Mathematical Operations: Notes और MCQ Practice

Explanation (आसान भाषा में):

Mathematical Operations में addition, subtraction, multiplication, division, percentage, ratio‑proportion, square‑root आदि basic calculations आते हैं। ये सभी G.I.R के quantitative सेक्शन में जल्दी‑से solve करने के लिए must‑know हैं.

Important Formulas / Rules

• Percentage = (Part ÷ Whole) × 100
• Percentage Change = (New − Old) ÷ Old × 100
• Ratio : If a:b = m:n then a/b = m/n
• Simple Interest = (P × R × T) ÷ 100
• Square of a number ending in 5 = (n×(n+1)) followed by 25, जहाँ n = tens digit
• √(a² + b²) = hypotenuse in right triangle (Pythagoras)

Quick Tricks

1. 5‑rule for multiplication – Any number ×5 = (number ÷2) then add a zero at end.
2. Cross‑multiply for ratio problems – a/b = c/d → a×d = b×c, instantly gives unknown.

Common Mistakes

1. Percentage Change में “Old ÷ New ×100” लिख देना, जबकि सही formula (New‑Old) ÷ Old ×100 है।
2. Square‑root निकालते समय negative number को positive मान लेना; √(−) undefined in real numbers.

Concept

Mathematical Operations in reasoning involve using basic arithmetic (addition, subtraction, multiplication, division), percentages, averages, ratio & proportion, profit‑loss, simple & compound interest, time‑&‑work, etc., to solve word problems quickly.

Important formulas / rules / facts

• Percentage = Part ÷ Whole × 100
• Profit % = Profit ÷ Cost Price × 100
• Loss % = Loss ÷ Selling Price × 100
• Simple Interest = P × R × T ÷ 100
• Compound Interest (annual) = P × [(1 + R/100)^T – 1]
• Average = Sum of numbers ÷ Number of terms
• Ratio: a:b = a ÷ b
• Time & Work: Work = Rate × Time
• Speed = Distance ÷ Time
• Distance = Speed × Time
• Combined work: 1/Total time = 1/Time A + 1/Time B + …
• Number of terms in AP = (Last – First) ÷ Common difference + 1

Solved examples

Example 1

A shopkeeper buys an article for Rs 1200 and sells it at a profit of 25 %. Find the selling price.

Solution:

Profit = 25 % of 1200 = 1200 × 25 ÷ 100 = 300

Selling Price = Cost Price + Profit = 1200 + 300 = Rs 1500

Example 2

Two pipes can fill a tank in 15 min and 20 min respectively. How long will it take to fill the tank if both are opened together?

Solution:

Work done per minute by pipe 1 = 1/15 tank

Work done per minute by pipe 2 = 1/20 tank

Combined work per minute = 1/15 + 1/20 = (4 + 3)/60 = 7/60 tank

Time = 1 ÷ (7/60) = 60/7 ≈ 8.57 min → 8 min 34 sec

Example 3

The average of 7 numbers is 56. If one number 68 is removed, what is the new average?

Solution:

Total of 7 numbers = 56 × 7 = 392

New total after removing 68 = 392 – 68 = 324

New average = 324 ÷ 6 = 54

Example 4

A sum of Rs 5000 is invested at 10 % per annum compound interest. Find the amount after 2 years.

Solution:

Amount = P × (1 + R/100)^T = 5000 × (1 + 10/100)^2 = 5000 × (1.1)^2 = 5000 × 1.21 = Rs 6050

Example 5

A car travels 150 km in 3 hours and then 90 km in 2 hours. Find its average speed for the whole journey.

Solution:

Total distance = 150 + 90 = 240 km

Total time = 3 + 2 = 5 h

Average speed = 240 ÷ 5 = 48 km/h

Shortcuts and tricks

• Percentage change shortcut: (New – Old) ÷ Old × 100. For quick mental calc, find difference first, then divide by old value.
• Profit/Loss on selling price: Use Loss % = Loss ÷ Selling Price × 100 → Selling Price = Cost Price ÷ (1 – Loss %).
• Combined work: Use reciprocals of times; add them, then take reciprocal of the sum.
• Average of a series: If numbers form an arithmetic progression, average = (First + Last) ÷ 2.
• Compound interest shortcut for 2 years: Amount = P + P×R + P×R×(1+R) (where R is in decimal).

Common mistakes to avoid

1. Mixing up cost price and selling price while calculating profit or loss percentages.
2. Forgetting to convert percentage to decimal before using in formulas (e.g., 10 % → 0.10).
3. Using average formula (sum ÷ n) without updating the total when a number is added or removed.
4. Ignoring the time unit consistency in speed‑distance‑time problems (mixing minutes with hours).
5. In compound interest, applying the simple interest formula for more than one year.