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Equalizing Strength / Ratio-Based Equations

Introduction

Equalizing Strength या Ratio-Based Equation problems में दो या अधिक solutions (या alloys) जिनकी strengths या concentrations अलग-अलग हों, उन्हें इस तरह मिलाया जाता है कि final mixture में एक desired uniform strength मिल सके। ये questions इसलिए महत्वपूर्ण हैं क्योंकि ये आपको percentage या ratio-based word statements को साफ़ algebraic equations में बदलना सिखाते हैं, जिन्हें step-by-step solve किया जा सकता है।

आपको ऐसे problems mixture, alloy और concentration वाले questions में अक्सर मिलेंगे - ये आपकी quantities को balance करने और proportional reasoning को सही तरह से apply करने की ability को test करते हैं।

Pattern: Equalizing Strength / Ratio-Based Equations

Pattern

मुख्य concept: हर component से आने वाली active substance की total मात्रा लिखें, उसे total quantity से divide करें ताकि overall concentration मिले, और उसे target strength या दी गई condition के बराबर रखें।

Follow करने के steps:
1. सभी percentages या ratios को decimal fractions में बदलें।
2. Unknown quantities को variables से represent करें।
3. एक total composition (mass-balance) equation बनाएं: (Sum of component parts) ÷ (Total quantity) = Target concentration
4. Equation को simplify करके unknown को solve करें।

Step-by-Step Example

Question

दो solutions A और B में क्रमशः 30% और 70% acid है। इन्हें इस तरह mix करना है कि final mixture की concentration 50% हो। अगर 12 litres solution A लिया गया है, तो solution B कितनी मात्रा में मिलाना होगा?

Solution

  1. Step 1: Define variables

    मान लेते हैं कि solution B की मात्रा x litres है।

  2. Step 2: Express total acid from both solutions

    Solution A से acid = 30% of 12 = 0.30 × 12 = 3.6 L.
    Solution B से acid = 70% of x = 0.70 × x = 0.7x L.

  3. Step 3: Write the equation for the target concentration

    Total acid / Total volume = Target concentration ⇒ (3.6 + 0.7x) / (12 + x) = 0.50

  4. Step 4: Simplify and solve

    3.6 + 0.7x = 0.50(12 + x) ⇒ 3.6 + 0.7x = 6 + 0.5x ⇒ 0.2x = 2.4 ⇒ x = 12 L.

  5. Step 5: Verify the result

    Total acid = 3.6 + 0.7×12 = 3.6 + 8.4 = 12 L. Total volume = 12 + 12 = 24 L. 12 ÷ 24 = 0.50 → 50% ✅

  6. Final Answer:

    Solution B required = 12 litres

  7. Quick Check:

    दोनों solutions को बराबर मात्रा में लेने पर concentration ठीक 50% आती है, क्योंकि यह 30% और 70% के बीच का midpoint है। ✅

Quick Variations

1. Target concentration दी हो तो किसी एक mixture की मात्रा निकालना।

2. किसी stronger या weaker solution को adjust करके desired strength पाना।

3. Unknowns को ratios (जैसे A : B = x : y) के रूप में express करके x या y find करना।

4. दो से अधिक mixtures होने पर multi-step equalization करना।

Trick to Always Use

  • Step 1: % को decimal form में convert करें, फिर formulas apply करें।
  • Step 2: एक साफ़ equation लिखें: (Sum of acid from each) ÷ (Total volume) = Target %.
  • Step 3: Denominator हटाने के लिए cross multiply करें और unknown solve करें।
  • Step 4: Result verify करें - final % हमेशा given % values के बीच होना चाहिए।

Summary

Summary

Equalizing Strength / Ratio-Based Equations pattern का इस्तेमाल concentrations या strengths को algebraic balance के ज़रिए equal करने के लिए किया जाता है।

  • सभी strengths को fractional या decimal form में बदलें।
  • एक formula याद रखें: (Sum of parts) ÷ (Total mixture) = Target concentration.
  • Cross-multiply करके simplify करें और unknown quantities निकालें।
  • Answer check करें - final strength हमेशा original strengths के बीच होगी।

Practice

(1/5)
1. Two acid solutions contain 30% and 60% acid respectively. In what ratio should they be mixed to obtain a 45% acid solution?
easy
A. 1 : 1
B. 2 : 1
C. 1 : 2
D. 3 : 2

Solution

  1. Step 1: Note concentrations

    Lower = 30%, Higher = 60%, Target = 45%.

  2. Step 2: Apply alligation

    (Higher - Target) : (Target - Lower) = (60 - 45) : (45 - 30) = 15 : 15 = 1 : 1.

  3. Final Answer:

    Mix in ratio 1 : 1 → Option A.

  4. Quick Check:

    Equal parts of 30% and 60% give (30 + 60)/2 = 45% ✅

Hint: Use (higher - target) : (target - lower) to get the ratio directly.
Common Mistakes: Swapping the two differences or forgetting to simplify the ratio.
2. A 20% salt solution is mixed with a 40% salt solution to get a 25% solution. Find the ratio of the two solutions.
easy
A. 1 : 3
B. 3 : 1
C. 2 : 3
D. 1 : 2

Solution

  1. Step 1: Note concentrations

    Lower = 20%, Higher = 40%, Target = 25%.

  2. Step 2: Apply alligation

    (Higher - Target) : (Target - Lower) = (40 - 25) : (25 - 20) = 15 : 5 = 3 : 1.

  3. Final Answer:

    Ratio (20% : 40%) = 3 : 1 → Option B.

  4. Quick Check:

    (3×20 + 1×40)/4 = (60 + 40)/4 = 100/4 = 25% ✅ - more of the 20% solution is needed because the target (25%) is closer to 20%.

Hint: Alligation gives parts of lower : higher as (higher - target) : (target - lower).
Common Mistakes: Reversing which difference corresponds to which component.
3. How many litres of 60% sugar solution must be mixed with 40 litres of 20% sugar solution to get a 40% sugar solution?
easy
A. 20 L
B. 30 L
C. 40 L
D. 50 L

Solution

  1. Step 1: Define variable

    Let x = litres of 60% solution.

  2. Step 2: Write concentration equation

    (0.6x + 0.2×40) / (x + 40) = 0.4

  3. Step 3: Simplify and solve

    0.6x + 8 = 0.4x + 16 → 0.2x = 8 → x = 40 L.

  4. Final Answer:

    40 L → Option C.

  5. Quick Check:

    (0.6×40 + 0.2×40)/(80) = (24 + 8)/80 = 32/80 = 0.4 = 40% ✅

Hint: Use (C1V1 + C2V2) / (V1 + V2) = Target to form a linear equation.
Common Mistakes: Forgetting to multiply concentrations by their volumes before dividing by total volume.
4. A 15% acid solution is mixed with a 60% acid solution so that the resulting mixture is 30% acid. If 12 litres of the 15% solution are used, find how much of the 60% solution is required.
medium
A. 3 L
B. 4 L
C. 5 L
D. 6 L

Solution

  1. Step 1: Let x = litres of 60% solution

  2. Step 2: Form equation

    (0.15×12 + 0.60×x) / (12 + x) = 0.30

  3. Step 3: Solve

    1.8 + 0.6x = 0.30(12 + x) → 1.8 + 0.6x = 3.6 + 0.3x → 0.3x = 1.8 → x = 6 L.

  4. Final Answer:

    6 L → Option D.

  5. Quick Check:

    Total acid = 1.8 + 3.6 = 5.4 L; total volume = 18 L; 5.4/18 = 0.30 → 30% ✅

Hint: Multiply each concentration by its volume, add, then divide by total volume and set equal to target.
Common Mistakes: Not multiplying the target by the total volume when rearranging the equation.
5. In what ratio must a 25% sugar solution be mixed with a 75% sugar solution to get a 60% solution?
medium
A. 3 : 7
B. 7 : 3
C. 1 : 3
D. 2 : 3

Solution

  1. Step 1: Apply alligation

    (Higher - Target) : (Target - Lower) = (75 - 60) : (60 - 25) = 15 : 35 = 3 : 7.

  2. Step 2: Interpret

    Parts correspond to (25% : 75%) = 3 : 7.

    3. Final Answer:

    3 : 7 → Option A.

  3. Quick Check:

    (25×3 + 75×7)/(3+7) = (75 + 525)/10 = 600/10 = 60% ✅

Hint: Alligation gives lower : higher as (higher - target) : (target - lower).
Common Mistakes: Reversing the parts or failing to simplify the ratio to smallest integers.

Mock Test

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