "per hundred". Symbol: %.
• Converting Fraction / Ratio to Percentage: $$\text{Percentage} = \text{Fraction or Ratio} \times 100$$
• Converting Percentage to Fraction: $$\text{Fraction} = \frac{\text{Percentage}}{100}$$

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2. Increase and Decrease Percentage (वृद्धि और कमी प्रतिशत)

• Percentage Increase: $$\text{Percentage Increase} = \left( \frac{\text{Amount of Increase}}{\text{Original Value}} \right) \times 100$$

• Percentage Decrease: $$\text{Percentage Decrease} = \left( \frac{\text{Amount of Decrease}}{\text{Original Value}} \right) \times 100$$

• New Value: $$\text{New Value} = \text{Original Value} \pm \text{Change Amount}$$

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3. Discount (बट्टा / छूट)

• Marked Price (MP / अंकित मूल्य): The price printed on an item.

• Selling Price (SP / विक्रय मूल्य): The price at which the item is actually sold.

• Discount Formula: $$\text{Discount} = \text{Marked Price (MP)} - \text{Selling Price (SP)}$$

• Discount Percentage: $$\text{Discount %} = \left( \frac{\text{Discount}}{\text{Marked Price}} \right) \times 100$$

• Finding SP using Discount %: $$\text{SP} = \text{MP} \times \frac{(100 - \text{Discount %})}{100}$$

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4. Profit and Loss (लाभ और हानि)

• Cost Price (CP / क्रय मूल्य): The buying price of an item plus overhead expenses (transportation, repair, etc.).
• Selling Price (SP / विक्रय मूल्य): The price at which the item is sold.

A. Profit (Profit happens when SP > CP)

• $\text{Profit} = \text{SP} - \text{CP}$
• $\text{Profit %} = \left( \frac{\text{Profit}}{\text{CP}} \right) \times 100$
• $\text{SP} = \text{CP} \times \frac{(100 + \text{Profit %})}{100}$
• $\text{CP} = \frac{\text{SP} \times 100}{100 + \text{Profit %}}$

B. Loss (Loss happens when CP > SP)

• $\text{Loss} = \text{CP} - \text{SP}$
• $\text{Loss %} = \left( \frac{\text{Loss}}{\text{CP}} \right) \times 100$
• $\text{SP} = \text{CP} \times \frac{(100 - \text{Loss %})}{100}$
• $\text{CP} = \frac{\text{SP} \times 100}{100 - \text{Loss %}}$

Note: Profit % and Loss % are always calculated on the Cost Price (CP) unless specified otherwise.

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5. Taxes: Sales Tax / VAT / GST

• Taxes are added to the selling price of the item before calculating the total bill.
• Formula: $$\text{Tax Amount} = \left( \frac{\text{Tax %}}{100} \right) \times \text{Selling Price}$$ $$\text{Total Bill Amount} = \text{Selling Price} + \text{Tax Amount}$$

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6. Simple Interest (SI) vs Compound Interest (CI)

Where:

• $P$ = Principal (मूलधन)
• $R$ = Rate of interest per annum (ब्याज दर % वार्षिक)
• $T$ or $n$ = Time period in years (समय)
• $A$ = Total Amount (मिश्रधन)

A. Simple Interest (साधारण ब्याज)

Interest calculated only on the principal amount throughout the period.

• $\text{SI} = \frac{P \times R \times T}{100}$
• $\text{Amount (A)} = P + \text{SI}$

B. Compound Interest (चक्रवृद्धि ब्याज)

Interest calculated on the accumulated principal plus interest of previous periods.

• 1. Compounded Annually (वार्षिक संयोजन): $$A = P \left(1 + \frac{R}{100}\right)^n$$ $$\text{CI} = A - P$$

• 2. Compounded Half-Yearly (अर्द्धवार्षिक संयोजन):

  • Time becomes double ($2n$)
  • Rate becomes half ($\frac{R}{2}$) $$A = P \left(1 + \frac{R}{200}\right)^{2n}$$

• 3. Compounded Quarterly (त्रैमासिक संयोजन):

  • Time becomes four times ($4n$)
  • Rate becomes one-fourth ($\frac{R}{4}$) $$A = P \left(1 + \frac{R}{400}\right)^{4n}$$

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7. Applications of Compound Interest Formula

A. Growth (Population / Bacteria Growth)

If population grows at a rate of $R%$ per year: $$\text{Population after } n \text{ years} = P_0 \left(1 + \frac{R}{100}\right)^n$$ (Where $P_0$ is initial population)

B. Depreciation (अवमूल्यन / घिसावट)

Decrease in the value of an asset (like machinery, vehicles) over time at rate $R%$: $$\text{Value after } n \text{ years} = V_0 \left(1 - \frac{R}{100}\right)^n$$ (Where $V_0$ is initial value)

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Quick Memory Tricks for MP Board Exams:

1. Overhead Charges: Always add repair, transport, or labor costs to CP to get the Total CP.
2. Discount is on MP, Profit/Loss is on CP, and Tax is on SP.
3. In Half-Yearly problems: Double the time, Half the rate.