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Equation-Based Simplification

Introduction

Equation-based simplification में आपको छोटी equations को rearrange करके solve करना होता है, जिनमें percentages, fractions, roots या simple algebraic expressions शामिल हो सकते हैं। Aptitude tests में ये अक्सर आते हैं जहाँ आपको inverse operations apply करके unknown को जल्दी isolate करना होता है।

यह pattern महत्वपूर्ण है क्योंकि यह आपको worded math को algebraic steps में बदलना और बिना लंबी algebra के unknown निकालना सिखाता है।

Pattern: Equation-Based Simplification

Pattern

मुख्य idea: Statement को equation में बदलें, inverse operations (पहले multiplications/divisions को undo करें), और step-by-step solve करें।

Common tactics:

  • Percent को fraction या decimal में बदलें (जैसे 45% = 45/100 = 0.45).
  • अगर expression product के रूप में है (जैसे A = x × B) तो x को isolate करने के लिए दोनों sides को B से divide करें।
  • Simple algebraic moves-addition/subtraction और multiplication/division-को reverse order में apply करें ताकि unknown isolate हो सके।

Step-by-Step Example

Question

45% of 600 = ? × 90
? का मान निकालें।

Options:

  • A) 2
  • B) 3
  • C) 4
  • D) 5

Solution

  1. Step 1: Percent को number में बदलें:

    45% = 45/100 = 0.45. इसलिए 45% of 600 = 0.45 × 600.

  2. Step 2: Left side calculate करें:

    0.45 × 600 = (45/100) × 600 = 45 × 6 = 270.

  3. Step 3: Equation साफ लिखें:

    270 = ? × 90.

  4. Step 4: Inverse operation से unknown isolate करें:

    ? = 270 ÷ 90 = 3.

  5. Final Answer:

    3 → Option B.

  6. Quick Check:

    3 × 90 = 270, जो left side (45% of 600) के बराबर है ✅

  7. Explanation for Beginners:

    Percent का मतलब होता है “per hundred,” इसलिए 45% of 600 = (45 ÷ 100) × 600. यह number मिलने के बाद equation 270 = ? × 90 को दोनों sides को 90 से divide करके solve किया जाता है।

Quick Variations

1. Percent right side में हो: 90 = 30% of x → percent convert करें और divide करके x निकालें।

2. Percent की जगह fraction हो: (3/4) of 200 = ? + 50 → simplify करके subtract करें।

3. Roots/exponents के साथ mix: √x × 5 = 35 → divide करें, फिर square करके x निकालें।

Trick to Always Use

  • Step 1 → Percent को ÷100 में बदलें और “of” को × समझें।
  • Step 2 → Unknown isolate करने से पहले known side को simplify कर लें।
  • Step 3 → Inverse operations को logic से apply करें ताकि unknown आसानी से निकले।

Summary

Summary

  • Percentages और fractions को पहले numbers में बदलें।
  • Known parts को पूरी तरह simplify करें फिर unknown solve करें।
  • Inverse operations को reverse order में लगाएँ।
  • हमेशा वापस substitute करके verify करें।

Practice

(1/5)
1. If 25% of 200 = ? × 25, find the value of ?.
easy
A. 2
B. 3
C. 4
D. 5

Solution

  1. Step 1: Compute the percentage

    25% of 200 = (25/100) × 200 = 0.25 × 200 = 50.

  2. Step 2: Set up the equation

    We have 50 = ? × 25.

  3. Step 3: Isolate the unknown

    ? = 50 ÷ 25 = 2.

  4. Final Answer:

    2 → Option A.

  5. Quick Check:

    2 × 25 = 50, which equals 25% of 200, so the answer is correct ✅

  6. Explanation for Beginners:

    Percent means 'per hundred'. So 25% of 200 is 25/100 times 200. After computing that you get a simple multiplication equation; undo the multiplication (divide) to find the unknown.

Hint: Convert percent to fraction (x/100), compute that value, then divide by the multiplier to get ?.
Common Mistakes: Forgetting to convert percent to /100 or dividing the wrong way (e.g., dividing 25 by 50).
2. If 60% of 150 = ? × 30, find the value of ?.
easy
A. 2
B. 3
C. 4
D. 5

Solution

  1. Step 1: Convert percent and compute

    60% of 150 = (60/100) × 150 = 0.6 × 150 = 90.

  2. Step 2: Write the equation

    90 = ? × 30.

  3. Step 3: Solve for ?

    ? = 90 ÷ 30 = 3.

  4. Final Answer:

    3 → Option B.

  5. Quick Check:

    3 × 30 = 90, which equals 60% of 150 ✅

  6. Explanation for Beginners:

    First find the numeric value of the percentage of the base. Then treat the result as a product and undo multiplication by dividing by the known multiplier to get the unknown.

Hint: Find percent value first (0.6×150), then divide by the given multiplier.
Common Mistakes: Treating 60% as 60 (not 0.6) or forgetting to divide by 30 at the end.
3. If (3/4 of 160) = ? × 30, find the value of ?.
easy
A. 3
B. 5
C. 4
D. 6

Solution

  1. Step 1: Compute the fraction of the number

    (3/4) of 160 = (3/4) × 160 = 3 × 40 = 120.

  2. Step 2: Form the equation

    120 = ? × 30.

  3. Step 3: Find ? by dividing

    ? = 120 ÷ 30 = 4.

  4. Final Answer:

    4 → Option C.

  5. Quick Check:

    4 × 30 = 120, equals (3/4) of 160 ✅

  6. Explanation for Beginners:

    When you see '3/4 of 160', multiply 160 by 3/4. That gives a number which is expressed as a product with ?. Divide by the given multiplier to find the unknown.

Hint: Compute the 'of' part first (fraction × number), then divide by the right-hand multiplier.
Common Mistakes: Forgetting to simplify 160×(3/4) by cancelling before multiplying which makes arithmetic harder.
4. If 1/2 of 144 = ? × 12, find the value of ?.
medium
A. 5
B. 7
C. 8
D. 6

Solution

  1. Step 1: Compute the half

    1/2 of 144 = (1/2) × 144 = 72.

  2. Step 2: Form the equation

    72 = ? × 12.

  3. Step 3: Solve for ?

    ? = 72 ÷ 12 = 6.

  4. Final Answer:

    6 → Option D.

  5. Quick Check:

    6 × 12 = 72, matches the left side ✅

  6. Explanation for Beginners:

    Halving a number is easy: divide by 2. After you get that value, treat the equation normally-divide by the multiplier to isolate ?.

Hint: Do simple fraction-of calculations first (half, third, quarter), then divide by the multiplier.
Common Mistakes: Misreading '1/2 of 144' as 1 ÷ (2×144) or mixing up numerator/denominator operations.
5. If 1/3 of 96 = ? × 4, find the value of ?.
medium
A. 8
B. 6
C. 9
D. 12

Solution

  1. Step 1: Compute the third

    1/3 of 96 = (1/3) × 96 = 32.

  2. Step 2: Set up the equation

    32 = ? × 4.

  3. Step 3: Solve for ?

    ? = 32 ÷ 4 = 8.

  4. Final Answer:

    8 → Option A.

  5. Quick Check:

    8 × 4 = 32, equals 1/3 of 96 ✅

  6. Explanation for Beginners:

    Divide the number by 3 to get one third, then divide that result by the multiplier to find the unknown. Reading '1/3 of 96' as (1/3)×96 avoids mistakes.

Hint: Compute the 'of' (fraction × number) first, then divide by the RHS multiplier.
Common Mistakes: Treating '1/3 of 96' as 96 ÷ 3 ÷ ? incorrectly or swapping division order.

Mock Test

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