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Equation-Based Simplification

Introduction

Equation-based simplification asks you to rearrange and solve short equations that mix percentages, fractions, roots, or simple algebraic expressions. These appear frequently in aptitude tests where you must isolate an unknown quickly by applying inverse operations.

This pattern is important because it trains you to translate worded math into algebraic steps and solve for the unknown without lengthy algebra.

Pattern: Equation-Based Simplification

Pattern

Key idea: Translate the statement into an equation, perform inverse operations (undo multiplications/divisions first), and solve step-by-step.

Common tactics:

  • Turn percent statements into fraction or decimal form (e.g., 45% = 45/100 = 0.45).
  • When an expression equals a product (like A = x × B) isolate x by dividing both sides by B.
  • Use simple algebraic moves-addition/subtraction, multiplication/division-in reverse order to isolate the unknown.

Step-by-Step Example

Question

45% of 600 = ? × 90
Find the value of ?.

Options:

  • A) 2
  • B) 3
  • C) 4
  • D) 5

Solution

  1. Step 1: Translate the percent to a number:

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

  2. Step 2: Compute the left side:

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

  3. Step 3: Write the equation clearly:

    270 = ? × 90.

  4. Step 4: Isolate the unknown using inverse operation:

    ? = 270 ÷ 90 = 3.

  5. Final Answer:

    3 → Option B.

  6. Quick Check:

    3 × 90 = 270, matches left side (45% of 600) ✅

  7. Explanation for Beginners:

    Percent means “per hundred,” so 45% of 600 is (45 ÷ 100) × 600. Once that number is calculated, you solve the simple equation 270 = ? × 90 by dividing both sides by 90.

Quick Variations

1. Percent on right side: 90 = 30% of x → convert percent and divide to find x.

2. Fractions instead of percent: (3/4) of 200 = ? + 50 → simplify and subtract.

3. Mixed with roots/exponents: √x × 5 = 35 → divide then square to find x.

Trick to Always Use

  • Step 1 → Convert percent to ÷100 and “of” to ×.
  • Step 2 → Simplify the known side before isolating the unknown.
  • Step 3 → Use inverse operations logically to isolate the unknown.

Summary

Summary

  • Convert percentages and fractions to numbers first.
  • Simplify known parts fully before solving for the unknown.
  • Apply inverse operations in reverse order.
  • Always verify by substituting back into the equation.

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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