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Equivalent Ratios (Missing Value)

Introduction

In many ratio problems, one of the terms is missing, and you are asked to find it using the concept of equivalent ratios. These questions test your ability to scale ratios up or down correctly.

The key is to remember that a ratio is like a fraction. If a : b = c : d, then a/b = c/d. Using this, you can easily find the missing value.

Pattern: Equivalent Ratios (Missing Value)

Pattern

If a : b = c : d, then a/b = c/d.

Cross multiply → a × d = b × c.

Use this equation to find the missing term.

Step-by-Step Example

Question

If 2 : 3 = 4 : x, find the value of x.

Solution

  1. Step 1: Write the ratio as a fraction.

    2 : 3 = 2/3, and 4 : x = 4/x

  2. Step 2: Set them equal.

    2/3 = 4/x

  3. Step 3: Cross multiply.

    2 × x = 3 × 4 → 2x = 12

  4. Step 4: Solve for x.

    x = 12 ÷ 2 = 6

  5. Step 5: Quick Check.

    Ratio = 4 : 6 = 2 : 3 ✅

Question

If x : 15 = 7 : 21, find x.

Solution

  1. Step 1: Write as fractions.

    x : 15 = x/15, and 7 : 21 = 7/21

  2. Step 2: Simplify if possible.

    7/21 = 1/3

  3. Step 3: Set them equal.

    x/15 = 1/3

  4. Step 4: Cross multiply.

    3x = 15 × 1 → 3x = 15

  5. Step 5: Solve for x.

    x = 15 ÷ 3 = 5

  6. Step 6: Quick Check.

    Ratio = 5 : 15 = 1 : 3 ✅

Quick Variations

Sometimes, ratios appear in word problems. Example: “If Rs. 500 is divided between A and B in the ratio 3 : 2, how much does A get?” → Total parts = 3 + 2 = 5 → A’s share = (3/5) × 500 = Rs. 300

Fractions and decimals can also appear, but the method remains the same: write as a fraction and cross multiply.

Trick to Always Use

  • Step 1: Write ratios as fractions.
  • Step 2: Cross multiply.
  • Step 3: Solve for the missing term.
  • Step 4: Always check by simplifying back to the original ratio.

Summary

Summary

The Equivalent Ratios pattern uses the relation:

a : b = c : d → a/b = c/d → a × d = b × c

  • Write the ratio as fractions.
  • Cross multiply to form an equation.
  • Solve for the missing value.
  • Double-check by reducing back to the given ratio.

With practice, missing value ratio questions become quick and straightforward!

Practice

(1/5)
1. If 2 : 3 = x : 9, find x.
easy
A. 5
B. 6
C. 7
D. 8

Solution

  1. Step 1: Write the proportion

    Given ratio: 2 : 3 = x : 9.

  2. Step 2: Cross-multiply

    2 × 9 = 3 × x.

  3. Step 3: Solve for x

    18 = 3x → x = 6.

  4. Final Answer:

    6 → Option B

  5. Quick Check:

    2/3 = 6/9 = 0.666... ✅
Hint: Use cross-multiplication: a:b = c:d → ad = bc.
Common Mistakes: Inverting terms or dividing instead of cross-multiplying.
2. If 7 : x = 21 : 9, find x.
easy
A. 2
B. 3
C. 4
D. 5

Solution

  1. Step 1: Write the proportion

    Given ratio: 7 : x = 21 : 9.

  2. Step 2: Cross-multiply

    7 × 9 = 21 × x.

  3. Step 3: Solve for x

    63 = 21x → x = 3.

  4. Final Answer:

    3 → Option B

  5. Quick Check:

    7/3 ≈ 2.333, 21/9 ≈ 2.333 ✅
Hint: Reduce a:b = c:d by simplifying one side if possible before cross-multiplying.
Common Mistakes: Mixing positions of terms when cross-multiplying (use ad = bc).
3. If 15 : 25 = 9 : x, find x.
medium
A. 12
B. 13
C. 14
D. 15

Solution

  1. Step 1: Write the proportion

    Given ratio: 15 : 25 = 9 : x.

  2. Step 2: Cross-multiply

    15 × x = 25 × 9.

  3. Step 3: Solve for x

    15x = 225 → x = 15.

  4. Final Answer:

    15 → Option D

  5. Quick Check:

    15/25 = 0.6, 9/15 = 0.6 ✅
Hint: Simplify 15:25 → 3:5 first to see expected scale (9 corresponds to 3→ x corresponds to 5).
Common Mistakes: Not reducing the given ratio first and missing easy mental checks.
4. If x : 8 = 5 : 10, find x.
medium
A. 3
B. 4
C. 5
D. 6

Solution

  1. Step 1: Write the proportion

    Given ratio: x : 8 = 5 : 10.

  2. Step 2: Cross-multiply

    x × 10 = 8 × 5.

  3. Step 3: Solve for x

    10x = 40 → x = 4.

  4. Final Answer:

    4 → Option B

  5. Quick Check:

    4/8 = 0.5, 5/10 = 0.5 ✅
Hint: Recognize 5:10 = 1:2 so x:8 should be 1:2 → x = 4.
Common Mistakes: Forgetting to reduce the known ratio before solving.
5. If 18 : 24 = x : 16, find x.
hard
A. 10
B. 11
C. 12
D. 13

Solution

  1. Step 1: Write the proportion

    Given ratio: 18 : 24 = x : 16.

  2. Step 2: Cross-multiply

    18 × 16 = 24 × x.

  3. Step 3: Solve for x

    288 = 24x → x = 12.

  4. Final Answer:

    12 → Option C

  5. Quick Check:

    18/24 = 0.75, 12/16 = 0.75 ✅
Hint: Reduce 18:24 → 3:4 to see x should be 3/4 of 16 = 12.
Common Mistakes: Attempting subtraction or other operations instead of cross-multiplication.

Mock Test

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