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Basic Ratio Simplification

Introduction

The most basic concept in Ratio problems is simplifying ratios. A ratio compares two or more quantities, similar to a fraction. Just like fractions, ratios can be simplified by dividing both parts by their greatest common divisor (GCD).

Mastering ratio simplification is important because most other ratio problems depend on it.

Pattern: Basic Ratio Simplification

Pattern

Ratio = Quantity₁ : Quantity₂

To simplify → divide both sides of the ratio by their GCD.

Ratio can also be expressed as a fraction → a : b = a/b.

Step-by-Step Example

Question

Simplify the ratio 50 : 100.

Solution

  1. Step 1: Write the ratio as numbers.

    Ratio = 50 : 100

  2. Step 2: Find the greatest common divisor (GCD).

    GCD of 50 and 100 = 50

  3. Step 3: Divide both numbers by GCD.

    50 ÷ 50 = 1
    100 ÷ 50 = 2

  4. Step 4: Simplified Ratio.

    Ratio = 1 : 2

  5. Step 5: Quick Check.

    Fraction form = 50/100 = 1/2 ✅

Question

Express the ratio 45 : 60 in simplest form.

Solution

  1. Step 1: Write the ratio.

    Ratio = 45 : 60

  2. Step 2: Find the GCD.

    GCD of 45 and 60 = 15

  3. Step 3: Divide both terms by GCD.

    45 ÷ 15 = 3
    60 ÷ 15 = 4

  4. Step 4: Simplified Ratio.

    Ratio = 3 : 4

  5. Step 5: Quick Check.

    Fraction = 45/60 = 3/4 ✅

Quick Variations

If decimals are given, first remove decimals by multiplying by 10, 100, etc., then simplify. Example: 2.5 : 5 → multiply both by 10 → 25 : 50 → simplified to 1 : 2.

If ratios involve units, convert them to the same unit before simplifying. Example: 2 m : 150 cm → 200 cm : 150 cm = 4 : 3.

Trick to Always Use

  • Step 1: Write ratio numbers clearly.
  • Step 2: Find GCD of the numbers.
  • Step 3: Divide both by GCD.
  • Step 4: Double-check by converting to fraction.

Summary

Summary

The Basic Ratio Simplification pattern is solved by dividing both terms of the ratio by their greatest common divisor (GCD).

  • Ratio a : b = fraction a/b.
  • Divide by GCD to simplify.
  • Convert units if required before simplifying.

Once you master this, ratio simplification becomes quick and effortless!

Practice

(1/5)
1. Simplify the ratio 24 : 36.
easy
A. 2 : 3
B. 3 : 5
C. 4 : 7
D. 6 : 11

Solution

  1. Step 1: Find the GCD

    GCD(24, 36) = 12.

  2. Step 2: Divide both terms by GCD

    24 ÷ 12 = 2 and 36 ÷ 12 = 3.

  3. Final Answer:

    2 : 3 → Option A.

  4. Quick Check:

    24/36 reduces to 2/3 exactly. ✅
Hint: Always divide both numbers by their GCD for fastest simplification.
Common Mistakes: Dividing by smaller factors like 6 instead of the GCD 12.
2. Express the ratio 56 : 84 in simplest form.
easy
A. 2 : 5
B. 3 : 4
C. 2 : 3
D. 5 : 7

Solution

  1. Step 1: Compute the GCD

    GCD(56, 84) = 28.

  2. Step 2: Divide both terms

    56 ÷ 28 = 2 and 84 ÷ 28 = 3.

  3. Final Answer:

    2 : 3 → Option C.

  4. Quick Check:

    56/84 reduces exactly to 2/3. ✅
Hint: Spot common multiples of 7 and 14 to quickly reach 28.
Common Mistakes: Reducing only to 28 : 42 and forgetting to simplify further.
3. Simplify the ratio 72 : 108.
easy
A. 4 : 5
B. 2 : 3
C. 3 : 7
D. 5 : 9

Solution

  1. Step 1: Find the GCD

    GCD(72, 108) = 36.

  2. Step 2: Divide

    72 ÷ 36 = 2, 108 ÷ 36 = 3.

  3. Final Answer:

    2 : 3 → Option B.

  4. Quick Check:

    72/108 simplifies exactly to 2/3. ✅
Hint: Break numbers into prime factors to spot the GCD quickly.
Common Mistakes: Leaving the ratio at 12 : 18 instead of fully reducing.
4. Reduce the ratio 90 : 150 to lowest terms.
easy
A. 3 : 5
B. 5 : 9
C. 2 : 7
D. 6 : 11

Solution

  1. Step 1: Determine the GCD

    GCD(90, 150) = 30.

  2. Step 2: Simplify

    90 ÷ 30 = 3 and 150 ÷ 30 = 5.

  3. Final Answer:

    3 : 5 → Option A.

  4. Quick Check:

    90/150 simplifies to 3/5. Perfect match. ✅
Hint: Divide numerator and denominator by 10 first to get 9 : 15, then simplify.
Common Mistakes: Stopping at 9 : 15 instead of 3 : 5.
5. Simplify the ratio 135 : 180.
easy
A. 2 : 5
B. 4 : 7
C. 3 : 4
D. 5 : 8

Solution

  1. Step 1: Compute the GCD

    GCD(135, 180) = 45.

  2. Step 2: Simplify

    135 ÷ 45 = 3 and 180 ÷ 45 = 4.

  3. Final Answer:

    3 : 4 → Option C.

  4. Quick Check:

    135/180 = 3/4 - matches perfectly. ✅
Hint: Prime-factorise: 135 = 3³×5, 180 = 2²×3²×5 → GCD = 3²×5 = 45.
Common Mistakes: Choosing 15 as GCD instead of 45.

Mock Test

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